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/