Rolling
Your first task for this chapter is to imagine two equal coins, such as quarters. One is held fast to a table, and the other is started at one point upon the circumference of the stationary quarter with the head showing upright, and is rolled once around the circumference of the in-place quarter back to the starting position. How many rotations do you think the rolling quarter will complete? Fig-11
Take two equal coins and try it. Where is the head in the upright position as started? Were you correct?
This question was on a college entrance examination test, and was challenged by a student that answered, two rotations, and was marked incorrect. He won his challenge. The coin goes through 720° of rotation to make it around the other equal coins circumference.
Any explanations I found did not satisfy me. So, I went into my investigative mode... I decided to roll a square about another equal square. After all, they might just be crude circles? Well, squares don't roll very well, but I flipped it around the other square, and it also rotated 720°! I tried it by starting at different points on the circumference of the stationary square. I started it positioned on a corner. I started it hanging over the corner... Always the same; twice, or 720°.
Yes, I did it with triangles, pentagons, hexagons, and more... with still the same results!
And guess what? You can fashion coordinate, and mix and match... Odd and even polygons, and the circle can all be rolled about each other at will with the only requirement__ their circumferences must be equal! It still comes up 720° of rotation!
I rolled polygons and coins on their respective circumferences laid out Horizontally Flat. The polygons and the coins (circles) only did 360° as expected.
I rolled polygons and coins on a concave arc of their respective circumferences. In this scenario I found it to be less than 360° of rotation. Hum?
When a square is rolled, it very noticeably flips about its own corner points, and flips around the corner points of the fixed square. In fact it is very easy to show the square, when in motion, flips about its own corners for 360° while going around the fixed square, and also flips 90˚ for each of the fixed four corners, also giving an additional 360° to total 720°. This is also the case with all the polygons. With more sides and corners for the polygons, there is less angle of flip at each corner, but more corners to flip___ still resulting in a total of 720°.
I mused about what the coin or circle was doing... ? By projecting my thoughts a bit, I know that a million sided regular polygon coin is going to be pretty round, (yes, I know round is relative... ), and I am now pretty sure what the polygon is doing. Flipping! So__ is the circle or coin flipping also? Wait, there is more...
Back to rolling polygons in a concave arc of equal length as the polygon circumference. It is also readily shown that there is a Skipping action occurring! The flat sides of the polygons bridge from corner to corner as they flip along and reduce the total rotation...
I have to tell you... in my mind I am pretty sure there is no such thing as a smooth rolling action. There appears to be only Flipping and Skipping! So I put it to a few more tests...
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When a wheel with a mounted hubcap is rolled across a table of length equal to the wheel circumference, both wheel and hubcap will make one 360˚ rotation. However, when the smaller hubcap is rolled across the same table length, it will do more rotations depending upon the hubcap size. Fig-13 How is this explained? I found out, not very well.
So I rolled (flipped) a square wheel with a square hubcap across a table the same length as the wheel circumference. And, I rolled (flipped) the square hubcap across as mentioned. I got the same effect. Fig-14
I investigated a bit more. I diagramed out regular polygons when rolled upon the flat. I shrunk the hubcap to just a center point for the polygons. I traced out the center point trajectory. When a polygon flips (rolls) it has a flip angle for each flip. This angle of flip equates to how much angle of rotation is accomplished by the center point, or a hubcap. the movement of the hubcap is a flipping action and not a rolling action across the length of the table when mounted on a wheel. Fig-15
The more sides added to a polygon, the smaller the flips, and the better it rolls. It does not take a great amount of sides to make something on an everyday scale, round. When there are quite a few sides the horizontal linear travel of the center axis point is pretty flat__ but there still must be some tiny flipping happening to allow only 360° of rotation for the mounted hubcap in one wheel circumference length of travel.
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I rolled polygons to emulate the actions of the circle, and analyze same. Such as the Cycloid, shown in Figure 16. I also did Curate Cycloids, Prolate Cycloids, Astroids (Hypocycloids), Epicloids, and etc. All these figures when plotted with polygons are very similar to when plotted with a circle. It is obvious that when the polygons have more sides, they are more similar to the circle.
As you see in Figure 17, I used an octagon to simulate the involute of a circle. This not a case of rolling or flipping the polygon, but a scenario of unrolling the circumference as if it were a string wrapped about the figure. And, again a close similarity.
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Exploring the seeming Paradox of Rolling or Flipping of polygons. If one rotation of a regular polygon is designated as the decimal number of one (1) instead of degrees; any fraction thereof is a decimal fraction. With any and all polygons, no matter how many sides or angles, or how small they might be... they must exist, or the figure is no longer a polygon.
Thus a formula can be used to give the angle of Flip when one side of a polygon is flat with the horizon. This decimal fraction angle is equal to one full rotation angle of one (1) divided by the number of polygon sides.
Thus we can formulate: One, divided by infinite equals infinitesimal. Or, possibly equals zero? On a calculator, 1 / 0 = 0. I reasoned, that if the angle is zero... it can no longer flip or roll, and in effect would be a flat line. But, we are left with how large or small is infinitesimal? Yet still, I wonder about the circle as an infinite sided or angled regular polygon, how does it roll?
A Paradox? At least an enigma? Maybe not. Our mind has the capability to imagine a perfectly straight line or circle, but in the reality of the Universe we live in, there is probably no such thing. The smallest micro particle is presently thought to be a quark. These go together to make larger atomic particles, then the atom, molecules and etc. Our Universe is known to be expanding, galaxies, planets, and other heavenly bodies all in motion. Atoms and their particles are in motion. Along with this, gravitation it thought to permeate all of the Universe. This means that everything is probably in motion, and in curved motion... however so slight. It is also very unlikely that the building blocks of matter are configured with angular sharp corners, or straight lines.
So what now? Well, I believe, if the circle circumference is thought of as micro particles that probably have curved, and probably flexible make-up, and more probably are just electromagnetic fields__ the answer is the circle edge is a moving flexible microscopic curved bumpy line! This would allow for the circle to progress forward as it tilts downward over micro sized moving flexing bumps as the circle (wheel) rolls, i.e. flips.
So, yes, a circle is probably not an infinite sided regular polygon!
It is a random multi-bumpy or jagged, irregular curved circumference polygon like figure, with an equal average mean radii about a single point.
But, many sided regular polygons do emulate the circle. Thus, I kept going with my math curiosity...
