Coin With 11 Sides and a Constant Diameter
Date: 10/20/2007 at 11:59:19 From: Benson Subject: Do 11 sided coins roll better than 10 sided coins I have been told by a friend that one of the reasons why a Canadian Loonie coin is 11 sided is that it rolls better than a 10 sided coin. Why is that? Is it also true that an odd numbered shape will roll better than a even numbered one? I think that the reason behind this has something to do with balance of the coin, if the coin has an even number of sides, the coin will be able to balance since there is always a side of the coin that is pointing up while the other is pointing down causing perfect symmetry.
Date: 10/20/2007 at 19:26:32 From: Doctor Tom Subject: Re: Do 11 sided coins roll better than 10 sided coins Hello Benson, Obviously, if a coin has more sides, it is closer to a circle and should roll better, but there is more to it than that. A Loonie, I think, does not have perfectly flat sides, but rather sides that are a little bit curved, so in fact, its surface is a shape of "constant diameter". This is critical for vending machines that measure which coin goes in by checking for the width, and a polygon with flat sides will measure slightly different widths, depending on how it goes in. But a shape with slightly curved sides, if constructed correctly, will have EXACTLY the same diameter, no matter how it's measured. Only shapes with an odd number of sides can have constant diameter, so the Loonie has to have 3, 5, 7, ... sides to work. Eleven sides makes a nice-shaped coin that's not too far from a circle. The other nice thing about the Loonie design that would cause it to roll MUCH better than a 10-sided coin is that since it has constant diameter, as it rolls, the center of mass (CM) does not have to go up and down. If you roll a 10-sided coin, its CM would rise and fall a tiny amount each time it passed a bump, so it would slow down and speed up a tiny amount for each bump; the Loonie would go at basically constant speed. For more information, look up "curves of constant width" on the Internet. You might also find it interesting to look up the "wankel rotary engine" that had "pistons" that were effectively 3-sided curves of constant diameter. Good question! - Doctor Tom, The Math Forum http://mathforum.org/dr.math/
Date: 08/11/2012 at 16:06:50 From: Trevor Subject: Re: Do 11 sided coins roll better than 10 sided coins Dear Dr. Math, I believe there is an error in the answer on this page. You wrote that because a loonie has constant diameter, its center of mass does not go up and down as it rolls. This implication does not hold in general, as the following animation of a rolling Reuleaux triangle makes apparent (ignoring the side-to-side motion): http://mathworld.wolfram.com/images/gifs/reuleaux.gif In fact, it is not hard to prove that the only convex shape whose center of mass (assuming uniform density) does not go up and down as it rolls is the circle. Also, for what it's worth, loonies roll poorly (and noisily!) Regards, Trevor
Date: 08/14/2012 at 11:21:06 From: Doctor Douglas Subject: Re: Do 11 sided coins roll better than 10 sided coins Hi, You are correct. So let's clarify the difficulty in defining what is meant by "rolling better." A Loonie will roll, but certainly not as smoothly as a circular coin, because its center of mass (CM) goes up and down. For a circular coin, the CM remains at the same height above the floor, so it doesn't require energy from the forward motion to raise and lower the CM, and so it rolls smoothly. Of course, a 10-sided coin will also suffer from the same jerky rolling action. On the other hand, if you try to roll a board across the floor atop two or more cylindrical logs, you would do well to use logs that have cross-sections that are circular, Loonie-shaped, or any curve of constant width. Otherwise, it will be a bumpy ride! - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
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