Physical π : Pi's relationship to 4
Re: Physical π : Pi's relationship to 4
Let me add something which may make it easier to visualize the essence. It is in the same video as mentioned, https://www.youtube.com/watch?v=CwZBX2XcAmđ, starting at 11:47 . Every circular motion has 2 separate components as can be observed and while the point travels around the circumference, it can be clearly observed that in the same time point travels 1 full rotation, on the Xaxis it covers 4R of distance and on the Yaxis it covers another 4R of distance. 4R+4R=8R

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Re: Physical π : Pi's relationship to 4
Sorry, Vexman  but you are missing the whole point of what that video tries to explain  with regards to how our eyes can (wrongly) perceive / interpret reality. The example presented (of an orbiting object, viewed from a 90° angle, moving up and down) shows precisely why someone may falsely conclude that the object travels a distance of 4R (up / down / and back up again)  whereas in fact the object completes a circular path / distance (approx 21% shorter than 8R).VexMan » October 7th, 2016, 9:09 am wrote:Let me add something which may make it easier to visualize the essence. It is in the same video as mentioned, https://www.youtube.com/watch?v=CwZBX2XcAmđ, starting at 11:47 . Every circular motion has 2 separate components as can be observed and while the point travels around the circumference, it can be clearly observed that in the same time point travels 1 full rotation, on the Xaxis it covers 4R of distance and on the Yaxis it covers another 4R of distance. 4R+4R=8R
To be sure, if JOE (red dot in my below diagram) wishes to walk from a to c, he will get to c approx 21% earlier (if he walks along the curved green path) than if he chooses to walk from a to b to c:
I hope this helps.
Re: Physical π : Pi's relationship to 4
You say: "To be sure, if JOE (red dot in my below diagram) wishes to walk from A to C, he will get to C approx 21% earlier (if he walks along the curved green path) than if he chooses to walk from A to B to C."
That is exactly what is refuted by cycloid , which is the actual distance traveled by the observed point while in motion. A point doesn't follow the green line, nor would a man actually travel that distance / part of circumference. Actually, you can't ignore one of the vectors (as i.e. point's Yaxis movement) and this is all I wanted to point with my reference to this video. I never discussed anything else, so it's overstretched to say I missed the point as my point was only what I just wrote again. Put in simpler form, although it looks as the point travels a "shortcut" which represents part of the circumference/your green line, it actually travels 21% more (to be precise, it's cca 21.5% => 3.14..../4),which means 1/4*2R*3.14 (equals 1.57*R) *versus what* the point actually travels 1R on the Xaxis + 1R on the Yaxis (equals 2R) . So it is 1.57R vs 2R , but here is the big "trick" > they both travel that distance in the same amount of time when we are observing a circle in motion.
*added by EDIT for more clarity
So according to that logic it is wrong to say that your Joe would arrive at the point C 21% earlier walking the circumference than if he took suggested ABC path. No, Joe would arrive there in the same amount of time. With important notion, that even though it looks like Joe is walking the circumference / green line when in motion, he's actually moving on a cycloid path. In other words, Joe is not walking the green line as he cannot separate his Xaxis movement from his Yaxis movement when he is in circular motion. That's the essence of it, he's walking cycloid line.
It would be easier to visualise, if you've added actual cycloid path that is traveled by the point/Joe in time while in motion.
That is exactly what is refuted by cycloid , which is the actual distance traveled by the observed point while in motion. A point doesn't follow the green line, nor would a man actually travel that distance / part of circumference. Actually, you can't ignore one of the vectors (as i.e. point's Yaxis movement) and this is all I wanted to point with my reference to this video. I never discussed anything else, so it's overstretched to say I missed the point as my point was only what I just wrote again. Put in simpler form, although it looks as the point travels a "shortcut" which represents part of the circumference/your green line, it actually travels 21% more (to be precise, it's cca 21.5% => 3.14..../4),which means 1/4*2R*3.14 (equals 1.57*R) *versus what* the point actually travels 1R on the Xaxis + 1R on the Yaxis (equals 2R) . So it is 1.57R vs 2R , but here is the big "trick" > they both travel that distance in the same amount of time when we are observing a circle in motion.
*added by EDIT for more clarity
So according to that logic it is wrong to say that your Joe would arrive at the point C 21% earlier walking the circumference than if he took suggested ABC path. No, Joe would arrive there in the same amount of time. With important notion, that even though it looks like Joe is walking the circumference / green line when in motion, he's actually moving on a cycloid path. In other words, Joe is not walking the green line as he cannot separate his Xaxis movement from his Yaxis movement when he is in circular motion. That's the essence of it, he's walking cycloid line.
It would be easier to visualise, if you've added actual cycloid path that is traveled by the point/Joe in time while in motion.
Last edited by VexMan on Fri Oct 07, 2016 11:04 am, edited 2 times in total.
Re: Physical π : Pi's relationship to 4
Well, MM's argument is that, no, they should arrive at the same time. The distance traveled as you walk from A to B to C is the same distance that you would travel along the curve from A to C. To be sure, if you walked a straight line from A to C (the hypotenuse of the triangle joining segments AB to BC), you would arrive earlier. But not if you walk the curve. And this gets very much to the heart of the matter:simonshack » October 7th, 2016, 12:11 pm wrote: To be sure, if JOE (red dot in my below diagram) wishes to walk from A to C, he will get to C approx 21% earlier (if he walks along the curved green path) than if he chooses to walk from A to B to C:
The consensus opinion now is that if you drew ever smaller triangles inside the larger one, the hypotenuses would shrink more and more. At the limit, they should converge to the length of the curve. But Miles is arguing that this is wrong. Because they are chords, they will never converge to the curve. And this is his main correction to Newton et al. The chords will always be shorter than the curve. Instead, it is the line segments AB and BC that will 'converge' to the curve at the limit (of getting smaller and smaller ad infinitum).
I was rereading his Manhattan metric paper, and there was a discussion there that I found helpful:
"I do understand why this paper on π has been so controversial. It not only conflicts with all
we have been taught, it conflicts with basic intuition and with our own eyeballs. Even after I did the
math and analysis, it was initially hard for me to accept. I am an artist and I rely on my eyes. I am
extremely visual. It simply doesn't seem right that these two figures should have the same
circumference [perimeter]:"
"I will be told that any idiot can see they don't have the same circumference. The circle has to have a
smaller circumference, since it encloses a smaller area. That is clear just from looking at the four
spaces in the corners, which are outside the circle but inside the square. I used to think this way, and it
is an intuitive way to think. However, it is mistaking circumference for area. Area is not a function of
circumference, although you would think it would be. I can show you this very easily."
"The red and blue lines have the same circumference, but very different areas. The area of red is ¾ the
area of blue. And we can make the area of B go down very quickly, keeping the circumferences the
same:"
"Red still has the same circumference as blue, but the area is approaching ½. This is precisely what is
happening with the circle, although it isn't what we are taught."
"Study the green line. By the same intuition that told us that the red line must have a smaller
circumference than the blue line, we should also expect the green line to have a smaller circumference
than the blue. But it doesn't! I don't have to prove that or take anything to a limit to show it. Green
has a much smaller area than blue, but exactly the same circumference.
"It is pretty obvious that if we increase the steps in the green line, we will approach the red line. As the
number of steps increases, the size of each step decreases, and by the logic of Newton, you could claim
that “it would ultimately vanish.” I don't like that wording and never have, since it isn't rigorously true.
The steps never “vanish,” they simply become negligible. “Without extreme magnification, they
vanish”: that would be a preferable wording."

If you're familiar with fractals, Simon, I think you'll see some similarities here. Like with the Mandelbrot set, where you can draw a figure with an infinite perimeter surrounding a finite area. That area can change, even as the perimeter remains infinite. In this case, as we redraw the boundary, the perimeter remains the same length even as the surface area changes. Since Miles argues that you can't get infinitely small in the real world (no infinitesimals for him!), he is not on board with fractal math as appropriate for describing real world objects, but you can hopefully see the analogy.

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Re: Physical π : Pi's relationship to 4
I've looked at this again, and realised I had been taken in by the labelling of the stills in the experiment. Where there are stills with captions added (about 4:30 onward), which are:
First still: Both balls at the '0' mark, (same speed calculated back to the previous mark)
Next still: The caption says 'Then circle ball is at 1 quarter' and 'When straight ball is at 1 diameter'.
Next still: 'Then circle ball is at 2 quarter' and 'When straight ball is at 2 diameter'
Next still: 'Then circle ball is at 3 quarter' and 'When straight ball is at 3 diameter'
Next still: 'Then circle ball is nowhere near the end' and 'When ball on straight path is at pi diameter'
Next still: 'When ball is at end of curved path' and 'Then ball on straight path is at 4'
Let's take the measurements as accurate and see how they compare just for the first quarter of travel.
Caption: 'Then circle ball is at 1 quarter' and 'When straight ball is at 1 diameter'.
One quarter of the circumference is (pi * diameter) / 4 which equals 0.79 x diameter.
1 diameter for the straight ball is 1 x diameter.
So, what the still with captions is saying is that at that point in time, the circle path ball had only travelled 79% of the distance that the straight path ball had travelled, i.e. 21% less.
Labelling the experiment so that the circle path ball has markers each quarter of a whole circle, and the straight path ball has markers at quarters of 4, doesn't really do anything other than imply some sort of equivalence of the balls being at their own markers at the same time. They are not equivalent at all. They are both bang on the first quarter markers, but that just verifies they are travelling at different speeds.
Whilst the Mathis paper claims the circle path ball does 'NOT' slow down, just comparing those first two stills tells us that the either:
a) The circle path ball in fact does slow down as a result of losing energy as the tube wall forces it to take the circular path.
Or….
b) It's travelling at the same speed, but somehow the fabric of spacetime has been warped by travelling in a circle.
To come back to a previous analogy  not the Wall of Death though, as you correctly pointed out, the downward force of gravity makes it not all that similar for comparison purposes  but the one where I am riding my motorcycle down a straight road with the throttle at a constant opening to maintain a given speed. When I turn into a bend, if I leave the throttle where it is, I will lose speed. If I wish to maintain constant speed then while turning I need to have the throttle open a little wider to do so. The conclusion from this is that to maintain that constant speed requires extra energy to be put in whilst cornering, and when I resume on the straight road afterwards, I will either speed up, or can maintain the constant speed but close the throttle a little bit.
If I take the motorcycle scenario and just keep the throttle constant all the way through, so we can remove friction from the interaction as I've already balanced it with the throttle, I will go into the bend and start losing speed throughout. This is exactly what the circle path ball in the experiment is doing.
Sorry to keep banging on, but I'm trying to make sense of some proposed measurements, as I firmly believe the ball will lose energy by being forced into a nonstraight path, and that is why it takes longer to cover the same distance as the straight path ball.
I've no doubt there is some whizzer fun to be had debating whether Pi should equal 4 for the purposes of calculating all sorts of stuff, but I don't think anything happening in this experiment contradicts the traditional geometric notion of pi being the ratio circumference/diameter. Literally the only thing in Mathis' paper and the video experiment that suggests otherwise is the claim that the circle path ball doesn't ('does NOT') slow down.
I wonder what method will be used to measure speed at the end of the next experiment? Presumably the circle should exit into a short straight section and have speed measured from start to finish of that straight, but let's wait and see.
At the moment, the only reason I can see for Mathis claiming (so vociferously) that the ball doesn't slow down, is to drag people into an endless rabbit hole of confusing abstractions where he can claim to be enlightening and groundbreaking.
Your mileage may vary.
First still: Both balls at the '0' mark, (same speed calculated back to the previous mark)
Next still: The caption says 'Then circle ball is at 1 quarter' and 'When straight ball is at 1 diameter'.
Next still: 'Then circle ball is at 2 quarter' and 'When straight ball is at 2 diameter'
Next still: 'Then circle ball is at 3 quarter' and 'When straight ball is at 3 diameter'
Next still: 'Then circle ball is nowhere near the end' and 'When ball on straight path is at pi diameter'
Next still: 'When ball is at end of curved path' and 'Then ball on straight path is at 4'
Let's take the measurements as accurate and see how they compare just for the first quarter of travel.
Caption: 'Then circle ball is at 1 quarter' and 'When straight ball is at 1 diameter'.
One quarter of the circumference is (pi * diameter) / 4 which equals 0.79 x diameter.
1 diameter for the straight ball is 1 x diameter.
So, what the still with captions is saying is that at that point in time, the circle path ball had only travelled 79% of the distance that the straight path ball had travelled, i.e. 21% less.
Labelling the experiment so that the circle path ball has markers each quarter of a whole circle, and the straight path ball has markers at quarters of 4, doesn't really do anything other than imply some sort of equivalence of the balls being at their own markers at the same time. They are not equivalent at all. They are both bang on the first quarter markers, but that just verifies they are travelling at different speeds.
Whilst the Mathis paper claims the circle path ball does 'NOT' slow down, just comparing those first two stills tells us that the either:
a) The circle path ball in fact does slow down as a result of losing energy as the tube wall forces it to take the circular path.
Or….
b) It's travelling at the same speed, but somehow the fabric of spacetime has been warped by travelling in a circle.
To come back to a previous analogy  not the Wall of Death though, as you correctly pointed out, the downward force of gravity makes it not all that similar for comparison purposes  but the one where I am riding my motorcycle down a straight road with the throttle at a constant opening to maintain a given speed. When I turn into a bend, if I leave the throttle where it is, I will lose speed. If I wish to maintain constant speed then while turning I need to have the throttle open a little wider to do so. The conclusion from this is that to maintain that constant speed requires extra energy to be put in whilst cornering, and when I resume on the straight road afterwards, I will either speed up, or can maintain the constant speed but close the throttle a little bit.
If I take the motorcycle scenario and just keep the throttle constant all the way through, so we can remove friction from the interaction as I've already balanced it with the throttle, I will go into the bend and start losing speed throughout. This is exactly what the circle path ball in the experiment is doing.
Sorry to keep banging on, but I'm trying to make sense of some proposed measurements, as I firmly believe the ball will lose energy by being forced into a nonstraight path, and that is why it takes longer to cover the same distance as the straight path ball.
I've no doubt there is some whizzer fun to be had debating whether Pi should equal 4 for the purposes of calculating all sorts of stuff, but I don't think anything happening in this experiment contradicts the traditional geometric notion of pi being the ratio circumference/diameter. Literally the only thing in Mathis' paper and the video experiment that suggests otherwise is the claim that the circle path ball doesn't ('does NOT') slow down.
I wonder what method will be used to measure speed at the end of the next experiment? Presumably the circle should exit into a short straight section and have speed measured from start to finish of that straight, but let's wait and see.
At the moment, the only reason I can see for Mathis claiming (so vociferously) that the ball doesn't slow down, is to drag people into an endless rabbit hole of confusing abstractions where he can claim to be enlightening and groundbreaking.
Your mileage may vary.

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Re: Physical π : Pi's relationship to 4
*
Well, I guess I'm just a bit dense  and will probably need another lifetime (or a brain transplant) in order to wrap my head around this Pi=4 thing...
Meanwhile, I would warmly appreciate if someone can explain  in simple words  why I can place 21 billiard balls between a > b > and c ...
...and only 17 billiard balls between a and c : [please excuse me for the rather crude graphics]
Thanks. I'll buy you a large beer if you'll put me out of my rather painful, current headache!
Well, I guess I'm just a bit dense  and will probably need another lifetime (or a brain transplant) in order to wrap my head around this Pi=4 thing...
Meanwhile, I would warmly appreciate if someone can explain  in simple words  why I can place 21 billiard balls between a > b > and c ...
...and only 17 billiard balls between a and c : [please excuse me for the rather crude graphics]
Thanks. I'll buy you a large beer if you'll put me out of my rather painful, current headache!

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Re: Physical π : Pi's relationship to 4
Dear bongostaple,bongostaple wrote:When I turn into a bend, if I leave the throttle where it is, I will lose speed. If I wish to maintain constant speed then while turning I need to have the throttle open a little wider to do so.
I think it should be also pointed out that the reason why your motorcycle will lose speed when turning  is due to your motorcycle (and its rider) becoming heavier due to the Gforces involved. Of course, the suddenly heavier vehicle will need more power to maintain its original straight line speed. Now, another thing is that motorcycles going around bends HAVE to slow down far more than, say, Formula 1 cars (for obvious reasons related to 'grip' and sheer balance). The 4wheeled, groundeffected Formula 1 cars can, on occasions, remain at full throttle (around long, notsotight bends)  yet the driver will always see his engine's RPMmeter losing / dropping down by a few revsperminute. This, because his vehicle becomes heavier than it is on a straight line.
Now, in the Oostdijk experiment, we do NOT have a selfpropelled object. We have a nonselfpropelled ballbearing which is 'coasting', as per dictionary definition:
Therefore, it is to be fully expected that ANY force counteracting the ballbearing's initial momentum & rectilineal direction will make it slow down. Needless to say, if the "Xforce propelled" ballbearing encountered a totally straight wall (of "force Y") it would immediately stop moving (and bounce back a little). Instead, in the Oostdijk experiment, it encounters a curved "force Y", which reaches its peak (being diametrically opposed to inertial "force X") as indicated in my below graphic:to coast
a: "To slide down an incline through the effect of gravity."
b: "To move without use of propelling power."
http://www.thefreedictionary.com/coast
Moreover, some degree of friction / slip (i.e. loss of speed) should be expected when the ballbearing meets the curve  as its initial rotational motion / direction will have to gradually 'learn' to rotate in increasingly different / opposed directions.
Re: Physical π : Pi's relationship to 4
Simon, I just got confused with your question....
Why or what is it with the billiard balls that is in question? The only thing that is out of order with billiard balls (BB) is that the ratio between them isn't proper. I guess due to less accurate positioning you did. If in the first picture there are precisely 21 BB, then in second picture there should be 21*0.785=16.485 BB. If in the second picture there are precisely 17 BB, then in the first picture there should be 17*1.215=20.655 . I believe the correct number of BB in both pictures is more probably approximately 21 and approximately 17 accordingly.
Other than that I can't find anything that is to be explained... Am I missing something?
Why or what is it with the billiard balls that is in question? The only thing that is out of order with billiard balls (BB) is that the ratio between them isn't proper. I guess due to less accurate positioning you did. If in the first picture there are precisely 21 BB, then in second picture there should be 21*0.785=16.485 BB. If in the second picture there are precisely 17 BB, then in the first picture there should be 17*1.215=20.655 . I believe the correct number of BB in both pictures is more probably approximately 21 and approximately 17 accordingly.
Other than that I can't find anything that is to be explained... Am I missing something?

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Re: Physical π : Pi's relationship to 4
That's some pretty heavy nitpicking, Vexman. Good grief...
Re: Physical π : Pi's relationship to 4
I don't get it, honestly... you pointed to your picture with billiard balls, 21 vs 17 . If you by any chance see this a nitpicking, that was not my intention in any possible way.
If I can I most certainly would take time and patience to try and help anybody. So, can I kindly ask you to rephrase your question and point for me exclusively? I'd like to be thought of as somebody else rather than a nit picking guy just because I honestly missed your question's point.
If I can I most certainly would take time and patience to try and help anybody. So, can I kindly ask you to rephrase your question and point for me exclusively? I'd like to be thought of as somebody else rather than a nit picking guy just because I honestly missed your question's point.
Re: Physical π : Pi's relationship to 4
Pi = 4, a common high school math problem. This post comes to you from a common high school math teacher.
The video at the end of this post shows the common mistakes with the idea that Pi could equal four. Vi Hart, the narrator, does a pretty good job of explaining how Pi ≠ 4 (Pi cannot = 4). This type of problem is a common calculus findtheerror type problem. I’m rather shocked Miles Mathis went on about it so. Miles Mathis claims to be knowledgeable in higher level mathematics but I see the opposite. He is spewing a lot of made up mumbo jumbo. I am confident in my formal and informal math training and math abilities to call Miles Mathis a math fraud. It's like Vortex Math all over again. Repeating nonsense does not make it true. Coming up with pretty patterns does not prove a thing. These fakers make up their own math rules that cannot be manifested in the real world. Good math, however, is manifested most magnificently as seen in the construction of giant monuments, math has allowed industry to become this enormous monster and math has given me the power to post here on CluesForum. Real math is very powerful if one knows how to exploit it.
What else is being exploited in the Pi=4 discussion? Lack of knowledge in mathematics. Vi Hart talks about how cutting corners repeatedly, while going from a square to a circle, does not change the length of the line. Length stays the same.
The Big Point: Removing corners means the line gets more zigzaggy, not smooth. A zigzaggy line does not equal a smooth line.
The result is an infinite zigzaggy line with the same length of four. If one stretched the line back out it would still equal four.
Here is another way to show the problem
Like Vortex Math, Miles Mathis is taking advantage of a cool math trick.
Rhapsody on the Proof of Pi = 4
full link: http://www.youtube.com/watch?v=D2xYjiL8yyE
Re: Physical π : Pi's relationship to 4
That makes sense. But the problem is that M.M. is talking about physics, not about maths. That is why I still think these experiments are useful. To decide if we are "zigzagging" or not when we are running in circles.
Edit:
A simple experiment would be the following. It is similar but simpler than the one posted on youtube. You only need one tube and one ball. Instead of a circle(spiral) you make only half a circle followed by a straight part. Like the letter U. At the end of the turn you measure a short length (=l) and mark both the end end the beginning. So you can time how much time the ball needs to cover the distance= time A. You do the same thing at the start of the straight part after the turn. Using exactly the same length. Here you can measure time B.
I would predict B>A. Because the further the ball has travelled the more it will have slowed down because of friction. But the difference will be small (depending on l)
M.M. would predict that A>B. Because according to him the ball has to cover more "distance" during A. This difference would be about 21%. A bit less because the ball will have slowed down at the end of A because of friction (depending on l)
Edit:
A simple experiment would be the following. It is similar but simpler than the one posted on youtube. You only need one tube and one ball. Instead of a circle(spiral) you make only half a circle followed by a straight part. Like the letter U. At the end of the turn you measure a short length (=l) and mark both the end end the beginning. So you can time how much time the ball needs to cover the distance= time A. You do the same thing at the start of the straight part after the turn. Using exactly the same length. Here you can measure time B.
I would predict B>A. Because the further the ball has travelled the more it will have slowed down because of friction. But the difference will be small (depending on l)
M.M. would predict that A>B. Because according to him the ball has to cover more "distance" during A. This difference would be about 21%. A bit less because the ball will have slowed down at the end of A because of friction (depending on l)
Last edited by Seneca on Sat Oct 08, 2016 8:12 am, edited 2 times in total.

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Re: Physical π : Pi's relationship to 4
Kham, thank you for posting, you explained what I just can't get my head around, i.e. it's a circular path and that's not the same as an infinitely zigzaggy line. I probably err too heavily on trying to find realworld examples that illustrate what seems like common sense to me, rather than get bogged down in the hoodoo...
And thanks also Simon for the motorsport angle  I only chose a motorcycle analogy because I ride one and I have personally carried out the experiment I describe, so I feel on fairly safe ground starting from that point. The point I was (not particularly efficiently) trying to get across was exactly what you described  making the ball go round a curved path involves some energy being expended, and it has to come from the ball, i.e. its kinetic energy. So it slows down.
If anyone is doubting that the slowing down is occurring in the video experiment, just have a look at the point where the kinetically exhausted circle path ball flops out of the end of the tube. At the same time, the straight path ball exits the straight tube and continues zooming along the table. It is very clear to me, at least.
And thanks also Simon for the motorsport angle  I only chose a motorcycle analogy because I ride one and I have personally carried out the experiment I describe, so I feel on fairly safe ground starting from that point. The point I was (not particularly efficiently) trying to get across was exactly what you described  making the ball go round a curved path involves some energy being expended, and it has to come from the ball, i.e. its kinetic energy. So it slows down.
If anyone is doubting that the slowing down is occurring in the video experiment, just have a look at the point where the kinetically exhausted circle path ball flops out of the end of the tube. At the same time, the straight path ball exits the straight tube and continues zooming along the table. It is very clear to me, at least.
Re: Physical π : Pi's relationship to 4
Kham, there is a huge point you did not include when trying to explain what you call a math trick.
If I quote Mathis again (http://milesmathis.com/manh.pdf:
What in your view looks like a math trick is actually :
If I quote Mathis again (http://milesmathis.com/manh.pdf:
To make it simple and straightforward : the point you tried to make to refute Mathis actually points that he is correct when considering Pi=4 in kinematic situations. Note the two latter words, since in any kinematic situation, we have a time variable implied, existing underneath any other vector analysis. To continue I'll just again quote Mathis as he puts it in a very simple wording:Problem is, we have two variant methods of approaching the circle, one with polygons and one with steps. You would think they would converge on one another, but they don't. They don't meet in the middle at all. By one method of going to a limit, we get π, by the other we get 4. And that is a huge difference.
Which is correct? Well, amazingly, we have never had to choose. Archimedes picked polygons early on, and history has followed his method. The step method never even came up, as far as most of us know. Even when Hilbert discovered or rediscovered the step method with his Manhattan metric, he never thought to apply it to the circle. Maybe he didn't wish to stir up the firestorm I have stirred up.
But, as I have shown, it is the step method that is correct. The red, blue and green lines above all have the same circumference. Very different areas, but equivalent circumferences. And the reason the step method is correct while the polygon method is incorrect has to do with the way the field or metric or space is defined. As I said above, measuring the circumference as a limit of polygon sides requires we take slants or diagonals to a limit, and that cannot be done. Polygon sides are always diagonals in our space, and diagonals are always compound variables or compound vectors. You cannot take compound vectors to a limit in an ordered way, because, with the underlying time variable, they are actually curves or accelerations. A rigorous field solution requires we take simple variables or vectors to limits, and that can only be done with orthogonal or rectilinear vectors. This is precisely what our steps represent in the field. They are simple vectors, one that can be decomposed no further. An acceleration can always be decomposed into velocities, for instance, but a velocity cannot be further decomposed. A velocity uses a single distance, and you cannot get less than “single.
But this time variable is a bit tricky, since it can mean two different things. We can apply time to x and y in two very different ways. We can either give x a time and y a time, or we can give them both the same time. In other words, we can let our taxicab travel x in time t1, then let it travel y in time t2. OR, we can let the taxicab travel x and y during the same time. That is, during the same interval. In the first case, we get a Manhattan grid, as with Hilbert. In that case the sum of the two distances could either give us a Manhattan distance or a Euclidean distance, depending on whether we summed along the grid or along the slant. But in the second case, the summing of the distances would actually be an integral. We would integrate the motions into the same interval, and would thereby obtain a curve. Instead of a slant, we would have a curve and an acceleration. Working backward, we can then measure this curve by redrawing the x's and y's. The curve is measured and defined by the orthogonalvectors, not by the slants or hypotenuses. And this applies not only to circles, but to all curves. The length of any curve must be found by analyzing its orthogonal vectors in a defined space, not by analyzing chords or hypotenuses. This is because curves are a series of arcs, and arcs never converge upon chords, or the reverse.
What in your view looks like a math trick is actually :
I'm reposting the video from few days ago, the Youtube link got unavailable again, so here is the new link : http://www.dailymotion.com/video/x2wiyaj , from 11:15 on . I quote Mathis:…measuring the circumference as a limit of polygon sides requires we take slants or diagonals to a limit, and that cannot be done. Polygon sides are always diagonals in our space, and diagonals are always compound variables or compound vectors. You cannot take compound vectors to a limit in an ordered way, because, with the underlying time variable, they are actually curves or accelerations. A rigorous field solution requires we take simple variables or vectors to limits, and that can only be done with orthogonal or rectilinear vectors. This is precisely what our steps represent in the field. They are simple vectors, one that can be decomposed no further. An acceleration can always be decomposed into velocities, for instance, but a velocity cannot be further decomposed. A velocity uses a single distance, and you cannot get less than “single.”
*to better understand where this all comes from I'd suggest reading this : http://milesmathis.com/lemma.html . In short:There are two other things worth mentioning while you have these animations before you. One, notice that the motions in x and y are both accelerations. Neither cursor is moving at a constant velocity. Each speeds up in the middle of each circuit and slows at both ends. This means that in this method, neither component is a velocity. In total, we have four velocities, two in x and two in y. Which makes the total circular motion a variable acceleration to the power 4. Two, the limit here is completely a matter of speed. The faster we move the cursors or the animation, the closer we come to any limit. Which means the limit is a matter of time t, not of x or y. By speeding up the animation, we are making the steps smaller, and taking the method closer to a smooth curve. But since we cannot speed up the animation infinitely, the curve is never smooth. It only looks smooth to us because we cannot see at those speeds. We can't even see at the speed of movie reels, which is relatively slow. The narrator at Caltech admits this, in a way, when he says that circular motion is a sort of trick of the eye:
Can the naked eye see reality perfectly? Not always. Sometimes perceptions of reality may be mere shadows of the real thing.
We see a curve where there is no curve, only a changing integration of x and y motions.«
Given that proof, we find that the circle is NOT composed at the limit of chords or hypotenuses, as in Archimedes, Newton, or anyone after. Given motion and the time variable, the circle is composed at the limit of the orthogonal vectors. In other words, it is composed of the two shorter sides of the right triangle, not the long side. Which means that real objects in orbit travel a path that is represented not by the limit of the Euclidean metric, but by the limit of the Manhattan metric. And this means that in the kinematic circle, π=4 and C = 8r.”
Re: Physical π : Pi's relationship to 4
[As I went to post this, I saw that VexMan had already answered. I will leave this as is, as I think our answers are complementary.]
Believe me, Simon, I understand where you're coming from  it is difficult to wrap your head around. Nevertheless, I have to admit that my patience has worn very thin at this point. Some have said that this Pi=4 thing is just meant to waste our time. But here's the thing: the only people who are wasting my time are you guys who can't be bothered to read his papers to try to understand his argument. I've spent countless hours poring over Miles's physics papers, and I have never thought for a moment that I was wasting my time. To the contrary: I felt that I had been given a gift of deep insight into how the physical universe works. His physics papers are logical, perspicacious and for the most part easy to follow. There is not a lot of dense equations and the like, as Miles believes that dense and abstract math tends to obscure more than illuminate. And they are also entertaining, because his critiques of mainstream theory are delivered with devastating wit.
So following my response here, I will only continue to respond to arguments and questions that deal with the substance of the papers I linked to in my earlier post (http://www.cluesforum.info/viewtopic.ph ... 0#p2401740) and below or are posed by someone who has clearly read those papers. If you're not willing to make a good faith effort to understand, I don't see why I should make a good faith effort to explain. If you really wish to understand what he has written, you should make a good faith effort to do so. Otherwise it will be clear that you are just trolling and deliberately trying to waste mine and other people's time (Simon, I'm including you in this). Again, it's not Miles that is wasting my/our time, it's posters like bongostaple who come in with halfbaked arguments. He even admits his only reason for joining the forum was to comment on this subject. I was hoping he might provide some clarity, or at least some intelligent criticism, but all I've seen from him is FUD. And really lazy, halfassed FUD at that.
OK, first Simon:
What you've done in your billiard ball diagrams is you've measured the lengths of the paths. So far so good. And you're wondering: since the length of the curved path is shorter than the length of the square path (17 vs. 21), how could it possibly take the same amount of time to travel both of them? Well, first of all, Miles would not tell you that your measurements are off. They are not. And measured in this way, you will get a value for Pi of 3.14 (circumference/diameter), and he will agree with that. But the problem is that you cannot apply this measurement of the circumference to physical objects in motion. That's his point. It's like comparing apples to oranges.
We generally think that physics equations describe things in the physical world, correct? That's what physics equations do. So we have equations that describe orbits, or real bodies moving in circles. The current equations that describe real bodies moving in circles use the value 3.14 for Pi. But he's saying that is a mistake. You need to use the value of 4. And later he discovered that we have the cycloid math (which is also describes objects moving in a circle), which is the appropriate math for describing objects moving in circles. The current widely accepted equations for describing objects moving in a circle (or curve) are wrong.
Then the question is: why are they wrong? Again, he explains it much better in his papers, but I'll answer it here because it also provides a good answer to Kham's post.
Answer to Kham (and continuation of reply to Simon):
I really have to laugh when I see you guys come in with all these lazy counterarguments to his work without having taken the time to read and digest it. How can you say he is spouting mumbo jumbo without reading his work? And if you have read it, why are you unable to argue what exactly he says is mumbo jumbo?? To quote from his Manhattan paper again:
In his pi2.html paper, he writes:
http://milesmathis.com/pi2.html
First paper on calculus:
http://milesmathis.com/are.html
(And here is a shorter, more reader friendly version of that paper: http://milesmathis.com/calcsimp.html)
Second paper on Newton's Lemmae:
http://milesmathis.com/lemma.html
Third paper on rewrite of orbital equation:
http://milesmathis.com/avr.html
And it's not enough to come back and say, "this guy couldn't possibly have found errors in Newton and Leibniz, etc. What an ego!" Or, "no, he doesn't understand calculus at all!" No more lazy rebuttals and handwaving, folks.You've got to show specifically where he's wrong and what about it he doesn't understand. He also comments in his calculus (and other) papers on how mathematicians and physicists have managed to achieve so much even though some of their fundamental equations are wrong.
In response to the video you posted, it's important to note that the video is about Euclidean geometry and lengths, not orbital equations. So we have to be careful about comparing apples to oranges. Nevertheless, the method of drawing ever smaller shapes is a way of thinking about how we both approximate Pi and a way of thinking about the calculus that underlies the orbital equations. (However, it should be noted that Miles's arguments applies to curves, and another issue with the video is that Miles argues that not physical object can ever get to 0. It can approach 0 at the limit, but it will never arrive at 0. So your math has got to match that reality. Without recognizing that, you get all kinds of paradoxes, like black holes. In fact, here is a nice paper he wrote on paradoxes that deals exactly with this issue of infinities and limits: http://milesmathis.com/zeno.html)
I find the argument about zigzaggy lines not being smooth to be a bit disingenuous, because the same exact argument applies to the accepted ways of approximating Pi, based on the method devised by Archimedes. You're saying that drawing ever smaller polygons on the outer perimeter of the circle in order to approximate the circumference is wrong, because they will always be zigzaggy not smooth, no matter how small. But the Archimedes method does the same thing but from the inside of the circle! Here is a video about Archimedes' method for calculating Pi with polygons:
full link: http://www.youtube.com/watch?v=DLZMZCT7YU
Notice at just under 4 minutes in he says increasing the number of N approximates the value of Pi (where the number of N is just the number of sides of the polygon). But just as the 'zigzaggy' lines that approach the circle from the outside are never actually smooth no matter how small they get, the edges of the polygon are always flat, never smooth, never curved. So they will also be jagged or however you want to describe it. If the edges of the polygon can get infinitely small and smooth, then the zigzags can get infinitely small and smooth too. But neither of them can ever actually get smooth  not in the real, physical world; only in abstract Euclidean geometry. What's interesting (and what VexMan pointed out in his answer), is that each method of approximation gives a different answer  so which one is right?
We can apply this same bit of reasoning to orbital velocity. Orbital velocity is currently understood as being the sum of the tangential velocity as an object travels in a circle. Here is a diagram, just ignore the triangle at the bottom:
So if you want to know how long it will take a physical object to go around the circle (what might be called its orbital velocity), what do you do? Well the current solution is to basically add all the velocities at all the tangents at every point around the circle. So, let's do a thought experiment: say you wanted to draw all the tangents yourself by hand. You might start by drawing a straight line at each quarter point. then each 1/8 point, then each 1/6th point, and so on. One way to imagine this would be to draw a square around the circle, Then draw a hexagon, then an dodecahedron, etc. It's similar to Archimedes's method but instead of drawing the polygon from the inside, you're drawing it from the outside. As the shape adds more sides (or as you add more lines to represent tangential velocity) the shape gets rounder and rounder. But just as with the inside polygons, you will never get a smooth curve. It will always be jagged. As it happens, these inner and outer polygon approximation methods basically get us to the same answer.But when we apply abstract Euclidean geometry where a point with no dimensions is physically possible and a calculus where values of 0 can be reached instantaneously to describe the movement of a real physical object over time, we are getting the wrong answer. (Because although in Euclidean geometry you can have a point with zero length; in the real physical world you can't.)
And just as you can't have a physical object with a length=0, nothing ever happens instantaneously. Movement requires time, and every distance traveled is a length covered in some amount time. That amount of time can never be 0. That time can be vanishingly small, but it can never actual vanish and be 0. Therefore, you have to be careful applying Euclidean geometry and calculus to describe real world physics, and in many cases, you can't.
Currently, if you want to know the velocity of an object traveling a curve, you look to find its tangential velocity. But Miles argues that, no, the velocity of a real object traveling in a curve is not its tangential velocity. He calls this other velocity, what we might call its true velocity, an orbital velocity (though strictly speaking it is an acceleration). Tangential velocity does not equal orbital velocity.
And one reason for this is the way you have to incorporate time into the equations, which has not been done correctly. [Vexman's answer also addresses this question.] The correct way to incorporate time in these equations is, as it turns out, with the cycloid math. And here I will cutandpaste from his pi2.html paper:
To circle back to the video Kham posted: when we're talking about real physical bodies traveling in circular paths, the "ever smaller zigzag" method is the appropriate way to describe the movement of that body, not the "polygon with an ever increasing number of sides" method. Remember: that's what physics is  equations that describe physical objects and their movement and interactions in the real physical world world.
As for the approximation method that starts with a triangle outside of the circle, you can hopefully see from Vexman's post that the problem with the triangle is similar to the problem of using diagonal hypotenuses: "diagonals are always compound variables or compound vectors. You cannot take compound vectors to a limit in an ordered way, because, with the underlying time variable, they are actually curves or accelerations. A rigorous field solution requires we take simple variables or vectors to limits, and that can only be done with orthogonal or rectilinear vectors."
Coming back to Simon: if you've read Vexman's reply, then perhaps one way to get your head around it is to think about it like this: when you're measuring a circle, then the circumference is best approximated via the Archimedian polygon method (the diagonals). When you're trying to describe movement in a circle, then the circumference is best approximated with the eversmaller zigzag method (the orthoganal vectors or sides). "Which means that real objects in orbit travel a path that is represented [or described or calculated] not by the limit of the Euclidean metric, but by the limit of the Manhattan metric."
A final reply to bongostaple:
I hope it's finally clear to you now why your motorcycle analogy fails. You've gotten so lazy that you just recycled your previous "critiques." Why don't you actually spend some time on reading and understanding his work rather than wasting our time having to clear the air from the smoke coming out of your ass? Or at least offer us some originality?
As for your point about friction. Again: if it was friction, the slow down would be gradual, not instantaneous. It doesn't matter what the apparent speed of the ball is when it leaves the tube. Anyway you have no way of gauging the speed of the ball as it exits (and is immediately stopped by the tube in front of it). Second, you are drawing people's attention away from observing the ball as it crosses the 3/4 mark of the circle. It hits its mark exactly as the straight ball hits the third diameter length marking, just as the ball hit the 1/4 mark at the same time as the straight ball hit the first diameter marking. The only reasonable conclusion to be drawn from this is that the ball in the circle was not experiencing enough additional friction to perceptibly slow it down. If it were, then the 3/4 mark would not have been reached later than the 3rd diameter mark. But it's not, so no, friction is not a reasonable argument, as I've already demonstrated to you in previous posts. Wishing it were so will not make it happen. If you want to tell me that the last 1/4 is bent even more, such that there is more friction and slowdown in just the last quarter, I will tell you it doesn't matter: the fact that the straight ball hits the Pi marker before the circular ball barely makes it past the 3/4 mark already demonstrates the argument. The last 1/4 is just icing on the cake. Now please, would you go blow smoke somewhere else?
I am going to reply now to Simon's comment, then the other comments that followed his.simonshack » October 7th, 2016, 2:34 pm wrote:Well, I guess I'm just a bit dense  and will probably need another lifetime (or a brain transplant) in order to wrap my head around this Pi=4 thing...
Meanwhile, I would warmly appreciate if someone can explain  in simple words  why I can place 21 billiard balls between a > b > and c ... and only 17 billiard balls between a and c?
Believe me, Simon, I understand where you're coming from  it is difficult to wrap your head around. Nevertheless, I have to admit that my patience has worn very thin at this point. Some have said that this Pi=4 thing is just meant to waste our time. But here's the thing: the only people who are wasting my time are you guys who can't be bothered to read his papers to try to understand his argument. I've spent countless hours poring over Miles's physics papers, and I have never thought for a moment that I was wasting my time. To the contrary: I felt that I had been given a gift of deep insight into how the physical universe works. His physics papers are logical, perspicacious and for the most part easy to follow. There is not a lot of dense equations and the like, as Miles believes that dense and abstract math tends to obscure more than illuminate. And they are also entertaining, because his critiques of mainstream theory are delivered with devastating wit.
So following my response here, I will only continue to respond to arguments and questions that deal with the substance of the papers I linked to in my earlier post (http://www.cluesforum.info/viewtopic.ph ... 0#p2401740) and below or are posed by someone who has clearly read those papers. If you're not willing to make a good faith effort to understand, I don't see why I should make a good faith effort to explain. If you really wish to understand what he has written, you should make a good faith effort to do so. Otherwise it will be clear that you are just trolling and deliberately trying to waste mine and other people's time (Simon, I'm including you in this). Again, it's not Miles that is wasting my/our time, it's posters like bongostaple who come in with halfbaked arguments. He even admits his only reason for joining the forum was to comment on this subject. I was hoping he might provide some clarity, or at least some intelligent criticism, but all I've seen from him is FUD. And really lazy, halfassed FUD at that.
OK, first Simon:
What you've done in your billiard ball diagrams is you've measured the lengths of the paths. So far so good. And you're wondering: since the length of the curved path is shorter than the length of the square path (17 vs. 21), how could it possibly take the same amount of time to travel both of them? Well, first of all, Miles would not tell you that your measurements are off. They are not. And measured in this way, you will get a value for Pi of 3.14 (circumference/diameter), and he will agree with that. But the problem is that you cannot apply this measurement of the circumference to physical objects in motion. That's his point. It's like comparing apples to oranges.
We generally think that physics equations describe things in the physical world, correct? That's what physics equations do. So we have equations that describe orbits, or real bodies moving in circles. The current equations that describe real bodies moving in circles use the value 3.14 for Pi. But he's saying that is a mistake. You need to use the value of 4. And later he discovered that we have the cycloid math (which is also describes objects moving in a circle), which is the appropriate math for describing objects moving in circles. The current widely accepted equations for describing objects moving in a circle (or curve) are wrong.
Then the question is: why are they wrong? Again, he explains it much better in his papers, but I'll answer it here because it also provides a good answer to Kham's post.
Answer to Kham (and continuation of reply to Simon):
I really have to laugh when I see you guys come in with all these lazy counterarguments to his work without having taken the time to read and digest it. How can you say he is spouting mumbo jumbo without reading his work? And if you have read it, why are you unable to argue what exactly he says is mumbo jumbo?? To quote from his Manhattan paper again:
I would say that describes you to a T, Kahm. Especially the part about high school.I also see many of my critics—one might say the bulk of them—claiming I don't understand calculus
when it is clear they don't understand it. Their criticisms look incredibly lazy to me, which is why I
haven't bothered to respond to them. What they commonly do is blast me because my analysis doesn't
match the soundbite analysis they were taught in school. They have developed some halfbaked
slipshod idea of calculus, most of it seemingly manufactured in their own minds from nothing, and
because my historical analysis conflicts with that, I am a crackpot. It is clear from studying their
criticisms that they have never bothered to study the historical progression of the calculus or any of the
original proofs. They are only concerned that my analysis doesn't match what they were taught in high
school, or doesn't match some video on youtube. But of course if they don't understand the original
proofs of the calculus, it is very unlikely they will be able to follow my critiques of those proofs. To
know whether a critique of A is right or wrong, you have to have a pretty good idea of what A is to start
with, and none of my critics has that. Because they don't, they can only stomp around and make a lot
of noise, hoping to keep others from looking at what I am saying.
In his pi2.html paper, he writes:
Here, allow me to offer you some lowhanging fruit by supplying you with links to those papers:this paper cannot stand alone. It is a mistake to start with this paper. Those who do start with this paper will very likely be led to believe I am simply doing the calculus wrong. To these people, I say that it is not I who am doing the calculus wrong. It is Newton and Leibniz and Cauchy and everyone since who has been doing the calculus wrong. I have earned the right to write this paper by first writing three important papers on the foundations of the calculus. The first shows that the derivative has been defined wrongly from the beginning, and that the derivative is a constant differential over a subinterval, not a diminishing differential as we approach zero. There is no necessary approach to zero in the calculus, and the interval of the derivative is a real interval. In any particular problem, you can find the time that passes during the derivative, so nothing in the calculus is instantaneous, either. This revolutionizes QED by forbidding the point particle and bypassing all need for renormalization. The second paper proves that Newton's first eight lemmae or assumptions in the Principia are all false. Newton monitors the wrong angle in his triangle as he goes to the limit, achieving faulty conclusions about his angles, and about the value of the tangent and arc at the limit. Finally, the third paper rigorously analyzes all the historical proofs of the orbital equation a=v2/r, including the proofs of Newton and Feynman, showing they all contain fundamental errors. The current equation is shown to be false, and the equation for the orbital velocity v=2πr/t is also shown to be false. Those who don't find enough rigor or math in this paper should read those three papers before they decide this is all too big a leap. I cannot rederive all my proofs in each paper, or restate all my arguments, so I am afraid more reading is due for those who really wish to be convinced. This paper cannot stand without the historical rewrite contained in those papers, and I would be the first to admit it.
http://milesmathis.com/pi2.html
First paper on calculus:
http://milesmathis.com/are.html
(And here is a shorter, more reader friendly version of that paper: http://milesmathis.com/calcsimp.html)
Second paper on Newton's Lemmae:
http://milesmathis.com/lemma.html
Third paper on rewrite of orbital equation:
http://milesmathis.com/avr.html
And it's not enough to come back and say, "this guy couldn't possibly have found errors in Newton and Leibniz, etc. What an ego!" Or, "no, he doesn't understand calculus at all!" No more lazy rebuttals and handwaving, folks.You've got to show specifically where he's wrong and what about it he doesn't understand. He also comments in his calculus (and other) papers on how mathematicians and physicists have managed to achieve so much even though some of their fundamental equations are wrong.
In response to the video you posted, it's important to note that the video is about Euclidean geometry and lengths, not orbital equations. So we have to be careful about comparing apples to oranges. Nevertheless, the method of drawing ever smaller shapes is a way of thinking about how we both approximate Pi and a way of thinking about the calculus that underlies the orbital equations. (However, it should be noted that Miles's arguments applies to curves, and another issue with the video is that Miles argues that not physical object can ever get to 0. It can approach 0 at the limit, but it will never arrive at 0. So your math has got to match that reality. Without recognizing that, you get all kinds of paradoxes, like black holes. In fact, here is a nice paper he wrote on paradoxes that deals exactly with this issue of infinities and limits: http://milesmathis.com/zeno.html)
I find the argument about zigzaggy lines not being smooth to be a bit disingenuous, because the same exact argument applies to the accepted ways of approximating Pi, based on the method devised by Archimedes. You're saying that drawing ever smaller polygons on the outer perimeter of the circle in order to approximate the circumference is wrong, because they will always be zigzaggy not smooth, no matter how small. But the Archimedes method does the same thing but from the inside of the circle! Here is a video about Archimedes' method for calculating Pi with polygons:
full link: http://www.youtube.com/watch?v=DLZMZCT7YU
Notice at just under 4 minutes in he says increasing the number of N approximates the value of Pi (where the number of N is just the number of sides of the polygon). But just as the 'zigzaggy' lines that approach the circle from the outside are never actually smooth no matter how small they get, the edges of the polygon are always flat, never smooth, never curved. So they will also be jagged or however you want to describe it. If the edges of the polygon can get infinitely small and smooth, then the zigzags can get infinitely small and smooth too. But neither of them can ever actually get smooth  not in the real, physical world; only in abstract Euclidean geometry. What's interesting (and what VexMan pointed out in his answer), is that each method of approximation gives a different answer  so which one is right?
We can apply this same bit of reasoning to orbital velocity. Orbital velocity is currently understood as being the sum of the tangential velocity as an object travels in a circle. Here is a diagram, just ignore the triangle at the bottom:
So if you want to know how long it will take a physical object to go around the circle (what might be called its orbital velocity), what do you do? Well the current solution is to basically add all the velocities at all the tangents at every point around the circle. So, let's do a thought experiment: say you wanted to draw all the tangents yourself by hand. You might start by drawing a straight line at each quarter point. then each 1/8 point, then each 1/6th point, and so on. One way to imagine this would be to draw a square around the circle, Then draw a hexagon, then an dodecahedron, etc. It's similar to Archimedes's method but instead of drawing the polygon from the inside, you're drawing it from the outside. As the shape adds more sides (or as you add more lines to represent tangential velocity) the shape gets rounder and rounder. But just as with the inside polygons, you will never get a smooth curve. It will always be jagged. As it happens, these inner and outer polygon approximation methods basically get us to the same answer.But when we apply abstract Euclidean geometry where a point with no dimensions is physically possible and a calculus where values of 0 can be reached instantaneously to describe the movement of a real physical object over time, we are getting the wrong answer. (Because although in Euclidean geometry you can have a point with zero length; in the real physical world you can't.)
And just as you can't have a physical object with a length=0, nothing ever happens instantaneously. Movement requires time, and every distance traveled is a length covered in some amount time. That amount of time can never be 0. That time can be vanishingly small, but it can never actual vanish and be 0. Therefore, you have to be careful applying Euclidean geometry and calculus to describe real world physics, and in many cases, you can't.
Currently, if you want to know the velocity of an object traveling a curve, you look to find its tangential velocity. But Miles argues that, no, the velocity of a real object traveling in a curve is not its tangential velocity. He calls this other velocity, what we might call its true velocity, an orbital velocity (though strictly speaking it is an acceleration). Tangential velocity does not equal orbital velocity.
And one reason for this is the way you have to incorporate time into the equations, which has not been done correctly. [Vexman's answer also addresses this question.] The correct way to incorporate time in these equations is, as it turns out, with the cycloid math. And here I will cutandpaste from his pi2.html paper:
...the arc of a cycloid is also 8r. That is, in the cycloid, π is replaced by 4, just as in the Manhattan metric. I don't know why I didn't think to include this before, since it is so obvious. We should have always asked more persistently why the arc of the cycloid is 8r while the circumference is 2πr. As a matter of kinematics, it makes no sense. The same point draws both, so why the 21% miss? I will be told that it is because with the circumference, the circle is not moving along an xaxis, but with the cycloid, it is. It is the difference between a rolling circle and a nonrolling circle. It is this lateral movement that adds the 21%. But whoever is telling me this is missing a very important point: in the kinematic circle I am talking about, the circle is also rolling. If you are in an orbit, for instance, the circle is not moving laterally, but a point on the circle is moving. The circle is rolling in place, and it is moving exactly like the point in the cycloid. Therefore, we see it is not the lateral motion that adds the 21%, it is the rolling alone. A static circle and a circle drawn by motion are not the same. The number π works only on the given static circle, in which there is no motion, no time, and no drawing. Any realworld circle drawn in time by a real object cannot be described with π.
If we study the generation of the cycloid closely, we find more evidence of this, since the arc of the cycloid isn't some sort of integration of the circumference with the distance rolled. It can't be, because some point on the circle is always contiguous with the flat surface. We would have to slide the circle in order to add any of the xdistance traveled. What is actually happening is that with the cycloid, the xtoy integration of distances is explicitly including time, as you see here:
So there you have it: go and study his papers first before announcing that he is a fraud who is wasting everyone's time. He can explain things much more clearly and deeply than I ever could.In that integral, we have three variables or functions: x, y, and t. Study the second and third lines of the math, where we are explicitly following the value of t. It is not the sine or cosine of x or y we are following, it is the cosine and then sine of t. In that integration, we have three degrees of freedom or a 3vector. So it is not the lateral motion that is causing the difference, it is the inclusion of time. To calculate the arc length of the cycloid, we need the integral which includes dt. But when we calculate the circumference, we don't include any dt. So the methods of calculation don't match. Despite that, we use the naive static circumference that includes π when we calculate orbits. This is illogical, since all orbits include time. They should be solved with integrals like the one above, not with the static circumference calculated from π.
You may wish to differentiate the two as I do in my long calculus paper. There I differentiate between length and distance. A length is a given parameter that does not include motion or time. It is geometric only. But a distance is a length traveled in some real time, so it requires motion. A length is not kinematic, while a distance is. The circumference 2πr is a length. The circumference 8r is a distance.
To circle back to the video Kham posted: when we're talking about real physical bodies traveling in circular paths, the "ever smaller zigzag" method is the appropriate way to describe the movement of that body, not the "polygon with an ever increasing number of sides" method. Remember: that's what physics is  equations that describe physical objects and their movement and interactions in the real physical world world.
As for the approximation method that starts with a triangle outside of the circle, you can hopefully see from Vexman's post that the problem with the triangle is similar to the problem of using diagonal hypotenuses: "diagonals are always compound variables or compound vectors. You cannot take compound vectors to a limit in an ordered way, because, with the underlying time variable, they are actually curves or accelerations. A rigorous field solution requires we take simple variables or vectors to limits, and that can only be done with orthogonal or rectilinear vectors."
Coming back to Simon: if you've read Vexman's reply, then perhaps one way to get your head around it is to think about it like this: when you're measuring a circle, then the circumference is best approximated via the Archimedian polygon method (the diagonals). When you're trying to describe movement in a circle, then the circumference is best approximated with the eversmaller zigzag method (the orthoganal vectors or sides). "Which means that real objects in orbit travel a path that is represented [or described or calculated] not by the limit of the Euclidean metric, but by the limit of the Manhattan metric."
A final reply to bongostaple:
I hope it's finally clear to you now why your motorcycle analogy fails. You've gotten so lazy that you just recycled your previous "critiques." Why don't you actually spend some time on reading and understanding his work rather than wasting our time having to clear the air from the smoke coming out of your ass? Or at least offer us some originality?
As for your point about friction. Again: if it was friction, the slow down would be gradual, not instantaneous. It doesn't matter what the apparent speed of the ball is when it leaves the tube. Anyway you have no way of gauging the speed of the ball as it exits (and is immediately stopped by the tube in front of it). Second, you are drawing people's attention away from observing the ball as it crosses the 3/4 mark of the circle. It hits its mark exactly as the straight ball hits the third diameter length marking, just as the ball hit the 1/4 mark at the same time as the straight ball hit the first diameter marking. The only reasonable conclusion to be drawn from this is that the ball in the circle was not experiencing enough additional friction to perceptibly slow it down. If it were, then the 3/4 mark would not have been reached later than the 3rd diameter mark. But it's not, so no, friction is not a reasonable argument, as I've already demonstrated to you in previous posts. Wishing it were so will not make it happen. If you want to tell me that the last 1/4 is bent even more, such that there is more friction and slowdown in just the last quarter, I will tell you it doesn't matter: the fact that the straight ball hits the Pi marker before the circular ball barely makes it past the 3/4 mark already demonstrates the argument. The last 1/4 is just icing on the cake. Now please, would you go blow smoke somewhere else?
Last edited by daddie_o on Sat Oct 08, 2016 7:43 pm, edited 2 times in total.