Proof that pi=4

This paper is here: http://milesmathis.com/pi7.pdf

There is something I misunderstood maybe. Let's call the ball that goes straight A and the ball that goes around the circle B.

t0: A and B start.
t1: A and B are at the speed mark.
t2: A is at second mark and B is at the start of the circle.

now things get a bit messed up:
t3: A is at d from the second mark and B is at 1/4 of the circle. If we straighten the circle, that mark is at 0.785*d from the second mark.
t4: A is at 2*d from the second mark and B is at 1.57*d.
t5: A is at 3*d from the second mark and B is at 2.355*d.
t6: A is at 3.14*d from the second mark and B is at 2.4649*d.
t7: A is at 4*d from the second mark and B is at 3.14*d.

We can se that there is a slowing down, instantaneous and not cumulative. Is it apparent? We proposed a new experiment that calculates the final speed of the two balls.

I think that for logistic reasons we must use half a circumference.

There are 2 possible outcomes:
1) A and B have the same speed
2) A is faster than B

But that is not the case I'm afraid. If 2 is true, then in general changing curvature changes the speed as an inverse proportion. That would result in B gaining speed when getting on the straight path. So both possibilities (speed loss and physical longer distance) will produce the same final speed.
Is there anything we can do to better understand this?

PS: I love Mathis rants about mainstreams physicists

There is something I misunderstood maybe. Let's call the ball that goes straight A and the ball that goes around the circle B.

t0: A and B start.
t1: A and B are at the speed mark.
t2: A is at second mark and B is at the start of the circle.

now things get a bit messed up:
t3: A is at d from the second mark and B is at 1/4 of the circle. If we straighten the circle, that mark is at 0.785*d from the second mark.
t4: A is at 2*d from the second mark and B is at 1.57*d.
t5: A is at 3*d from the second mark and B is at 2.355*d.
t6: A is at 3.14*d from the second mark and B is at 2.4649*d.
t7: A is at 4*d from the second mark and B is at 3.14*d.

We can se that there is a slowing down, instantaneous and not cumulative. Is it apparent? We proposed a new experiment that calculates the final speed of the two balls.

The two balls:1) A on the straight track ,and; 2) B on the circular track – appear to hit their marks simultaneously. This rules out fiction. Adding a final straight section to circular track B would allow us to compare and calculate straight to straight final velocities.

Loop velocity is real and measurable. It also turns out to be pi/4 slower when compared to the same ‘straight’ velocity. The apparent part of loop velocity is calling it a velocity, when it is actually an acceleration.

I think that for logistic reasons we must use half a circumference.

Based on what? You need to justify your statement.

There are 2 possible outcomes:
1) A and B have the same speed
2) A is faster than B

But that is not the case I'm afraid. If 2 is true, then in general changing curvature changes the speed as an inverse proportion. That would result in B gaining speed when getting on the straight path. So both possibilities (speed loss and physical longer distance) will produce the same final speed.

Ignoring friction, I’m certain that an additional final straight would show that B’s speed would increase from the loop velocity to match A’s final straight velocity. Of course I may be wrong, but there are now plenty of people looking at this.  

Is there anything we can do to better understand this?

Steve was able to prove pi=4 using just rolling velocity. The classical understanding is overturned. If we could generate a constant velocity we can conduct straight and curved comparisons. We can also accept the kinematic results and race imaginary spheros around a continuous track.
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I was starting to think that if instead of the curved circular tube -- the "Pi = 3.14xx" metal ball traveled along a isosceles triangular strip only 1 atom wide at the 3 edges...that curved perfectly at the key points for perfect balance.... would the entire surface area traveled by the "Pi=3.14xxx" ball be less than the surface area traveled by the "straight path = 4" ball? The different orbital turns in the "Pi=3.14" ball path creates less ball surface area traveled?

http://mathworld.wolfram.com/SurfaceofRevolution.html

Ciaolo wrote:
We can se that there is a slowing down, instantaneous and not cumulative. Is it apparent?

I think it is important to realise that we only think it slows down because we think in terms of pi. The fact that it is instantaneous tells us that it is not any sort of friction or surface area relation because those would be cumulative. The fact that the velocity on exit matches the velocity of the straight path also tells us that it is the actual distance that is different because we know it can't be time changing and since v=d/t, they are the only 2 things that can change.

Ciaolo wrote:
I think that for logistic reasons we must use half a circumference.

There are 2 possible outcomes:
1) A and B have the same speed
2) A is faster than B

But that is not the case I'm afraid. If 2 is true, then in general changing curvature changes the speed as an inverse proportion. That would result in B gaining speed when getting on the straight path. So both possibilities (speed loss and physical longer distance) will produce the same final speed.
Is there anything we can do to better understand this?

A and B have the same speed but B is traveling a distance that is different to expectation. You can only think that it slows down if you think that pi=3.14. Recalculate using pi=4 and you will find that it does not slow down.

We only have 3 variables to juggle, v, d and t. We know that t isn't changing so we can rule that out, leaving us with only v or d. You can pick whichever one you want and apply the change to it but you can't pick both and it doesn't really make any sense to pick v, since it is a derived value and all logic tells us that part of the velocity is not going to just disappear while in the curve and suddenly reappear once out of it. These are not self-propelled objects which is a huge plus for this experiment. So the only valid conclusion is that the distance is longer.

The extra distance is the sum of the changes made at each dt (delta t). To be more precise, it is the sum of the lengths of the change vectors. A change vector is the difference between the initial velocity vector, where it would go without curvature, and the resultant vector, where it did actually go. You just subtract one from the other. If you added the length of all of those change vectors you will find that it matches the difference between pi and 4. I say matches, not equals, because you have to take size differences into consideration before you can find an equality.

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Cr6 wrote:I was starting to think that if instead of the curved circular tube -- the "Pi = 3.14xx" metal ball traveled along a isosceles triangular strip only 1 atom wide at the 3 edges...that curved perfectly at the key points for perfect balance.... would the entire surface area traveled by the "Pi=3.14xxx" ball be less than the surface area traveled by the "straight path = 4" ball? The different orbital turns in the "Pi=3.14" ball path creates less ball surface area traveled?

Sorry Cr6, I’m not seeing it. Please define ‘isosceles triangular strip’. The sphere traces a rolling contact point - a line. In what sense can the tube’s loop or straight sections’ surfaces be replaced with an ‘isosceles triangular strip’?  Also, ‘surface area traveled’ doesn’t make sense to me. We are working in line distances, there are no surface areas involved that I can see.

Nevyn wrote: we only think it (the ball) slows down because we think in terms of pi.

Clearly, compared side-by-side, the ball moves slower in the loop than in the straight. I’ve mistakenly said ‘speeding up’ or ‘slowing down’, the ball has only two speeds, straight or curved (the curve speed varies with curvature). Since v=d/t, we can logically expect distance along the curve be longer than the straight. This agrees with our new understanding, the distance associated with curved motion is 4/pi longer than previously thought.

Nevyn wrote:A and B have the same speed but B is traveling a distance that is different to expectation. You can only think that it slows down if you think that pi=3.14. Recalculate using pi=4 and you will find that it does not slow down.

We only have 3 variables to juggle, v, d and t. We know that t isn't changing so we can rule that out, leaving us with only v or d. You can pick whichever one you want and apply the change to it but you can't pick both and it doesn't really make any sense to pick v, since it is a derived value and all logic tells us that part of the velocity is not going to just disappear while in the curve and suddenly reappear once out of it. These are not self-propelled objects which is a huge plus for this experiment. So the only valid conclusion is that the distance is longer.

Nevyn, You argue that the speeds are equal. Wouldn’t that make the distances equal too? We cannot see them as equal because they are based on two different metrics. In the straight metric (which includes hypotenuses and pi), distances are equal to their geometric length. In the curved (or motion, or manhattan) metric, the hypotenuse is the c-path which is not used and pi is replaced with 4. The motion metric is new and opens up the question of where is all that additional distance coming from?  

“Juggling 3 variables v, d and t”, as in velocity, distance and time? I use v=d/t above myself, but I don’t think it’s always correct anymore. For example, d is normally a distance, a hypotenuse, the source of Pi geometry.  

The velocity vector may be described by, V = a(t)X + b(t)Y.  Where X and Y are the orthogonal velocity components. The functions a(t), and b(t) can describe a loop or a straight line.  

Nevyn wrote:The extra distance is the sum of the changes made at each dt (delta t). To be more precise, it is the sum of the lengths of the change vectors. A change vector is the difference between the initial velocity vector, where it would go without curvature, and the resultant vector, where it did actually go. You just subtract one from the other. If you added the length of all of those change vectors you will find that it matches the difference between pi and 4. I say matches, not equals, because you have to take size differences into consideration before you can find an equality.

I like your description of the additional 4/pi as the sum of the change vectors made at each dt. I'm still entertaining the idea that the orthogonal direction to the circular loop involves an unseen 180deg spin rotation.

Miles wrote:Because, to solve this with math, we have to integrate both a and b with time t. In other words, we don't integrate a with b, we integrate a with time t and b with the same time t. If you don't like the word integrate—because it is a calculus term—I can say it this way: we have to track how a changes with time and the way b changes with time. That's how we include time in the problem.

I think we can nail this down.
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LongtimeAirman wrote:.

Cr6 wrote:I was starting to think that if instead of the curved circular tube -- the "Pi = 3.14xx" metal ball traveled along a isosceles triangular strip only 1 atom wide at the 3 edges...that curved perfectly at the key points for perfect balance.... would the entire surface area traveled by the "Pi=3.14xxx" ball be less than the surface area traveled by the "straight path = 4" ball? The different orbital turns in the "Pi=3.14" ball path creates less ball surface area traveled?

Sorry Cr6, I’m not seeing it. Please define ‘isosceles triangular strip’. The sphere traces a rolling contact point - a line. In what sense can the tube’s loop or straight sections’ surfaces be replaced with an ‘isosceles triangular strip’?  Also, ‘surface area traveled’ doesn’t make sense to me. We are working in line distances, there are no surface areas involved that I can see.
....
I think we can nail this down.

Sorry LTAM, I had more this in mind.  The apex of one-side of the wire is just 1 atom thick. The ball rolls along it on both the straight and "curved" paths.

LongtimeAirman wrote:
Nevyn, You argue that the speeds are equal. Wouldn’t that make the distances equal too?

Yes, that is the point of the experiment. Two objects with the same velocity will cover the same distance in the same time. Using velocity to measure the distance removes all need to rely on mathematical arguments. Some may argue that v=d/t does not describe angular velocity but they are missing the point. We are trying to determine the length of the circumference which means 'what is the length of this curve if it was straight'. If the velocities are equal when the ball enters the curve and they are equal when it exits the curve, then we must assume that they are equal through the curve. Anything else is absurd.

It reminds me of the old saying 'If a tree falls in the forest and no-one is around to hear it, does it make a sound?'

Well of course it does. To assume that it doesn't just because it is not being measured is absurd.

LongtimeAirman wrote:
“Juggling 3 variables v, d and t”, as in velocity, distance and time? I use v=d/t above myself, but I don’t think it’s always correct anymore. For example, d is normally a distance, a hypotenuse, the source of Pi geometry.

Yes, v=d/t means velocity = change in distance over change in time. The d is always a distance. It is not a hypotenuse because distance does not require a triangle and it is not the source of pi (=3.14) because pi is determined by pure geometry which does not use distance. Miles' argument is that if you determine pi using distance, then it will equal 4 and this experiment confirms that by keeping v and t the same so that we can measure d.

We are measuring the curved path by using the straight path. You guys seem to be fixated on the curved path and looking at it in isolation but that is not the point of this experiment. The straight path tells us all of the information, v, d and t. We then use that to compare to the curved path since we can look at it at any dt and compare it to where the ball is on the straight path.

LongtimeAirman wrote:
The velocity vector may be described by, V = a(t)X + b(t)Y.  Where X and Y are the orthogonal velocity components. The functions a(t), and b(t) can describe a loop or a straight line.  

You have moved up to a 2D vector by splitting the X and Y components out into individual terms. But I don't agree that a(t) and b(t) can describe a curve and it still be called a velocity. A velocity is straight, never curved, if it is curved then it is an acceleration which is the addition of multiple velocity vectors. In order to maintain the equality, you have to limit the functions a and b to be linear. A linear function maintains the same relationship between any 2 consecutive output values if the input values have the same difference.

I don't think I have said that very well so I will try to explain it in math.

Let's say we have a function and we call it f(x). We calculate a value for x=1 and for x=2 and we subtract one from the other giving us the value k(1,2).

k(1,2) = f(2) - f(1)

We then do this again with x=3 and x=4 (it could be any 2 consecutive values but let's keep it simple).

k(3,4) = f(4) - f(3)

If f(x) is linear then k(1,2) = k(3,4).

Let's try it out with some real numbers.

Let f(x) = 2x.

x	f(x)
1	2
2	4
3	6
4	8

f(2) - f(1)
= 4 - 2
= 2

f(4) - f(3)
= 8 - 6
= 2

So f(x) = 2x is linear.

If f(x) = x^2 then:

x	f(x)
1	1
2	4
3	9
4	16

f(2) - f(1)
= 2 - 1
= 1

f(4) - f(3)
= 16 - 9
= 7

So f(x) = x^2 is not linear.

LongtimeAirman wrote:
I like your description of the additional 4/pi as the sum of the change vectors made at each dt. I'm still entertaining the idea that the orthogonal direction to the circular loop involves an unseen 180deg spin rotation.

Any spin of the ball is irrelevant to this experiment and its conclusions. We are not measuring the distance traveled by a point on the surface of the ball, we are measuring the distance traveled by the ball itself. The spin imparted to the ball may be an interesting avenue for study but it does not impact this problem.

Nevyn wrote:      

Longtime Airman wrote:
Nevyn, You argue that the speeds are equal. Wouldn’t that make the distances equal too?

Yes, that is the point of the experiment.

Sorry Nevyn, Let me add “I agree” to my full quote here.  

Nevyn, You argue that the speeds are equal, I agree. Wouldn’t that make the distances equal too? We cannot see them as equal because they are based on two different metrics. In the straight metric (which includes hypotenuses and pi), distances are equal to their geometric length. In the curved (or motion, or manhattan) metric, the hypotenuse is the c-path which is not used and pi is replaced with 4. The motion metric is new and opens up the question of where is all that additional distance coming from?  

Or let me add to next sentence after yours: We cannot see them as equal, even though they are, because they are based on two different metrics.

The velocities and distances may be the same in a logical or absolute number sense, but they cannot be truly equal since only one is a velocity and the other an acceleration. The equality is not complete.

Nevyn wrote:v=d/t means velocity = change in distance over change in time. The d is always a distance.

Good point. Miles is also a little sloppy with d as a hypotenuse in Proof that Pi=4, http://milesmathis.com/pi7.pdf.

Nevyn wrote:
We are measuring the curved path by using the straight path. You guys seem to be fixated on the curved path and looking at it in isolation but that is not the point of this experiment. The straight path tells us all of the information, v, d and t. We then use that to compare to the curved path since we can look at it at any dt and compare it to where the ball is on the straight path.

They are not completely the same, one is a velocity and the other an acceleration. v=d/t is based on the geometric metric that works in the physical world - do you believe it should describe a curve, or a proportion of the curve?

Thanks for clarifying the numbers. I don't believe Miles has described how to properly integrate these curves yet. You may be right, we may not need any special transformation.

Thanks for the discussion.
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LongtimeAirman wrote:Sorry Nevyn, Let me add “I agree” to my full quote here.  

Nevyn, You argue that the speeds are equal, I agree. Wouldn’t that make the distances equal too? We cannot see them as equal because they are based on two different metrics. In the straight metric (which includes hypotenuses and pi), distances are equal to their geometric length. In the curved (or motion, or manhattan) metric, the hypotenuse is the c-path which is not used and pi is replaced with 4. The motion metric is new and opens up the question of where is all that additional distance coming from?  

Or let me add to next sentence after yours: We cannot see them as equal, even though they are, because they are based on two different metrics.

Yes, that is correct.

LongtimeAirman wrote:
The velocities and distances may be the same in a logical or absolute number sense, but they cannot be truly equal since only one is a velocity and the other an acceleration. The equality is not complete.

They are comparable so the equality remains. You can reduce both velocity and acceleration to distance and time.

v = d/t
a = v/t
  = d/t/t
  = d/t^2

But those equations are a little misleading so let me expand them into their proper forms.

Let d1, d2 and d3 be measurements of distance at t1, t2 and t3 respectively.

v = (d2 - d1)/(t2 - t1)
a = (v2 - v1)/(t2 - t1)
  = ( (d3 - d2)/(t3 - t2) - (d2 - d1)/(t2 - t1) ) / (t3 - t2)
since (t3 - t2) = (t2 - t1)
a = ( (d3 - d2)/(t2 - t1) - (d2 - d1)/(t2 - t1) ) / (t2 - t1)
  = ( (d3 - d2) - (d2 - d1) ) / (t2 - t1) / (t2 - t1)
  = ( d3 - 2*d2 + d1 ) / (t2 - t1)^2

The experiment is looking for a distance so we can absolutely compare velocities with accelerations. My analysis using the change vectors did exactly that. The ball has a velocity as it enters the curve, at the next dt (or frame) it has deviated from that vector. We look at that deviation, in isolation, which is a velocity itself. That is the second velocity, along with the initial velocity, that forms the acceleration. Both of those velocities have a length and the sum of those lengths over the complete curve gives us the distance of the curve. I could even say that straight line motion has the same 2 vectors, but one of them will always have 0 length.

LongtimeAirman wrote:

Nevyn wrote:v=d/t means velocity = change in distance over change in time. The d is always a distance.

Good point. Miles is also a little sloppy with d as a hypotenuse in Proof that Pi=4, http://milesmathis.com/pi7.pdf.

Sorry, I wasn't very clear. What I meant, in a very strict sense, is that a distance is not an hypotenuse. What I didn't say is that an hypotenuse can be a distance. An hypotenuse can be many things and distance is only one of them. An hypotenuse is really a relationship and it doesn't really care what the members of that relationship are. But a distance is a real thing. It is a measurable quantity and can not be transferred to other things like a relationship can.

LongtimeAirman wrote:

Nevyn wrote:
We are measuring the curved path by using the straight path. You guys seem to be fixated on the curved path and looking at it in isolation but that is not the point of this experiment. The straight path tells us all of the information, v, d and t. We then use that to compare to the curved path since we can look at it at any dt and compare it to where the ball is on the straight path.

They are not completely the same, one is a velocity and the other an acceleration. v=d/t is based on the geometric metric that works in the physical world - do you believe it should describe a curve, or a proportion of the curve?

No, v=d/t does not describe a curve, however, it can be used to measure a curve and that is because we only care about the distance part of the equation. We keep t and v equal and we measure d against a straight line path. Using it this way you can not break it down into little dt's. But if we just look at the full distance, then we can absolutely use v=d/t to compare them. We want the complete distance around the curve, not the distance between each dt.

Another way to look at it is that we are never measuring the curved path. We are measuring the straight path and comparing the curved path to it. We can definitely use v=d/t on the straight path and we want to know the length of the curved path if it was straight so we can use it there too.

LongtimeAirman wrote:
Thanks for clarifying the numbers. I don't believe Miles has described how to properly integrate these curves yet. You may be right, we may not need any special transformation.

Intergration just means to add things at each step along the path. In this case we need to add the a-path with the b-path (from Miles most recent pi paper). For each dt, we start at A and the initial velocity is represented by b. The ball ends up at B so the change vector is a. The acceleration is a+b which if you do this vector addition you would get the path c, but the length of c does not represent the correct distance traveled because the ball had both a and b applied to it, not c. Simply put, reality does not use vector addition, it just applies all vectors. There is no a+b, there is only a and b.

LongtimeAirman wrote:
Thanks for the discussion.  
.

Always a pleasure.

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Slumming again. I posted on A Simple Experiment Proves π = 4
http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?f=10&t=16448
.

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I posted this earlier today and wanted to include it here too.
http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?f=10&t=16448&p=115428#p115428

Hello Aardwolf, I need to think about lost momentum. I would say that the pi calculation is used to determine distance around the circle, not time. Nevertheless I guess I need to think about that too.

Plotting distance versus time for both Pi=3.14 and Pi =4 may help everyone. I should have done this earlier.

Balls enter both tracks at the same initial speed (1.2m/s) and hit their 4 marks simultaneously:

1. The ball in the circular track is slower than the straight track. There is friction slowing the ball in the circular track pi/4 when compared to the straight track.
.


or

2. Both tracks are the same length when the curve is measured with Pi=4.
.

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Double posting - My post from TB a little earlier today.
http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?f=10&t=16448&p=115637#p115637
...........................
A Simple Experiment Proves π = 4
http://milesmathis.com/pi7.pdf
...........................

Airman said. That’s what the experiment shows. We have two apparently different constant velocities: circular - 94.3cm/s; and straight - 120cm/s; that are both equal to 120cm/s when pi=4.

lw1990. Are you serious right now
They 'appear' to be different velocities because they are.
Your false idols of Mathis and his student have made you a foolish believer in the superstition of pi=4
By convincing you that the very real phenomenon of speed of the ball does not matter, but whatever bull they feed you, like 'distance warping magic' affects the experiment instead.

How do we know that a ball in any given experiment is traveling at any given speed, how do we know it is not just 'appearing to do so'? This is the kind of problem you get when you invent things with no basis in reality.

Airman. To calculate velocity, one needs to know distance and time.

Straight paths allow the single dimensional equation, (rate)*(time) = distance.
Unfortunately, the formula doesn’t “fit” curves, since curves are two dimensional.

Pi = Circumference/diameter, was invented in order to solve that. Pi=3.14 is the curve to linear metric that allows us to measure any curved path as a linear distance.

Steven’s experiment includes that. The circular track length is based on diameter 17.6cm.
Pi * diameter = 3.14 * 17.6cm = 55.3cm.

A steel ball is launched at 120cm/s into the circular track and completes it at a constant speed 0.5867 seconds later. The circular track velocity is thus: distance/time = 55.3cm/0.5867s = 94.3cm/s

Another steel ball is launched at the exact same time and initial velocity into a separate straight track, four diameters long (4*17.6cm), 70.4cm. Both balls travel for the same amount of time. The straight track velocity was a constant 120cm/s.

Two apparently different velocities: circular - 94.3cm/s; and straight - 120cm/s.

You might say there’s no problem here. The velocities cannot be equal because the distances 55.3cm and 70.4cm are different. If you watch the experiment (hypothetical you) you might even believe, as most would agree, the ball appears to travel slower through the circular track. The time may be the same but the two distances are different.

There’s a big problem. Steven was kind enough to include the 55.3cm distance (the Pi mark) on the straight track. When the ball in the circular track completes its circle, the straight track ball should be at the Pi mark, and not at the 4 mark, as is clearly shown in the video. A dilemma. This violates the known geometrical distance calculation (pi = 3.14 = 55.3cm/17.6cm = circumference / diameter). Repeat, geometry is broken here.

The two balls: 1) started with the same initial velocity; 2) experienced no significant friction; and 3) traveled the same amount of time. It’s not unreasonable to suspect that the two distances and velocities are actually the same.

A new motion metric is needed. The same experiment that demonstrates a failure in pi=3.14 geometry also shows us the correction, replace 3.14 with 4. We then see that the two distances and velocities are equal (70.4cm and 120cm/sec).

For objects in curved motion. Pi=4.

We should observe that we’ve always been underestimating certain distances and velocities. The ball is traveling ((4-3.14)/4) or 21 percent further and faster through the circular track than we thought.

...........................

Maol. Well, this certainly solves the paradoxical question of "What is the radius of a square?"

Airman. Pi are square.

Maol. Well, and I'm sure you easily see, Pi are square solves for the radii of a rectangle, which is sort of an ellipse with corners, like the difference between the tracks at Indy and Talladega, but only to an approximation, or several approximations, as required. Pi are cubed then solves for the multi-inclusive radii of an ellipsoid and requires more approximations, many more.

Airman. It’s not easy, but I think I see. Pi are hypercubed diagonalizes the universe, allowing us to solve for the radius in any corner of space and time. Thanks.
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Nice presentation LTAM. Steven's experiment gives even the old-timers something to chew on. I bet they all have their own "spin" on the ball for a complete explanation.

If the ball traveled along a steel ribbon instead of a "tube" what would be the shape of the "curved" path? Would it twist?

Cr6. Nice presentation LTAM. Steven's experiment gives even the old-timers something to chew on. I bet they all have their own "spin" on the ball for a complete explanation.

If the ball traveled along a steel ribbon instead of a "tube" what would be the shape of the "curved" path? Would it twist?

Hi Cr6. It's fair to say our imaginations were spinning up too.

I believe that Steven said the ball in the curve "banks a few tens of degrees". I currently interpret that "fact" to mean that we can replace the circular track with large radius gently concave surface, like a shallow serving bowl. If we were accurate enough, a gently banked steel ribbon along a plate's rim would work just as well. Simply replacing the steel balls with patterned glass marbles would show that there weren't any additional rotations present; the experimental velocities may vary, but the outcome would remain the same. It's still fun.
.

And again, today's post:
http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?f=10&t=16448&p=115660#p115660
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Airman wrote. There’s a big problem. Steven was kind enough to include the 55.3cm distance (the Pi mark) on the straight track. When the ball in the circular track completes its circle, the straight track ball should be at the Pi mark, and not at the 4 mark, as is clearly shown in the video. A dilemma. This violates the known geometrical distance calculation (pi = 3.14 = 55.3cm/17.6cm = circumference / diameter). Repeat, geometry is broken here.

lw1990 wrote.

It is not a dilemma if the speed of one ball is slower than another ball, even if the speed slowly decelerates through the track, this still means normal physics is happening and nothing is broken or wrong. The experiment does not even come close to showing a shred of evidence that the speeds of the two balls are identical throughout the track. If it did, you might have a point, but it absolutely does not.

You're calculating 'velocity' based on a start distance and end distance of marks on the track, rather than calculating if there's actually a change in velocity while going through the track (like deceleration or even acceleration difference). ...

There is a whole lot of horrible science here, but let's start with the fact that you don't measure velocity through the track except the overall start point-end mark. ...

... The point here is that you have to measure the speed of the balls throughout the track to find out why, ...

Airman. Steven uses 13 marks – a baker’s dozen – in order to calculate, compare and monitor for a change of speed in the two tracks. Not, you might notice, just the “overall start point-end mark”.

The circular track length of 55.3cm is marked into quarters (0,1,2,3,4) (5 marks inclusive). The straight track has 6 marks, (-1,0,1,2,3,4) with unit separations of 17.6cm; 17.6 the straight track also has a “PI mark” at 3.14 showing where the circular track’s original 55.3cm distance would end. The -1 (-17.6cm) and 0 marks allow one to calculate and compare each ball’s observed initial velocity of 120cm/s. A time vs distance plot of the balls’ path through the track markers are shown.


The horizontal values 0.0000s, 0.1467s, 0.2933s, 0.4400s, 0.5867s, are the observed times the steel balls crossed their marks (0,1,2,3,4) in both tracks. The corresponding distances are indicated on the plot lines.

Five measured points for each track lets us calculate 4 interval (quarter track) velocities. The velocity calculation for each interval is given by:

velocity = change of distance over the change in time, or
velocity(current) = (currentPosition – previousPosition)/(currentTime-previuosTime)

Circular track:
(0,1). 1st interval. (13.8-0)/(0.1467-0) = 94.3cm/s
(1,2). 2nd interval. (27.7-13./(0.2933-0.1467) = 94.3cm/s
(2,3). 3rd interval. (41.5-27.7)/(0.4400-0.2933) = 94.3cm/s
(3,4). 4th interval. (55.3-41.5)/0.5867-0.4400) = 94.3cm/s

Straight track:
(0,1). 1st interval. (17.6-0)/(0.1467-0) = 120cm/s
(1,2). 2nd interval. (35.2-17.6)/(0.2933-0.1467) = 120cm/s
(2,3). 3rd interval. (52.8-35.2)/(0.4400-0.2933) = 120cm/s
(3,4). 4th interval. (70.4-52./0.5867-0.4400) = 120cm/s

As you can see, our two apparently different velocities of 94.3cm/s and 120cm/s occurred over each of our quarter intervals.


All velocities, including our initial velocities are 120cm/s when Pi=4.

We also have a camera with frame rate. Measurement marks may be placed along the entire length of transparent tubing. Make that a future refinement. It isn’t necessary, the demonstration is clear.

My previous post referred to constant velocities. That’s what these plots show. There are no decelerations observed. Any friction present is acting over both tracks equally and is at least an order of magnitude less than the observed curve-to-linear (pi=3.14) metric error.

Pi=4 for bodies in curved motion.
.

Here is a video showing how the motions of the circle correspond with the motion on the two axes. Hosted on Vimeo, the link should take you there directly.

https://vimeo.com/189106809

Thanks Jared.

Unfortunately, it's a bit too unwieldy. 8 seconds is too short, each time it stops we have to view the vimeo site application in order to start it up again. Too many graph lines and numbers are unnecessary.  

The only image app I have is windows superpaint.

Can we turn this into a gif?
.

Sure, but I'll have to get to it tonight. I have all the big dog programs for my CAD and arch/viz career, I just never had much chance to do animations in Maya so mine are still pretty rough. I'll dress them up and make them pretty later on, once things are "right" for any particular topic. I'll do an animated .gif tonight.

Don't rush. It's just something good to have. Thanks

http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?f=10&t=16448&p=115713#p115713
.

Siggy_G. We actually discussed this topic a few years back here: Mathis and pi
http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?f=8&t=3183

Airman. Thanks for the reading material, what an eye-opener. I wasn’t a member yet and hadn’t seen it. Mathis and pi is a sensitive issue here and I think I understand that and a few of the earlier responses in this discussion much better now.

Thanks too, in a cosmic sense. I see TB member StevenO was also part of the discussion.
http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?f=8&t=3183&start=15#p34628
StevenO wrote: Re: Mathis and pi

Hi Siggy,

As I stated before, to get a good grasp of this dynamic circle vs. geometric circle problem, you should really study Miles' papers on the calculus. The mistakes in the calculus spill over to a lot of area's in physics and lead to many misconceptions. A math description is always less complete than the actual physics.

Airman. After reading that whole discussion I don’t know if you did or not. You may have won the discussion at the time, unfortunately the whole Mathis and pi thing has developed into a schism. This board has suffered because of it.

That discussion is history. Everyone needs to move on.

StevenO went and created an actual demonstration of the dynamic circle vs. geometric circle problem. I daresay you played a part in StevenO’s motivation (hence cosmic).

Siggy_G. As for the video (maybe it already has been mentioned), I believe the bent tube will add a slight amount of friction to the motion due to the radial redirectioning.
The original rolling motion of the ball will meet some resistance from the tube walls, at some points counter-roll the motion. If the ball had color patterns, this would show up. If the tube was very elastic, the ball would clearly lose energy to tube deformation.

Airman. "(maybe it already has been mentioned)" – yes, I’ll review:

Steven Oostdijk, A Simple Experiment Proves π = 4
http://milesmathis.com/pi7.pdf

StevenO‘s experiment “shows that, while Pi as a distance is 3.14, Pi as a distance/time is 4”

Two transparent pvc tube tracks, side-by-side;
Length. 1) circular and 2) straight; cut to two different lengths:
1) 55.3cm=3.14 * 17.6cm and 2) 70.4cm=4 * 17.6cm; and marked into quarters (0,1,2,3,4).
Motion. Two balls are released into the tracks; same initial velocities – 120cm/s; same start(0.0000), middle (0.1467s, 0.2933s, 0.4400s,) and stop(0.5867s) times; Speeds are constant throughout the experiment; 1) 94.3cm/s and 2) 120cm/s; indicating no apparent friction in either track.

Airman. About the apparent speed difference (94.3cm/s and 120cm/s) between the two tracks I wrote:

More than 20% different? With a ratio involving 3.14 and 4? They started at the same speed, and their track speeds remained constant. Friction’s been eliminated. Motion thru both tracks are synchronized. But why call it a slowdown? A constant velocity difference ratio of 20% without a change in time isn’t a slowdown, it’s a distance scaling change.

And so the solution is simple. The experiment shows us a direct comparison of curved and linear motion that happens to conflict with our current understanding of geometry. A new motion metric where we are shown that both distances are equal (70.4cm) and both velocities are equal (120cm/s). For bodies in motion pi=4.

I agree the experiment could be improved. I’ve remarked that colored marbles can be used to see whether any strange rotations are present. Nevertheless, for those who will see, it is clear. The fact of the matter is that this wonderful, world changing demonstration was created by a TB member as a result of discussion here.

For bodies in curved motion (accelerations comprised of orthogonal velocities) Pi=4
.

http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?f=10&t=16448&p=115745#p115745

Thanks Jared! It worked perfectly.

by LongtimeAirman » Sun Oct 30, 2016 11:35 am
.
Cycloids are not clear examples of curved motion. Calculating their paths, however, do require two variables and time (total of 3 variables).
.................

A circle is more than a circle when it is being created by an object moving through space. Curved motion means that an object is subject to two orthogonal velocities*.

"Here is a video showing how the motions of the circle correspond with the motion on the two axes. Hosted on Vimeo, the link should take you there directly."

https://vimeo.com/189106809


* Oops, no, an object is subject to forces resulting in orthogonal velocities.

I’m still learning.
.

Here is a newer, cleaner version of that animation that should loop, although I don't think Vimeo gives you the choice. Download it if you like.

https://vimeo.com/189647953

Great. The bars give a whole 'nother view. I've posted too many times today so I've gotta quit soon.

Question - not a tasking - I recall the video that was there and gone again, a scene showing a silhouette where two cyclically moving orthogonally overlapping bars presented a moving circular image. I'm not sure the memory is real or not right now.

I would cite you but you seemed to indicate otherwise. Dragon Face is due for a lot more hits.

Thanks for sharing.

Here is a short video of the cycloid, showing Pi=3.14 "racing" it down below.

Pi Cycloid Comparison



Hmm, I did just remove my previous Pi video (on my Vimeo page), but I don't know if that's the one you were talking about. I think these two are better.

Jared,
I'm confused by your cycloid example. Why is there a velocity difference between the yellow and gray balls? I see that the gray is moving 4/pi faster than the yellow ball, but why? The cycloid has nothing to do with the yellow ball.

I see that your original circular motion vimeo that I posted at TB ( https://vimeo.com/189106809 ) is dead. I can't go back to edit it. I'll post your new circular motion with bars ( https://vimeo.com/189647953 ).

I'm still breaking this little app in, hope you don't mind; I'll just leave this here.

.

Yeah, sorry about that cycloid mess-up. I was so deep into it that I wasn't checking my math and time variables, got confused. I'll delete that shortly and replace it with the proper timing, etc..

I love that animated .GIF! Works great, thanks for doing that. I honestly just didn't feel like tossing it into Photoshop or whatever (kinda tedious, from Maya) and glad you beat me to it in this case.

.

We turned a page, so I'll put it at the top.
.

LongtimeAirman wrote:
I believe that Steven said the ball in the curve "banks a few tens of degrees". I currently interpret that "fact" to mean that we can replace the circular track with large radius gently concave surface, like a shallow serving bowl. If we were accurate enough, a gently banked steel ribbon along a plate's rim would work just as well. Simply replacing the steel balls with patterned glass marbles would show that there weren't any additional rotations present; the experimental velocities may vary, but the outcome would remain the same. It's still fun.

The ball in the curve banks a few tens of degrees because the circular path has the centripetal force next to gravity while the straight path has only gravity.
I made some detailed video's of the banking to see if that could prove whether Miles formula a=v^2/2r or Newtons formula a=v^2/r is correct, but the difference between 20 degrees or 40 degrees was too hard to determine from a side view. Perhaps it can be determined more accurately with a half-circle setup. The banking was in the order of 20-40 degrees though, which was consistent with both formula's. Sometimes the banking was oscillating, guess that had to do with how the ball enters the circular path. The oscillation did not seem to affect the outcome.

soostdijk wrote. The ball in the curve banks a few tens of degrees because the circular path has the centripetal force next to gravity while the straight path has only gravity.

airman. I agree. I hadn’t worked with any centripetal force ‘analysis’, and so I avoided the subject, making the general observation to the effect that the additional track length due to banking was insignificant compared to the 20% distance increase of the curved motion metric.

Steven, Welcome. Congratulations for demonstrating the new motion metric, pi=4.

You’ve answered many comments, good and bad, again and again. I’m surprised that some people will not see the new metric, insisting (despite the evidence) that the ball is slower in the circular track. Could you share some thoughts on the reactions to your experiment?

Your circular and straight tracks are elegant, sufficient and necessary to prove the motion metric. One complaint is the inability to monitor final velocities. After our site discussion above, I (Robert M) left a comment at the top of your short video comments section pointing out the obvious. The discussion occurred at the long video. You’ve mentioned track alternatives versus the seasonal lack of sunlight. Any updates there?

soostdijk wrote. I made some detailed video's of the banking to see if that could prove whether Miles formula a=v^2/2r or Newtons formula a=v^2/r is correct, but the difference between 20 degrees or 40 degrees was too hard to determine from a side view. Perhaps it can be determined more accurately with a half-circle setup. The banking was in the order of 20-40 degrees though, which was consistent with both formula's. Sometimes the banking was oscillating, guess that had to do with how the ball enters the circular path. The oscillation did not seem to affect the outcome.

I assumed a rolling contact. How can oscillations occur? Good, unexpected extra data.

I became wrapped up in a 180 twist along the length of the 360deg pvc loop track – a spiraling tangent, but that has nothing to do with the motion metric.
   
Given your breakthrough, I was delighted to post at TBolts as your proponent. I wasn’t good enough to carry off a winning angular momentum argument and so had mainly avoided that subject too. Nevertheless, I think the crowd there is ready for you.

I'm convinced your work makes our future brighter.
.

LongtimeAirman wrote:

soostdijk wrote. The ball in the curve banks a few tens of degrees because the circular path has the centripetal force next to gravity while the straight path has only gravity.

airman. I agree. I hadn’t worked with any centripetal force ‘analysis’, and so I avoided the subject, making the general observation to the effect that the additional track length due to banking was insignificant compared to the 20% distance increase of the curved motion metric.

Steven, Welcome. Congratulations for demonstrating the new motion metric, pi=4.

Thanks for the welcome and the kind words. Credit where credit is due, I got the idea through Miles from his description of Jeff's experiment with the toy trains. I did some experiments with electricity tube and metal balls to convince myself and was lucky to find the transparent PVC tubing so it could be shown in a clear video. I'm an engineer, like to make things work, happy to be able to help Miles here.

LongtimeAirman wrote:You’ve answered many comments, good and bad, again and again. I’m surprised that some people will not see the new metric, insisting (despite the evidence) that the ball is slower in the circular track. Could you share some thoughts on the reactions to your experiment?

Question: is the ball really slower? That is what most people cannot get their head around (including me ). You mention a new "metric", but that is a bit too mainstream for me. People have been confused into thinking that there are three orthogonal axes with an arbitrary origin spanning space, but distance and time are both derived from motion, so we first have to understand motion better.
Think we have to start there by updating our understanding on momentum. The momentum of the ball seems to change instantly and is conserved around the circular motion. That is both curious.
I have also done the experiment with the 180 degree circle setup and then the exit velocity of the circular ball is 3.14/4 of the entry velocity, so the "b" momentum as you call it is then lost. It is logical as the tube cannot deliver a pushing force, only a restrictive force, but as long as we cannot show without doubt what happens with the momentum of the ball at entry and exit I will not post it. Let people absorb this experiment first.

LongtimeAirman wrote:Your circular and straight tracks are elegant, sufficient and necessary to prove the motion metric. One complaint is the inability to monitor final velocities. After our site discussion above, I (Robert M) left a comment at the top of your short video comments section pointing out the obvious. The discussion occurred at the long video. You’ve mentioned track alternatives versus the seasonal lack of sunlight. Any updates there?

soostdijk wrote. I made some detailed video's of the banking to see if that could prove whether Miles formula a=v^2/2r or Newtons formula a=v^2/r is correct, but the difference between 20 degrees or 40 degrees was too hard to determine from a side view. Perhaps it can be determined more accurately with a half-circle setup. The banking was in the order of 20-40 degrees though, which was consistent with both formula's. Sometimes the banking was oscillating, guess that had to do with how the ball enters the circular path. The oscillation did not seem to affect the outcome.

I assumed a rolling contact. How can oscillations occur? Good, unexpected extra data.

For the updates see above.

About the oscillations, think they happen if the spin of the ball is not aligned closely with the curving of the tube so the ball overshoots.

LongtimeAirman wrote:I became wrapped up in a 180 twist along the length of the 360deg pvc loop track – a spiraling tangent, but that has nothing to do with the motion metric.
   
Given your breakthrough, I was delighted to post at TBolts as your proponent. I wasn’t good enough to carry off a winning angular momentum argument and so had mainly avoided that subject too. Nevertheless, I think the crowd there is ready for you.

I'm convinced your work makes our future brighter.
.

TB might be ready but I do not have time to waste on trolls

LongtimeAirman wrote:
Given your breakthrough, I was delighted to post at TBolts as your proponent. I wasn’t good enough to carry off a winning angular momentum argument and so had mainly avoided that subject too. Nevertheless, I think the crowd there is ready for you.

I waded through the insults on the TB forum to see if there is anything that I could clarify but I think it all has been done excellently by you already.

There is two things I could suggest to mention in the discussion:

1. Pulling a "velocity" around a circle is not the same as pulling a solid ruler around a circle. The velocity vector will need to be rotated along the path otherwise it would not follow the path. This rotation takes time. So it would not be logical that the length of a circular path measured with a velocity would be the same as measured with a ruler. For all these people shouting that Pi will always remain 3.14, sure, but only for geometric ratios.

2. There was a discussion whether the definition of length would need to change but that is not the case. Measuring the length of the circular path starting with the origin on the path will yield 4 as is shown in the video. It would yield 3.14 measured from the virtual center but only if you ignore the rotating reference frame of the ball (which is interestingly enough actually admitted by the mainstream ).

soostdijk wrote. I waded through the insults on the TB forum to see if there is anything that I could clarify but I think it all has been done excellently by you already.

There is two things I could suggest to mention in the discussion:

1. Pulling a "velocity" around a circle is not the same as pulling a solid ruler around a circle. The velocity vector will need to be rotated along the path otherwise it would not follow the path. This rotation takes time. So it would not be logical that the length of a circular path measured with a velocity would be the same as measured with a ruler. For all these people shouting that Pi will always remain 3.14, sure, but only for geometric ratios.

2. There was a discussion whether the definition of length would need to change but that is not the case. Measuring the length of the circular path starting with the origin on the path will yield 4 as is shown in the video. It would yield 3.14 measured from the virtual center but only if you ignore the rotating reference frame of the ball (which is interestingly enough actually admitted by the mainstream  ).

airman. Thanks for the positive review. Obviously I was learning during the discussion. Debating trolls with slowly improving arguments may not be ideal but it works.

You’re right. I haven’t considered the “rotating reference frame of the ball” and how it’s necessary to properly calculate the motion distance. I will do so.

I also need to spend a great deal more time examining the angular velocity aspect. How many degrees of banking differentiates Newton’s from Miles’ formulas?

Of course, truth be told, we’ve had our own discussions here. Nevyn gave me the confidence to take on TB with the rate*time=distance argument; keeping it simple. I’m sure he’s ready for a rotating frame reference discussion.
.

.
Another Update to pi7
http://milesmathis.com/pi7.pdf

Miles wrote. November 8, 2016: I have had one of my online buddies create a gif for me that matches part of the
video they are suppressing. I have never imported a gif into a paper before, so hopefully this will
work, including the motion. See below.

If that doesn't move for you, you can go to his video at Vimeo

Jared, At this point we have a cheer (please forgive the fact that it's military). It goes,

HooAhh

The rest of the update is an eye-opener too.
.

.
The materials:


CatFoodClampHosedCakeContainer.



By some quirk, my working diameter is 31.4cm.
.

.
Steven O was careful to keep his ramp support structure outside his youtube video image. I can understand why, it's a distraction from the pi=4 demonstration. Hiding it only works once. I must improvise something - Oogly.


A height adjustable ramp support structure.


I haven’t cut the track lengths yet. One can easily straighten coiled pvc using rope. Start with a feather (or steel ball), and thread. Or ask an electrician.


Thank you Serendipity, all but the steel 5/8X8in bolt, washers and nuts were in my junk box. Only the threaded U bolts’ half circle needs improving/modifying. The U bolts clamps’ slightly concave/convex surfaces allow plenty of smooth controlled swivel about the tripod’s leg angle changes with little pvc pinching. An old plumbing coupler is used here as a heavy spacer to get past the bolt’s shank. The release mechanism must be here somewhere.


I'm grateful for the distraction.
.

.


The following R code is submitted for your approval.

HINT - Feedback is greatly appreciated.

Code:


library(ggplot2)

######################################
##                                  ##
##               Pi=4               ##
##                                  ##
######################################
##       Intended to recreate       ##
##         Steven Oostdijk's        ##
## Pi=4 experiment as described in  ##          
## http://milesmathis.com/pi7.pdf   ##
## -------------------------------- ##
##            Posted at             ##
## http://milesmathis.forumotion.com/ ##
## -------------------------------- ##
##    Contributing to a future      ##
##     Miles Mathis R project       ##
######################################
##   Input data used:               ##
##   1) The unit circle's diameter  ##
##   2) InsideTrackWidth            ##
##   3) OutsideTrackWidth           ##
##   4) Table length and width      ##
## -------------------------------- ##
##   Replace with your data to      ##
##     plot your own tracks         ##
## -------------------------------- ##
##       measurements in cm         ##
##     unless noted otherwise       ##
######################################
##  The tracklengths are scaled  
##     to the working diameter
##     CHANGE a_diameter HERE
a_diameter <- 31.4
a <- a_diameter
a_diaInInches <- a_diameter/2.54
a_diaInInches
## [1] 12.3622

##  CHANGE INSIDE AND OUTSIDE DIAMETERS HERE
## The pvc inside diameter is the trackWidthIn.
## 5/8" converted to centimeters
b_trackWidthIn <- 2.54*(5/8)
bi <- b_trackWidthIn
## The pvc outside diameter is the trackWidthOut.
## 7/8" converted to centimeters
b_trackWidthOut <- 2.54*(7/8)
bo <- b_trackWidthOut

## The circle track length, 0-4, is the circular
## track's centerline circumference
c_circumference <- 3.14159*a_diameter
c314 <- c_circumference
## c314=98.6459
## The radii needed to plot the circular track:
## 14.590, 14.910, 15.700, 16.490, 16.810 (sorted)
r <- a_diameter/2
rOut <- r + bo/2
rIn <- r - bo/2
rOut2 <- r + bi/2
rIn2 <- r - bi/2

## For rough (unknown ramp) overall length estimate,
## the total length will be:
## 1) ramp(a_diameter), 2)initNegOneToZero(a_diameter),
## and 3) straight track length (4*a_diameter).
## A total rough estimate plot length of 6*a_diameter
d_straight <- 6*a_diameter
## d_straight = 188.4

##  CHANGE TABLE DIMENSIONS (T1 and T2) HERE
## My workspace limit, table dimensions, is 29.5X71.25in.
## Convert to cm.
T1 <- 29.5*2.54
## T1 = 74.93
T2 <- 71.25*2.54
## T2 = 180.975
xCoordsInit <- c(-T2/2,-T2/2,T2/2,T2/2,-T2/2)
yCoordsInit <- c(-T1/2,T1/2,T1/2,-T1/2,-T1/2)
## With a_diameter=31.4, d_straight = 188.4 > T2 = 180.975.
## I've exceeded my table length limit by a few centimeters.
## I can gain some centimeters by aligning the straight
## track to the table's diagonal.

## For my special case, to determine a diagonal layout
## Calculate table rotation
MyTable <- cbind(xCoordsInit,yCoordsInit)
MyTable
##       xCoordsInit yCoordsInit
##[1,]    -90.4875     -37.465
##[2,]    -90.4875      37.465
##[3,]     90.4875      37.465
##[4,]     90.4875     -37.465
##[5,]    -90.4875     -37.465
## Rotation angle equals neg arcTan((T1/2)/(T2/2))
## = neg arcTan(37.465/90.4875)
Angle <- (-1)*atan(T1/T2)
## Angle in radians
Angle
## -0.3925...
AngInDeg <- Angle*360/(2*3.14159)
AngInDeg
## -22.49deg
## x'<- x*cos(theta)+ysin(theta)
## y'<- x*(-1)sin(theta)+ycos(theta)
ZRotation <- matrix(c(cos(Angle),sin(Angle),(-1)*sin(Angle),cos(Angle)),nrow = 2,ncol = 2,byrow = TRUE)
ZRotation
dim(MyTable)
TR <- MyTable%*%ZRotation
TR
##        [,1]      [,2]
##[1,] -97.93679   0.00000
##[2,] -69.27286  69.23066
##[3,]  97.93679   0.00000
##[4,]  69.27286 -69.23066
##[5,] -97.93679   0.00000
## I need a better way to transfer both table contents,  
## (initial and final) for the ggplot geom_path function.
xCoordsFinal <- c(-97.93678513/2,-69.27286231/2,97.93678513/2,69.27286231/2,-97.93678513/2)
yCoordsFinal <- c(-0,69.23066104,0,-69.23066104,-0)
## plot(xCoordsFinal,yCoordsFinal)
## The ramp's horiz distance must be under 31.4cm to
## gain any more cm's. It's looking ugly for Oogly.

ggplot(data=data.frame(x=1,y=1),aes(x,y))+
  ggtitle("Pi=4 Track Layout on My Almost Too Small Table\nEach unit length here is 31.4cm (or 12.4in)")+
  ## tabletop:
  geom_path(data=data.frame(x=c(-T2/2,-T2/2,T2/2,T2/2,-T2/2),y=c(-T1/2,T1/2,T1/2,-T1/2,-T1/2)), colour = "Red")+
  geom_path(data=data.frame(x=c(TR[1,1],TR[2,1],TR[3,1],TR[4,1],TR[5,1]),y=c(TR[1,2],TR[2,2],TR[3,2],TR[4,2],TR[5,2])))+
  ## Track Centerlines are blue
  ## ramps:
  ## circle track
  geom_path(data=data.frame(x=c(-3*a,-2*a),y=c(bo,bo)), colour = "Blue")+
  geom_path(data=data.frame(x=c(-3*a,-2*a,-2*a,-3*a,-3*a),y=c(bo/2,bo/2,1.5*bo,1.5*bo,bo/2)))+
  geom_path(data=data.frame(x=c(-3*a,-2*a,-2*a,-3*a,-3*a),y=c(bo+bi/2 ,bo+bi/2,bo-bi/2,bo-bi/2,bo+bi/2)))+
  ## straight track  
  geom_path(data=data.frame(x=c(-3*a,-2*a),y=c(0,0)), colour = "Blue")+
  geom_path(data=data.frame(x=c(-3*a,-2*a,-2*a,-3*a,-3*a),y=c(bo/2,bo/2,-bo/2,-bo/2,bo/2)))+
  geom_path(data=data.frame(x=c(-3*a,-2*a,-2*a,-3*a,-3*a),y=c(0+bi/2,0+bi/2,0-bi/2,0-bi/2,0+bi/2)))+
  annotate("text", x = -2.5*a, y = 8, label = "Ramps") +
  annotate("text", x = -63, y = -5, label = "-1") +
  ## NegOneToZero:
  ## circle track
  geom_path(data=data.frame(x=c(-2*a,-1*a),y=c(bo,bo)), colour = "Blue")+
  geom_path(data=data.frame(x=c(-2*a,-1*a,-1*a,-2*a,-2*a),y=c(bo/2,bo/2,1.5*bo,1.5*bo,bo/2)))+
  geom_path(data=data.frame(x=c(-2*a,-1*a,-1*a,-2*a,-2*a),y=c(bo+bi/2,bo+bi/2,bo-bi/2,bo-bi/2,bo+bi/2)))+
  ## straight track  
  geom_path(data=data.frame(x=c(-2*a,-1*a),y=c(0,0)), colour = "Blue")+
  geom_path(data=data.frame(x=c(-2*a,-1*a,-1*a,-2*a,-2*a),y=c(bo/2,bo/2,-bo/2,-bo/2,bo/2)))+
  geom_path(data=data.frame(x=c(-2*a,-1*a,-1*a,-2*a,-2*a),y=c(0+bi/2,0+bi/2,0-bi/2,0-bi/2,0+bi/2)))+
  annotate("text", x = -31, y = -5, label = "0") +
  ## ZeroToOne:
  ## circle track
  geom_path(data=data.frame(x=c(-1*a+(15700:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((15.7)^2-c(15700:(0)/1000,0)^2))),aes(x=x,y=y), colour = "Blue")+
  geom_path(data=data.frame(x=c(-1*a+(14590:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((14.59)^2-c(14590:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a+(16490:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((16.49)^2-c(16490:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a+(16810:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((16.81)^2-c(16810:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a+(14910:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((14.91)^2-c(14910:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a+14.59,-1*a+16.81),y=c(15.7+bo,15.7+bo)))+
  annotate("text", x = -21, y = 18, label = "1") +
  ## straight track  
  geom_path(data=data.frame(x=c(-1*a,0*a),y=c(0,0)), colour = "Blue")+
  geom_path(data=data.frame(x=c(-1*a,0*a,0*a,-1*a,-1*a),y=c(bo/2,bo/2,-bo/2,-bo/2,bo/2)))+
  geom_path(data=data.frame(x=c(-1*a,0*a,0*a,-1*a,-1*a),y=c(0+bi/2,0+bi/2,0-bi/2,0-bi/2,0+bi/2)))+
  annotate("text", x = 0, y = -5, label = "1") +
  ## OneToTwo:
  ## circle track
  geom_path(data=data.frame(x=c(-1*a+(15700:(0)/1000),-1*a),y=c((15.7+bo)+sqrt((15.7)^2-c(15700:(0)/1000,0)^2))),aes(x=x,y=y), colour = "Blue")+
  geom_path(data=data.frame(x=c(-1*a+(14590:(0)/1000),-1*a),y=c((15.7+bo)+sqrt((14.59)^2-c(14590:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a+(16490:(0)/1000),-1*a),y=c((15.7+bo)+sqrt((16.49)^2-c(16490:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a+(16810:(0)/1000),-1*a),y=c((15.7+bo)+sqrt((16.81)^2-c(16810:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a+(14910:(0)/1000),-1*a),y=c((15.7+bo)+sqrt((14.91)^2-c(14910:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a,-1*a),y=c(31.4+bo/2,2*16.81+bo/2)))+
  annotate("text", x = -31, y = 29, label = "2") +
  ## straight track  
  geom_path(data=data.frame(x=c(0,a),y=c(0,0)), colour = "Blue")+
  geom_path(data=data.frame(x=c(0,a,a,0*a,0*a),y=c(bo/2,bo/2,-bo/2,-bo/2,bo/2)))+
  geom_path(data=data.frame(x=c(0,a,a,0*a,0*a),y=c(0+bi/2 ,0+bi/2,0-bi/2,0-bi/2,0+bi/2)))+
  annotate("text", x = 32, y = -5, label = "2") +
  ## TwoToThree:
  ## circle track
  geom_path(data=data.frame(x=c(-1*a-(15700:(0)/1000),-1*a),y=c((15.7+bo)+sqrt(15.7^2-c(15700:(0)/1000,0)^2))),aes(x=x,y=y), colour = "Blue")+
  geom_path(data=data.frame(x=c(-1*a-(14590:(0)/1000),-1*a),y=c((15.7+bo)+sqrt((14.59)^2-c(14590:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a-(16490:(0)/1000),-1*a),y=c((15.7+bo)+sqrt((16.49)^2-c(16490:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a-(16810:(0)/1000),-1*a),y=c((15.7+bo)+sqrt((16.81)^2-c(16810:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a-(14910:(0)/1000),-1*a),y=c((15.7+bo)+sqrt((14.91)^2-c(14910:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a-14.59,-1*a-16.81),y=c(15.7+bo,15.7+bo)))+
  annotate("text", x = -42, y = 18, label = "3") +
  ## straight track  
  geom_path(data=data.frame(x=c(a,2*a),y=c(0,0)), colour = "Blue")+
  geom_path(data=data.frame(x=c(a,2*a,2*a,a,a),y=c(bo/2,bo/2,-bo/2,-bo/2,bo/2)))+
  geom_path(data=data.frame(x=c(a,2*a,2*a,a,a),y=c(0+bi/2 ,0+bi/2,0-bi/2,0-bi/2,0+bi/2)))+
  annotate("text", x = 63, y = -5, label = "3") +
  ## ThreeToFour:
  ## circle track
  geom_path(data=data.frame(x=c(-1*a-(15700:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((15.7)^2-c(15700:(0)/1000,0)^2))),aes(x=x,y=y), colour = "Blue")+
  geom_path(data=data.frame(x=c(-1*a-(14590:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((14.59)^2-c(14590:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a-(16490:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((16.49)^2-c(16490:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a-(16810:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((16.81)^2-c(16810:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a-(14910:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((14.91)^2-c(14910:(0)/1000,0)^2))),aes(x=x,y=y))+
  annotate("text", x = -31, y = 8, label = "4") +
  ## straight track  
  geom_path(data=data.frame(x=c(2*a,3*a),y=c(0,0)), colour = "Blue")+
  geom_path(data=data.frame(x=c(2*a,3*a,3*a,2*a,2*a),y=c(bo/2,bo/2,-bo/2,-bo/2,bo/2)))+
  geom_path(data=data.frame(x=c(2*a,3*a,3*a,2*a,2*a),y=c(0+bi/2,0+bi/2,0-bi/2,0-bi/2,0+bi/2)))+
  geom_path(data=data.frame(x=c(c314-a,c314-a),y=c(-bo/2,bo/2)))+
  annotate("text", x = c314-a, y = 6, label = "PI") +
  annotate("text", x = 3*a, y = 6, label = "4") +
  ## fix aspect ratio to 1:1
  coord_fixed()



Oh, and the table corners are rounded 6.35cm (2.5in).
.

Sorry not to comment sooner...

Looking really good LTAM. Nice craftsmanship.  
Excited to see the "pair" roll together down your set up! Maybe add some colored smoke down the tube at some point to see how it drifts out? Just a thought...sorry to add still another suggestion. I personally know how annoying extra-suggestions can sometimes be when you have completed a real piece of work and then someone stops by at the last minute and says --  "hey looks good but why don't you do this and that to it too?".

The R script looks pretty top notch as well I might add. It is non-trivial IMHO at this point - but I'm just a hack myself.

.
Cr6,  Thanks. I should have been clearer. I'll admit to too much psychic stress over the last month. Mostly concerning our country's ongoing train wreck. This project is a positive relief. Craftsmanship? I prefer thrift shops and junk piles. Colored smoke? Oh my.

I'm looking for suggestions on the R code. Criticism is better than compliments. Hack it. What would make you use it?

Nevyn, I don't like coding trash. Please mark errors or make suggestions. Also, you indicated an opportunity to study R. Any insights or lessons learned?  

I'm good with constraints. Clearly, R is most suitable for large data groups, which is almost opposite from the Pi=4 setup, or any other Miles R project that comes to my mind. Another R package, ggvis, is supposed to make browser interaction possible. I'll look into it.
.

Hey Airman, I can't offer too much advice specific to R because I didn't get that far into it but I can offer some suggestions about coding standards. As R is a language for mathematicians rather than programmers, you are going to find a lot of examples of bad coding style. I would even go so far as to say that the language itself promotes it.

Take this line for example:

Code:

geom_path(data=data.frame(x=c(-1*a-(15700:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((15.7)^2-c(15700:(0)/1000,0)^2))),aes(x=x,y=y), colour = "Blue")

It is doing too much and is confusing. You might know what it is doing today but will you understand it in a years time? Make use of variables for clarity. Even just a bit of whitespace can make it so much more readable. For example, even without using variables I would write that line like this:

Code:

geom_path( data=data.frame( x=c( -1*a - ( 15700:(0)/1000 ), -1*a ), y=c( ( 15.7+bo ) - sqrt( (15.7)^2 - c( 15700:(0)/1000, 0 )^2 ) ) ), aes( x=x, y=y ), colour = "Blue" )

Make whitespace your friend. There are no prizes for doing the most in the smallest file size but you will thank yourself later for well formatted code.

You have a few -1*a statements in there which should use the unary operator which is just -a. No need for an expensive multiplication just to make something negative. If you are using it in an expression you can use whitespace to specify exactly what you want, for example: 1--a won't work but 1 - -a will work. You could also do it once and assign it to a variable to save even more computations.

This is using a function called 'c'. Not a good function name. What does it do? There is no way to tell that just from a single letter name. A function name should be descriptive which is why I have called R an ugly language in the past. Since I can't see the function definition in your code I assume it is being pulled in from a library or is a core R function so it may not be your fault but you could still make your own code more readable by assigning the function to a better name.

Turns out 'c' is a core function that concatenates values into an array. See how if they had called it 'concat' it would be self-descriptive.

You should declare your own functions to make things clear. Take this section as an example:

Code:

 ## circle track
  geom_path(data=data.frame(x=c(-1*a+(15700:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((15.7)^2-c(15700:(0)/1000,0)^2))),aes(x=x,y=y), colour = "Blue")+
  geom_path(data=data.frame(x=c(-1*a+(14590:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((14.59)^2-c(14590:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a+(16490:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((16.49)^2-c(16490:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a+(16810:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((16.81)^2-c(16810:(0)/1000,0)^2))),aes(x=x,y=y))+
  geom_path(data=data.frame(x=c(-1*a+(14910:(0)/1000),-1*a),y=c((15.7+bo)-sqrt((14.91)^2-c(14910:(0)/1000,0)^2))),aes(x=x,y=y))+


This could be written as a function that takes an array as input. Try something like this:

Code:


// declare a function
// I assume a is an angle, not sure what bo is
createCircleTrack <- function( a, bo, values, colour )
{
  n = 0 // not sure what to initialize this too as I don't know what geom_path is returning but it should be the equivalent of zero, not null
  na = -a  // negative a
  for( i in values )
  {
    // use variables to reduce clutter
    x = c( na + ( i:(0)/1000), na )
    y = c( i/1000 + bo - sqrt( (i/1000)^2-c( i:(0)/1000, 0 )^2 )
    n += geom_path( data=data.frame( x=x, y=y ), aes( x=x, y=y ), colour=colour )
    // not sure if += operator is supported but it just means n = n+<some value>
  }
  return n
}
// call the function
path <- createCircleTrack( a, bo, c( 15700, 14590, 16490, 16810, 14910 ), "Blue" )


Now, you might be thinking that it took more lines of code to do the same thing but it becomes so much easier to fix when you find an error in it and if you call it more than once then you have saved lines of code. It also makes is much easier to copy into another project.

I hope that helps a little bit. Liberal use of whitespace, meaningful variable names and good function use go a long way to creating good, maintainable, re-usable code.

There are a few packages for formatting. Nothing fantastic yet but they work:

https://cran.r-project.org/web/packages/formatR/formatR.pdf
library(formatR)

messy <- readLines("C:\\Apache\\PI4.R")
#tidy_source(messy)
## the formatted version
tidy_source(text = messy)

Code:

library(ggplot2)

###################################### ## Pi=4 ## ## Intended to recreate ## Steven Oostdijk's ## Pi=4 experiment as described in ##
###################################### http://milesmathis.com/pi7.pdf ## -------------------------------- ## Posted at ##
###################################### http://milesmathis.forumotion.com/ ## -------------------------------- ## Contributing to a future ##
###################################### Miles Mathis R project ## Input data used: ## 1) The unit circle's diameter ## 2) InsideTrackWidth
###################################### ## 3) OutsideTrackWidth ## 4) Table length and width ## -------------------------------- ## Replace
###################################### with your data to ## plot your own tracks ## -------------------------------- ## measurements in cm
###################################### ## unless noted otherwise ## The tracklengths are scaled to the working diameter CHANGE a_diameter
###################################### HERE
a_diameter <- 31.4
a <- a_diameter
a_diaInInches <- a_diameter/2.54
a_diaInInches
## [1] 12.3622

## CHANGE INSIDE AND OUTSIDE DIAMETERS HERE The pvc inside diameter is the trackWidthIn.  5/8'
## converted to centimeters
b_trackWidthIn <- 2.54 * (5/8)
bi <- b_trackWidthIn
## The pvc outside diameter is the trackWidthOut.  7/8' converted to centimeters
b_trackWidthOut <- 2.54 * (7/8)
bo <- b_trackWidthOut

## The circle track length, 0-4, is the circular track's centerline circumference
c_circumference <- 3.14159 * a_diameter
c314 <- c_circumference
## c314=98.6459 The radii needed to plot the circular track: 14.590, 14.910, 15.700, 16.490, 16.810
## (sorted)
r <- a_diameter/2
rOut <- r + bo/2
rIn <- r - bo/2
rOut2 <- r + bi/2
rIn2 <- r - bi/2

## For rough (unknown ramp) overall length estimate, the total length will be: 1) ramp(a_diameter),
## 2)initNegOneToZero(a_diameter), and 3) straight track length (4*a_diameter).  A total rough
## estimate plot length of 6*a_diameter
d_straight <- 6 * a_diameter
## d_straight = 188.4

## CHANGE TABLE DIMENSIONS (T1 and T2) HERE My workspace limit, table dimensions, is 29.5X71.25in.
## Convert to cm.
T1 <- 29.5 * 2.54
## T1 = 74.93
T2 <- 71.25 * 2.54
## T2 = 180.975
xCoordsInit <- c(-T2/2, -T2/2, T2/2, T2/2, -T2/2)
yCoordsInit <- c(-T1/2, T1/2, T1/2, -T1/2, -T1/2)
## With a_diameter=31.4, d_straight = 188.4 > T2 = 180.975.  I've exceeded my table length limit by a
## few centimeters.  I can gain some centimeters by aligning the straight track to the table's
## diagonal.

## For my special case, to determine a diagonal layout Calculate table rotation
MyTable <- cbind(xCoordsInit, yCoordsInit)
MyTable
## xCoordsInit yCoordsInit [1,] -90.4875 -37.465 [2,] -90.4875 37.465 [3,] 90.4875 37.465 [4,] 90.4875
## -37.465 [5,] -90.4875 -37.465 Rotation angle equals neg arcTan((T1/2)/(T2/2)) = neg
## arcTan(37.465/90.4875)
Angle <- (-1) * atan(T1/T2)
## Angle in radians
Angle
## -0.3925...
AngInDeg <- Angle * 360/(2 * 3.14159)
AngInDeg
## -22.49deg x'<- x*cos(theta)+ysin(theta) y'<- x*(-1)sin(theta)+ycos(theta)
ZRotation <- matrix(c(cos(Angle), sin(Angle), (-1) * sin(Angle), cos(Angle)), nrow = 2, ncol = 2, byrow = TRUE)
ZRotation
dim(MyTable)
TR <- MyTable %*% ZRotation
TR
## [,1] [,2] [1,] -97.93679 0.00000 [2,] -69.27286 69.23066 [3,] 97.93679 0.00000 [4,] 69.27286
## -69.23066 [5,] -97.93679 0.00000 I need a better way to transfer both table contents, (initial and
## final) for the ggplot geom_path function.
xCoordsFinal <- c(-97.93678513/2, -69.27286231/2, 97.93678513/2, 69.27286231/2, -97.93678513/2)
yCoordsFinal <- c(-0, 69.23066104, 0, -69.23066104, -0)
## plot(xCoordsFinal,yCoordsFinal) The ramp's horiz distance must be under 31.4cm to gain any more
## cm's. It's looking ugly for Oogly.

ggplot(data = data.frame(x = 1, y = 1), aes(x, y)) + ggtitle("Pi=4 Track Layout on My Almost Too Small Table\nEach unit length here is 31.4cm (or 12.4in)") +
  ## tabletop:
  geom_path(data = data.frame(x = c(-T2/2, -T2/2, T2/2, T2/2, -T2/2), y = c(-T1/2, T1/2, T1/2, -T1/2, -T1/2)),
            colour = "Red") + geom_path(data = data.frame(x = c(TR[1, 1], TR[2, 1], TR[3, 1], TR[4, 1], TR[5,
                                                                                                          1]), y = c(TR[1, 2], TR[2, 2], TR[3, 2], TR[4, 2], TR[5, 2]))) + ## Track Centerlines are blue ramps: circle track
  geom_path(data = data.frame(x = c(-3 * a, -2 * a), y = c(bo, bo)), colour = "Blue") + geom_path(data = data.frame(x = c(-3 *
                                                                                                                            a, -2 * a, -2 * a, -3 * a, -3 * a), y = c(bo/2, bo/2, 1.5 * bo, 1.5 * bo, bo/2))) + geom_path(data = data.frame(x = c(-3 *
                                                                                                                                                                                                                                                    a, -2 * a, -2 * a, -3 * a, -3 * a), y = c(bo + bi/2, bo + bi/2, bo - bi/2, bo - bi/2, bo + bi/2))) +
  ## straight track
  geom_path(data = data.frame(x = c(-3 * a, -2 * a), y = c(0, 0)), colour = "Blue") + geom_path(data = data.frame(x = c(-3 *
                                                                                                                          a, -2 * a, -2 * a, -3 * a, -3 * a), y = c(bo/2, bo/2, -bo/2, -bo/2, bo/2))) + geom_path(data = data.frame(x = c(-3 *
                                                                                                                                                                                                                                            a, -2 * a, -2 * a, -3 * a, -3 * a), y = c(0 + bi/2, 0 + bi/2, 0 - bi/2, 0 - bi/2, 0 + bi/2))) + annotate("text",
                                                                                                                                                                                                                                                                                                                                                    x = -2.5 * a, y = 8, label = "Ramps") + annotate("text", x = -63, y = -5, label = "-1") + ## NegOneToZero: circle track
  geom_path(data = data.frame(x = c(-2 * a, -1 * a), y = c(bo, bo)), colour = "Blue") + geom_path(data = data.frame(x = c(-2 *
                                                                                                                            a, -1 * a, -1 * a, -2 * a, -2 * a), y = c(bo/2, bo/2, 1.5 * bo, 1.5 * bo, bo/2))) + geom_path(data = data.frame(x = c(-2 *
                                                                                                                                                                                                                                                    a, -1 * a, -1 * a, -2 * a, -2 * a), y = c(bo + bi/2, bo + bi/2, bo - bi/2, bo - bi/2, bo + bi/2))) +
  ## straight track
  geom_path(data = data.frame(x = c(-2 * a, -1 * a), y = c(0, 0)), colour = "Blue") + geom_path(data = data.frame(x = c(-2 *
                                                                                                                          a, -1 * a, -1 * a, -2 * a, -2 * a), y = c(bo/2, bo/2, -bo/2, -bo/2, bo/2))) + geom_path(data = data.frame(x = c(-2 *
                                                                                                                                                                                                                                            a, -1 * a, -1 * a, -2 * a, -2 * a), y = c(0 + bi/2, 0 + bi/2, 0 - bi/2, 0 - bi/2, 0 + bi/2))) + annotate("text",
                                                                                                                                                                                                                                                                                                                                                    x = -31, y = -5, label = "0") + ## ZeroToOne: circle track
  geom_path(data = data.frame(x = c(-1 * a + (15700:(0)/1000), -1 * a), y = c((15.7 + bo) - sqrt((15.7)^2 -
                                                                                                  c(15700:(0)/1000, 0)^2))), aes(x = x, y = y), colour = "Blue") + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                        a + (14590:(0)/1000), -1 * a), y = c((15.7 + bo) - sqrt((14.59)^2 - c(14590:(0)/1000, 0)^2))), aes(x = x,
                                                                                                                                                                                                                                                                                                          y = y)) + geom_path(data = data.frame(x = c(-1 * a + (16490:(0)/1000), -1 * a), y = c((15.7 + bo) -
                                                                                                                                                                                                                                                                                                                                                                                                  sqrt((16.49)^2 - c(16490:(0)/1000, 0)^2))), aes(x = x, y = y)) + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        a + (16810:(0)/1000), -1 * a), y = c((15.7 + bo) - sqrt((16.81)^2 - c(16810:(0)/1000, 0)^2))), aes(x = x,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          y = y)) + geom_path(data = data.frame(x = c(-1 * a + (14910:(0)/1000), -1 * a), y = c((15.7 + bo) -
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  sqrt((14.91)^2 - c(14910:(0)/1000, 0)^2))), aes(x = x, y = y)) + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        a + 14.59, -1 * a + 16.81), y = c(15.7 + bo, 15.7 + bo))) + annotate("text", x = -21, y = 18, label = "1") +
  ## straight track
  geom_path(data = data.frame(x = c(-1 * a, 0 * a), y = c(0, 0)), colour = "Blue") + geom_path(data = data.frame(x = c(-1 *
                                                                                                                        a, 0 * a, 0 * a, -1 * a, -1 * a), y = c(bo/2, bo/2, -bo/2, -bo/2, bo/2))) + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                                                        a, 0 * a, 0 * a, -1 * a, -1 * a), y = c(0 + bi/2, 0 + bi/2, 0 - bi/2, 0 - bi/2, 0 + bi/2))) + annotate("text",
                                                                                                                                                                                                                                                                                                                                                x = 0, y = -5, label = "1") + ## OneToTwo: circle track
  geom_path(data = data.frame(x = c(-1 * a + (15700:(0)/1000), -1 * a), y = c((15.7 + bo) + sqrt((15.7)^2 -
                                                                                                  c(15700:(0)/1000, 0)^2))), aes(x = x, y = y), colour = "Blue") + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                        a + (14590:(0)/1000), -1 * a), y = c((15.7 + bo) + sqrt((14.59)^2 - c(14590:(0)/1000, 0)^2))), aes(x = x,
                                                                                                                                                                                                                                                                                                          y = y)) + geom_path(data = data.frame(x = c(-1 * a + (16490:(0)/1000), -1 * a), y = c((15.7 + bo) +
                                                                                                                                                                                                                                                                                                                                                                                                  sqrt((16.49)^2 - c(16490:(0)/1000, 0)^2))), aes(x = x, y = y)) + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        a + (16810:(0)/1000), -1 * a), y = c((15.7 + bo) + sqrt((16.81)^2 - c(16810:(0)/1000, 0)^2))), aes(x = x,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          y = y)) + geom_path(data = data.frame(x = c(-1 * a + (14910:(0)/1000), -1 * a), y = c((15.7 + bo) +
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  sqrt((14.91)^2 - c(14910:(0)/1000, 0)^2))), aes(x = x, y = y)) + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        a, -1 * a), y = c(31.4 + bo/2, 2 * 16.81 + bo/2))) + annotate("text", x = -31, y = 29, label = "2") +
  ## straight track
  geom_path(data = data.frame(x = c(0, a), y = c(0, 0)), colour = "Blue") + geom_path(data = data.frame(x = c(0,
                                                                                                              a, a, 0 * a, 0 * a), y = c(bo/2, bo/2, -bo/2, -bo/2, bo/2))) + geom_path(data = data.frame(x = c(0,
                                                                                                                                                                                                              a, a, 0 * a, 0 * a), y = c(0 + bi/2, 0 + bi/2, 0 - bi/2, 0 - bi/2, 0 + bi/2))) + annotate("text",
                                                                                                                                                                                                                                                                                                        x = 32, y = -5, label = "2") + ## TwoToThree: circle track
  geom_path(data = data.frame(x = c(-1 * a - (15700:(0)/1000), -1 * a), y = c((15.7 + bo) + sqrt(15.7^2 -
                                                                                                  c(15700:(0)/1000, 0)^2))), aes(x = x, y = y), colour = "Blue") + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                        a - (14590:(0)/1000), -1 * a), y = c((15.7 + bo) + sqrt((14.59)^2 - c(14590:(0)/1000, 0)^2))), aes(x = x,
                                                                                                                                                                                                                                                                                                          y = y)) + geom_path(data = data.frame(x = c(-1 * a - (16490:(0)/1000), -1 * a), y = c((15.7 + bo) +
                                                                                                                                                                                                                                                                                                                                                                                                  sqrt((16.49)^2 - c(16490:(0)/1000, 0)^2))), aes(x = x, y = y)) + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        a - (16810:(0)/1000), -1 * a), y = c((15.7 + bo) + sqrt((16.81)^2 - c(16810:(0)/1000, 0)^2))), aes(x = x,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          y = y)) + geom_path(data = data.frame(x = c(-1 * a - (14910:(0)/1000), -1 * a), y = c((15.7 + bo) +
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  sqrt((14.91)^2 - c(14910:(0)/1000, 0)^2))), aes(x = x, y = y)) + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        a - 14.59, -1 * a - 16.81), y = c(15.7 + bo, 15.7 + bo))) + annotate("text", x = -42, y = 18, label = "3") +
  ## straight track
  geom_path(data = data.frame(x = c(a, 2 * a), y = c(0, 0)), colour = "Blue") + geom_path(data = data.frame(x = c(a,
                                                                                                                  2 * a, 2 * a, a, a), y = c(bo/2, bo/2, -bo/2, -bo/2, bo/2))) + geom_path(data = data.frame(x = c(a,
                                                                                                                                                                                                                  2 * a, 2 * a, a, a), y = c(0 + bi/2, 0 + bi/2, 0 - bi/2, 0 - bi/2, 0 + bi/2))) + annotate("text",
                                                                                                                                                                                                                                                                                                            x = 63, y = -5, label = "3") + ## ThreeToFour: circle track
  geom_path(data = data.frame(x = c(-1 * a - (15700:(0)/1000), -1 * a), y = c((15.7 + bo) - sqrt((15.7)^2 -
                                                                                                  c(15700:(0)/1000, 0)^2))), aes(x = x, y = y), colour = "Blue") + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                        a - (14590:(0)/1000), -1 * a), y = c((15.7 + bo) - sqrt((14.59)^2 - c(14590:(0)/1000, 0)^2))), aes(x = x,
                                                                                                                                                                                                                                                                                                          y = y)) + geom_path(data = data.frame(x = c(-1 * a - (16490:(0)/1000), -1 * a), y = c((15.7 + bo) -
                                                                                                                                                                                                                                                                                                                                                                                                  sqrt((16.49)^2 - c(16490:(0)/1000, 0)^2))), aes(x = x, y = y)) + geom_path(data = data.frame(x = c(-1 *
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        a - (16810:(0)/1000), -1 * a), y = c((15.7 + bo) - sqrt((16.81)^2 - c(16810:(0)/1000, 0)^2))), aes(x = x,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          y = y)) + geom_path(data = data.frame(x = c(-1 * a - (14910:(0)/1000), -1 * a), y = c((15.7 + bo) -
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  sqrt((14.91)^2 - c(14910:(0)/1000, 0)^2))), aes(x = x, y = y)) + annotate("text", x = -31, y = 8,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            label = "4") + ## straight track
  geom_path(data = data.frame(x = c(2 * a, 3 * a), y = c(0, 0)), colour = "Blue") + geom_path(data = data.frame(x = c(2 *
                                                                                                                        a, 3 * a, 3 * a, 2 * a, 2 * a), y = c(bo/2, bo/2, -bo/2, -bo/2, bo/2))) + geom_path(data = data.frame(x = c(2 *
                                                                                                                                                                                                                                      a, 3 * a, 3 * a, 2 * a, 2 * a), y = c(0 + bi/2, 0 + bi/2, 0 - bi/2, 0 - bi/2, 0 + bi/2))) + geom_path(data = data.frame(x = c(c314 -
                                                                                                                                                                                                                                                                                                                                                                      a, c314 - a), y = c(-bo/2, bo/2))) + annotate("text", x = c314 - a, y = 6, label = "PI") + annotate("text",
                                                                                                                                                                                                                                                                                                                                                                                                                                                                          x = 3 * a, y = 6, label = "4") + ## fix aspect ratio to 1:1
  coord_fixed()


.
Ready.



We’re looking down at two 7/16in steel ball bearings. Pull down that string on the left. It opens upward a cabinet hinge penetrated by the highest of three pipe clamp u-bolts in the assembly (I reamed the original screw holes open slightly to fit). The string gives the hinge its ‘fixed’ bolt position as it also sits on the next u-bolt just below it.  The top u-bolt is rubber banded to a full 5/8x2in bolt and half a clamp (not visible here). As you can see, the bolt holds 4 large and 9 smaller washers that make up the top of an initial raceway.  Just below, the ramp openings are at 15cm above the table.



Gone.

Below you can see things are coming together, in some off center fashion. Speed trials are next. I’ll lay out two long straights and try to determine what my ramp’s vertical as well as horizontal components should be before I make any final track cuts.  As I show with the R code plot, the complete layout for my diameter (31.4cm) should fit on the tabletop (please forgive the corner extension) with just a couple of centimeters to spare.



We should note that the balls’ paths over the small washer raceway involves curved motion – about 40deg - before a brief freefall. The ramps also force the balls through a 90deg change in direction to get to the tabletop, more curved motion. I’m not at all sure I can calculate the velocity changes complicated as they are by the compound velocities of gravitational acceleration, though I believe Steven O suggested I try. I suspect he used straight inclined ramps with a small final angle change – about 15cm high, ok, but how long? Of course it doesn’t matter what causes the initial velocity since we are only interested in the motion through the marked tracks. Using the quick pair of tracks here (held down with the weight of two wrenches), the steel balls exiting the tubes travel upward to reach a peak suggesting, as we might expect, pi/4 the height of the ready position.



//////////////////////////////////////////////////////////////////////////////////////

Pi = 4 R Code Update.

Thanks for helping. I’ve cleaned, white-spaced and added a variable list to the code. I don’t know how to program with the geom_path function yet, so that must wait. I see that I’ve made short and long variable names, or more, for the same object - the long names are mainly for the reader. The formatR product above is just plain awful; though how would I know?

Code:



library(ggplot2)

################################################
##                                            
##                     Pi=4                  
##                                            
################################################
##                                            
##             Intended to recreate          
##              Steven Oostdijk's            
##       Pi=4 experiment as described in              
##       http://milesmathis.com/pi7.pdf      
## ------------------------------------------
##                  Posted at                
##       http://milesmathis.forumotion.com/    
## ------------------------------------------
##          Contributing to a future          
##           Miles Mathis R project          
##                                            
################################################
##         Input data used:                  
##         1) The unit circle's diameter      
##         2) InsideTrackWidth                
##         3) OutsideTrackWidth              
##         4) Table length and width          
## ------------------------------------------
##          Replace with your data to        
##             plot your own tracks          
## ------------------------------------------
##             measurements in cm            
##           unless noted otherwise          
################################################
##
## Variables Used:
##
## a, a_diameter, a_diaInInches - The unit diameter
## Angle, AngInDeg - Table rotation aligning horiz straight
##                   track to the table diagonal
## bi, b_trackWidthIn - Track inside diameter
## bo, b_trackWidthOut - Track outside diameter
## c314, c_circumference - Equals 3.14*a
## r - Equals a_diameter/2
## rIn - Equals r - b_trackWidthOut/2
## rIn2 - Equals r - b_trackWidthIn/2
## rOut - Equals r + b_trackWidthIn/2
## rOut2 - Equals r + b_trackWidthOut/2
## straightEst - Equals 6*a, longest straight length
## T1 - Table width
## T2 - Table length
## xCoordsInit - Table x coordinates
## yCoordsInit - Table y coordinates
## MyTable - Matrix of table coordinates
## ZRotation - Matrix used to calculate rotation coordinates
## TR - Matrix multiplication product of MyTable and ZRotation
## xCoordsFinal - Rotated table x coordinates
## yCoordsFinal - Rotated table y coordinates
##
################################################
##
##  The tracklengths are scaled  
##     to the working diameter
##     CHANGE a_diameter HERE

a_diameter <- 31.4
a_diameter
##[1] 31.4

## The varible "a" is used in in place of "a_diameter" in the
## geom_path plot commands in order to reduce the plotting code.
a <- a_diameter
a
## [1] 31.4

## show "a_diameter" and "a" in inches.
a_diaInInches <- a_diameter/2.54
a_diaInInches
## [1] 12.3622


#############################################
##
##  CHANGE INSIDE AND OUTSIDE TRACK DIAMETERS HERE
##

## b_trackWidthIn is the pvc inside diameter.
## My pvc inside diameter is 5/8", converting to centimeters
b_trackWidthIn <- 2.54*(5/8)
b_trackWidthIn
## [1] 1.5875

## "bi" equals "b_trackWidthIn", to simplify the geom_path commands.
bi <- b_trackWidthIn
bi
## [1] 1.5875

## b_trackWidthOut is the pvc Outside diameter. Enter and convert 7/8
## inches into centimeters.
b_trackWidthOut <- 2.54*(7/8)
b_trackWidthOut
## [1] 2.2225

## "bo" equals "b_trackWidthOut", again, to simplify the geom_path commands.
bo <- b_trackWidthOut
bo
## [1] 1.5875


#############################################
##
##  Calculate the pi=3.14 circumference for the working diameter

## The circle track length, 0-4, is the circular
## track's centerline circumference
c_circumference <- 3.14159*a_diameter
c_circumference
## [1] 98.64593

## "c314" equals "c_circumference". Used in the geom_path commands.
c314 <- c_circumference
c314
## [1] 98.64593


#############################################
##
##  Calculate the the five radii used to plot
##  each quadrant of the circular track.

## "r" is the centerline radius, half our working diameter
r <- a_diameter/2
r
## [1] 15.7

rIn <- r - b_trackWidthOut/2
rIn
## [1] 14.58875

rIn2 <- r - b_trackWidthIn/2
rIn2
##[1] 14.90625

rOut <- r + b_trackWidthIn/2
rOut
## [1] 16.81125

rOut2 <- r + b_trackWidthOut/2
rOut2
## [1] 16.81125


#############################################
##
##  CHANGE TABLE DIMENSIONS (T1 and T2) HERE

## My workspace limit, table dimensions, is 29.5X71.25in.
## Convert to cm.
T1 <- 29.5*2.54
T1
## [1] 74.93

T2 <- 71.25*2.54
T2
## [1] 180.975

## The table coordinates:
xCoordsInit <- c(-T2/2,-T2/2,T2/2,T2/2,-T2/2)
yCoordsInit <- c(-T1/2,T1/2,T1/2,-T1/2,-T1/2)


#############################################
##
##  Calculate the rough (unknown ramp) overall length estimate
##
## The total straight length is found by adding the following sections
## (along with their corresponding "a" lengths): 1) ramp(a_diameter);
## 2)initNegOneToZero(a_diameter); and 3) track length (4*a_diameter).
##  
## A total rough estimate plot length of 6*a_diameter
straightEst <- 6*a_diameter
straightEst
## [1] 188.4


#############################################
## First reality check.
## With a_diameter=31.4, straightEst = 188.4 > T2 = 180.975.
## I've exceeded my table length limit by a few centimeters.
## I can gain some centimeters by aligning the straight
## track to the table's diagonal.
##
## Calculate table rotation

## Table coordinates as a matrix
MyTable <- cbind(xCoordsInit,yCoordsInit)
MyTable
##       xCoordsInit yCoordsInit
##[1,]    -90.4875     -37.465
##[2,]    -90.4875      37.465
##[3,]     90.4875      37.465
##[4,]     90.4875     -37.465
##[5,]    -90.4875     -37.465

## Rotation angle equals neg arcTan((T1/2)/(T2/2))
## = neg arcTan(37.465/90.4875). Angle is in radians.
Angle <- -atan(T1/T2)
Angle
## [1] -0.3925467

## Show the angle in degrees
AngInDeg <- Angle*360/(2*3.14159)
AngInDeg
## [1] -22.49129

## Build a z-rotation matrix from the rotation identities:
## x'<- x*cos(theta)+ysin(theta)
## y'<- x*(-1)sin(theta)+ycos(theta)
ZRotation <- matrix(c(cos(Angle),sin(Angle),(-1)*sin(Angle),cos(Angle)),nrow = 2,ncol = 2,byrow = TRUE)
ZRotation
##        [,1]       [,2]
## [1,] 0.9239378 -0.3825427
## [2,] 0.3825427  0.9239378

## Matrix multiply MyTable and ZRotation to find the
## rotated table coordinates "TR".
TR <- MyTable%*%ZRotation
TR
##        [,1]      [,2]
##[1,] -97.93679   0.00000
##[2,] -69.27286  69.23066
##[3,]  97.93679   0.00000
##[4,]  69.27286 -69.23066
##[5,] -97.93679   0.00000

## Here I'll break TR into separate x and y coordinate strings  
## (initial and final) for the ggplot geom_path function.
xCoordsFinal <- c(-97.93678513/2,-69.27286231/2,97.93678513/2,69.27286231/2,-97.93678513/2)
yCoordsFinal <- c(-0,69.23066104,0,-69.23066104,-0)

## The ramp's horiz distance must be under 31.4cm to
## gain any more cm's. It's looking ugly for Oogly.


#############################################
##
## Plot tracks, table and rotated table

## CHANGE TITLE HERE
ggplot( data=data.frame( x=1, y=1 ),aes( x,y ) ) +
  ggtitle( "Pi=4 Track Layout on My Almost Too Small Table\nEach unit length here is 31.4cm (or 12.4in)" ) +

  ## Tabletop (shown in red):
  geom_path( data=data.frame( x=c( -T2/2, -T2/2, T2/2, T2/2, -T2/2), y=c( -T1/2, T1/2, T1/2, -T1/2, -T1/2 ) ), colour = "Red" ) +
  ## Rotated tabletop:
  geom_path( data=data.frame( x=c( TR[1,1], TR[2,1], TR[3,1], TR[4,1], TR[5,1]),y=c(TR[1,2], TR[2,2], TR[3,2], TR[4,2], TR[5,2] ) ) ) +
  
  ## Plot both tracks in six sections each:
  ## 1)Ramp; 2)NegOneToZero; 3)ZeroToOne; 4)OneToTwo; 5)TwoToThree: and 4)ThreeToFour.
  ## Track Centerlines are blue.
  ## Text annotations are added in their appropriate sections.

  ## ramps:
  ## circle track
  geom_path( data=data.frame( x = c( -3*a, -2*a ), y = c( bo,bo ) ), colour = "Blue" ) +
  geom_path( data=data.frame( x = c( -3*a, -2*a, -2*a, -3*a, -3*a ), y = c( bo/2, bo/2, 1.5*bo, 1.5*bo, bo/2) ) ) +
  geom_path( data=data.frame( x = c( -3*a, -2*a, -2*a,-3*a,-3*a ), y = c( bo + bi/2, bo + bi/2, bo - bi/2, bo - bi/2, bo + bi/2) ) ) +
  ## straight track  
  geom_path( data=data.frame( x = c( -3*a, -2*a ), y = c( 0,0 ) ), colour = "Blue" )+
  geom_path( data=data.frame( x = c( -3*a, -2*a, -2*a, -3*a, -3*a ), y = c( bo/2, bo/2, - bo/2, - bo/2, bo/2 ) ) )+
  geom_path( data=data.frame( x = c( -3*a, -2*a, -2*a, -3*a, -3*a ), y = c( bi/2, bi/2, - bi/2, - bi/2, bi/2 ) ) ) +
  annotate( "text", x = -2.5*a, y = 8, label = "Ramps" ) +
  annotate( "text", x = -63, y = -5, label = "-1" ) +

  ## NegOneToZero:
  ## circle track
  geom_path( data=data.frame( x = c( -2*a, -a ) , y = c( bo, bo ) ), colour = "Blue" )+
  geom_path( data=data.frame( x = c( -2*a, -a, -a, -2*a, -2*a ), y = c( bo/2, bo/2, 1.5 * bo, 1.5 * bo, bo/2 ) ) ) +
  geom_path( data=data.frame( x = c( -2*a, -a, -a, -2*a, -2*a), y = c( bo + bi/2, bo + bi/2, bo - bi/2, bo - bi/2, bo + bi/2 ) ) ) +
  ## straight track  
  geom_path( data=data.frame( x = c( -2*a, -a ), y = c( 0, 0 ) ), colour = "Blue" ) +
  geom_path( data=data.frame( x = c( -2*a, -a, -a, -2*a, -2*a), y = c( bo/2, bo/2, -bo/2, -bo/2, bo/2 ) ) ) +
  geom_path( data=data.frame( x = c( -2*a, -a, -a, -2*a, -2*a), y = c( bi/2, bi/2, -bi/2, -bi/2, bi/2 ) ) ) +
  annotate( "text", x = -31, y = -5, label = "0") +

  ## ZeroToOne:
  ## circle track
  geom_path( data=data.frame( x = c( - a + ( 15700:(0)/1000 ), - a), y = c( 15.7 + bo - sqrt( ( 15.7)^2 - c( 15700:(0)/1000,0 )^2 ) ) ), aes( x =x, y = y ), colour = "Blue" )+
  geom_path( data=data.frame( x = c( - a + ( 14590:(0)/1000 ), - a), y = c( 15.7 + bo - sqrt( ( 14.59)^2 - c( 14590:(0)/1000,0 )^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a + ( 16490:(0)/1000 ), - a), y = c( 15.7 + bo - sqrt( ( 16.49)^2 - c( 16490:(0)/1000,0 )^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a + ( 16810:(0)/1000 ), - a), y = c( 15.7 + bo - sqrt( ( 16.81)^2 - c( 16810:(0)/1000,0 )^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a + ( 14910:(0)/1000 ), - a), y = c( 15.7 + bo - sqrt( ( 14.91)^2 - c( 14910:(0)/1000,0 )^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a + 14.59, - a + 16.81 ), y = c( 15.7 + bo, 15.7 + bo ) ) ) +
  annotate( "text", x = -21, y = 18, label = "1" ) +
  ## straight track  
  geom_path( data=data.frame( x = c( -a, 0 ), y = c( 0, 0 ) ), colour = "Blue" ) +
  geom_path( data=data.frame( x = c( -a, 0, 0, -a, -a ), y = c( bo/2, bo/2, -bo/2, -bo/2, bo/2 ) ) ) +
  geom_path( data=data.frame( x = c( -a, 0, 0, -a, -a ), y = c( bi/2, bi/2, -bi/2, -bi/2, bi/2 ) ) ) +
  annotate( "text", x = 0, y = -5, label = "1") +

  ## OneToTwo:
  ## circle track
  geom_path( data=data.frame( x = c( - a + ( 15700:(0)/1000 ), - a ), y = c( 15.7 + bo + sqrt( (15.7)^2 - c( 15700:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ), colour = "Blue" ) +
  geom_path( data=data.frame( x = c( - a + ( 14590:(0)/1000 ), - a ), y = c( 15.7 + bo + sqrt( (14.59)^2 - c( 14590:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a + ( 16490:(0)/1000 ), - a ), y = c( 15.7 + bo + sqrt( (16.49)^2 - c( 16490:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a + ( 16810:(0)/1000 ), - a ), y = c( 15.7 + bo + sqrt( (16.81)^2 - c( 16810:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a + ( 14910:(0)/1000 ), - a ), y = c( 15.7 + bo + sqrt( (14.91)^2 - c( 14910:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a, -a ), y = c( 31.4 + bo/2, 2*16.81 + bo/2 ) ) ) +
  annotate( "text", x = -31, y = 29, label = "2" ) +
  ## straight track  
  geom_path( data=data.frame( x = c( 0, a ), y = c( 0, 0 ) ), colour = "Blue" ) +
  geom_path( data=data.frame( x = c( 0, a, a, 0, 0 ), y = c( bo/2, bo/2, -bo/2, -bo/2, bo/2 ) ) ) +
  geom_path( data=data.frame( x = c( 0, a, a, 0, 0 ), y = c( bi/2, bi/2, -bi/2, -bi/2, bi/2 ) ) ) +
  annotate( "text", x = 32, y = -5, label = "2" ) +

  ## TwoToThree:
  ## circle track
  geom_path( data=data.frame( x = c( - a - ( 15700:(0)/1000 ), - a ), y = c( 15.7 + bo + sqrt( 15.7^2 - c( 15700:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ), colour = "Blue" ) +
  geom_path( data=data.frame( x = c( - a - ( 14590:(0)/1000 ), - a ), y = c( 15.7 + bo + sqrt( 14.59^2 - c( 14590:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a - ( 16490:(0)/1000 ), - a ), y = c( 15.7 + bo + sqrt( 16.49^2 - c( 16490:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a - ( 16810:(0)/1000 ), - a ), y = c( 15.7 + bo + sqrt( 16.81^2 - c( 16810:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a - ( 14910:(0)/1000 ), - a ), y = c( 15.7 + bo + sqrt( 14.91^2 -c( 14910:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a - 14.59, - a - 16.81), y = c( 15.7 + bo, 15.7 + bo ) ) ) +
  annotate( "text", x = -42, y = 18, label = "3" ) +
  ## straight track  
  geom_path( data=data.frame( x = c( a, 2*a ), y = c( 0, 0 ) ), colour = "Blue" ) +
  geom_path( data=data.frame( x = c( a, 2*a, 2*a, a, a ), y = c( bo/2, bo/2, -bo/2, -bo/2, bo/2 ) ) ) +
  geom_path( data=data.frame( x = c( a, 2*a, 2*a, a, a ), y = c( bi/2, bi/2, -bi/2, -bi/2, bi/2 ) ) ) +
  annotate( "text", x = 63, y = -5, label = "3" ) +

  ## ThreeToFour:
  ## circle track
  geom_path( data=data.frame( x = c( - a - ( 15700:(0)/1000 ), - a ), y = c( 15.7 + bo - sqrt( 15.7^2 -c( 15700:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ), colour = "Blue" ) +
  geom_path( data=data.frame( x = c( - a - ( 14590:(0)/1000 ), - a ), y = c( 15.7 + bo - sqrt( 14.59^2 -c( 14590:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a - ( 16490:(0)/1000 ), - a ), y = c( 15.7 + bo - sqrt( 16.49^2 -c( 16490:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a - ( 16810:(0)/1000 ), - a ), y = c( 15.7 + bo - sqrt( 16.81^2 -c( 16810:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  geom_path( data=data.frame( x = c( - a - ( 14910:(0)/1000 ), - a ), y = c( 15.7 + bo - sqrt( 14.91^2 -c( 14910:(0)/1000,0)^2 ) ) ), aes( x = x, y = y ) ) +
  annotate( "text", x = -31, y = 8, label = "4" ) +
  ## straight track  
  geom_path( data=data.frame( x = c( 2*a, 3*a ),y = c( 0, 0 ) ), colour = "Blue" ) +
  geom_path( data=data.frame( x = c( 2*a, 3*a, 3*a, 2*a, 2*a ), y = c( bo/2, bo/2, -bo/2, -bo/2, bo/2 ) ) ) +
  geom_path( data=data.frame( x = c( 2*a, 3*a, 3*a, 2*a, 2*a ), y = c( bi/2, bi/2, -bi/2, -bi/2, bi/2 ) ) ) +
  geom_path( data=data.frame( x = c( c314 - a, c314 - a ), y = c( -bo/2, bo/2 ) ) ) +
  annotate( "text", x = c314-a, y = 6, label = "PI" ) +
  annotate( "text", x = 3*a, y = 6, label = "4" ) +

  ## fix aspect ratio to 1:1
  coord_fixed()




.

.


Pvc tubes held by pvc tape onto a pvc table. *

Milestone update. My first complete track layout, marked with 31.4cm diameter units on both the table (taped surface) as well as the track tubes. The video - not ready for prime time - shows a 10cm out-of-synch difference reaching the 4 mark, with the circle track ball winning.

I removed the table’s circular track drawing and marks before this photo because of the unmatched confusion with the actual track. My ramp is as long as my current setup allows, placing my tube ends 21cm above the table. As such the ramps are too snug, and both tracks as well, having pushed past their table marks overnight despite plenty of tape. I’ll shorten the ramps another cm or so.



Here we see an actual working diameter - in the way. This circle track is too tear shaped to be acceptable – the tubes are most rigid at curvature ends. The zero and four marks cannot meet face-to-face, and so an accurate circle is physically impossible. Next time I’ll lose the container and let the circular track overlap (spiral in).

I’ve been very cautious, gently working suspended tracks with ropes and rags to straighten and mark them. Any unnecessary changes in direction reduces rolling efficiency. Taped (or fixed) track beside untaped positions can form areas of tension and compression resulting in unanticipated bends.



Many unanticipated bends. I must improve my methodology. The circle is trying to lift off the table. Here we can see the straight track 4 mark being pulled up off the table surface while the track tries to snake behind it.

Switching to the more flexible, completely transparent pvc is easily done at this point. I was disappointed when I saw the transparent pvc deformed and even folded, over-packed onto spools at alternate local hardware stores, while the adjacent braided pvc tubes clearly kept their circular cross sections. The stronger braided pvc allow us to view the steel balls positions’, though at some loss in visibility. The braids can serve as surface coordinates - nice perpendicular and parallel guidelines.

Still in progress.

*Polyvinyl chloride

.
Plenty of practice, but no joy.

My straight track is slightly slower than the circular. Both starting velocities are about 137 cm/s. Using braided pvc, the ball bearings aren’t always visible even in daylight. I’ve also cut a set of clear pvc tracks which I’ve hung up to straighten before final marking and cuts.


.

Are you sure it's a circular path? It seems slightly oval to me. Is there a way to check and correct it?

.

Ciaolo wrote:Are you sure it's a circular path? It seems slightly oval to me. Is there a way to check and correct it?

Hi Ciaolo, Thanks for the observation. In my 22Dec post above I include this comment -

This circle track is too tear shaped to be acceptable – the tubes are most rigid at curvature ends. The zero and four marks cannot meet face-to-face, and so an accurate circle is physically impossible. Next time I’ll lose the container and let the circular track overlap (spiral in).

In my latest image you can see a container, my working diameter, in the bottom left corner positioned to catch balls leaving the straight track. I actually use the container as working diameter/circumference to secure my circular track's 0,1, 2 and 3 marks to the table. I allow the circle's end, the four mark, to lift off the table, above the zero mark. I then remove the container and secure the 4 mark onto the table's surface as close to the zero mark as the track allows, thereby 'spiraling-in' to the 4 mark; the circular track is no longer a circle. You may have noticed that Steven's circular track is also slightly deformed. How does one correct for that?

Curved motion works with spirals and ovals as well as with circles. I could make the entire circular track a spiral. I would plot the spiral image onto the tabletop and lay the track onto it, but then I wouldn't be able to use my 'working diameter' container.

Any ideas?   
.

Actually I have an idea: use half a circumference instead of the whole one. This not only removes the problems in creating a perfect shape, but also allows you to measure the final speed.

Cheers

.
Ciaolo, I like your idea. I look forward to it.

It'll just take me a while as I've committed myself to recreating Steven's demonstration first. I hope I'm not testing everyone's patience.
.

LongtimeAirman wrote:.
Plenty of practice, but no joy.

My straight track is slightly slower than the circular. Both starting velocities are about 137 cm/s. Using braided pvc, the ball bearings aren’t always visible even in daylight. I’ve also cut a set of clear pvc tracks which I’ve hung up to straighten before final marking and cuts.
.

Is the straight tube track pvc too corrugated or uneven which would then impact the ball enough to slow it down slightly? Basically, is it a non-flat rolling surface inside the pvc tube.

.
Cr6, Correct. The problem is an uneven internal surface within the straight track caused by the coiling tendency.

I've purchased pvc conduit in coils of under a foot diameter, 10 feet long. Pvc sold by the foot off of 100 foot coils is not as pretty in maintaining a circular cross section. The tracks must be straightened sufficiently so that they can be properly measured. The exception - braided pvc allows someone to cut accurate lengths without straightening by using the braids as a measure.

Ok. Two tracks cut to length. One naturally lays the track beginning with the ramp, curved side downward - i.e. the track would coil in the vertical plane. With the circular track one might include a 90deg twist between the ramp and the track in order to ensure that the circular track's coil tendency would remain horizontal on the table surface. The circular track is therefore able to maintain a constant table pressure and cross section.

Laying the straight track is the problem. After the ramp, it also wants to coil back into the vertical plane. As one secures points along the track, the linear areas between the secured points do not maintain a constant pressure with the table. The ball's path, the bottom of the pvc's internal cross section varies between too flat (and wide) and narrow (tall). The ball in the straight track thus has both vertical and horizontal changes that slow it down. The straight track should, eventually, lose its coil, but that may take weeks or months.

It may be easier to lay the straight track such that the coil would remain in the horizontal plane. I believe I would also need a wall board alongside the straight path to secure it too and maintain a straight path direction.

That’s the way it looks to me. The clear pvc is softer and may not spring back into the coil so easily. I should find out.
.

LongtimeAirman wrote:.
Ciaolo, I like your idea. I look forward to it.

It'll just take me a while as I've committed myself to recreating Steven's demonstration first. I hope I'm not testing everyone's patience.

You're a scientist. Testing things is what scientists do.