What does it mathematically mean to roll and is there other objects that one can use to roll around with?
So what makes a circle a convenient shape to roll?
Well lets us first think of why you would ride on a bicycle with circular wheels instead of one with square wheels? Well the answer is it is smooth. You only move forward each time the circle rotates. The square on the other hand has a problem. When it rotates it moves the person forward but it also moves the person vertically up. This vertical upward movement kills the smoothness of the rolling action.
Here is a graphical visualisation:
So this is the same problem with most other shapes. This problem applies for triangles, hexagons, pentagons etc. etc. But the more sides vertices you add the smaller the vertical movement. If make a shape with infinite amount a vertices then it would become a circle thus no vertical movement.
So now let’s clarify, rolling is the act of moving a shape horizontally by rotating the object. To role smooth is the act of moving the shape horizontally by rotating the object and during this process it does not change its height.
With this definition now we can ask the following question, is there a shape other than a circle that one can rotate smoothly? Well the answer is surprisingly yes! These shapes are called shapes of constant width.
A circle is a shape of constant with because if you rotate its height and width/ height stays the same.
It turns out there is many of these shapes of constant width. One of these shapes is the Reuleaux triangle. It looks something like this:
Now this object looks like a triangle with rounded edges and does not look like an object that will roll, but it does! Here is a link to a site that shows you how to make one! Cut two of those out and try and roll it will ruler on top. You will notice how it rolls just like round objects would have!
Now here is the big question, why don’t we make wheels with that type of shape? Well there is another problem with it when it comes to making wheels like that. Its centre point will move vertically up and down when this shape rotates. Thus, it might be good to throw a path of the stuff and let a ruler rotate over it, but it would be bad to make a bike out of it.
But what about our non constant width shape like a square? Is there a way to manipulate the environment so that the shape can rotate smoothly for a bicycle? Well the answer is of course yes. If we could trace the height of a rotating square like this:
And if we use it for the surface to rotate the square on, then yes its centre point will stay a constant height. You can do this for all the non constant width shapes and cause it to roll smoothly!
Now this post was all about the 2D shapes. In my next post we will discuss the rolling of 3D shapes like spheres etc.
Thank you for all the support and reading! This was a short fun post! picture still made by me except for the Reuleaux triangle and the thumb nail picture.
As always thanks for reading!
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