New Solids of Constant Width

Jul 30, 2019By Archis Gore

When everyone’s special, no one is

You’ve probably seen this video before.

Aside from a 10-second gimmick, what benefit do these objects hold for us non-mathematicians?

Why we use Wheels and Balls

Spheres are nature’s preferred shape. When we need something to roll on, we think of wheels and balls.

We have tried other shapes every so often with entertaining results:

Spheres will be spheres…

For all the benefits of a sphere, it has one huge drawback: its symmetry.

Every time I look away…

Spheres make everyone special. Whether they want to play marbles, croquet, fetch or billiards, they will serve you faithfully. When you don’t want them rolling everywhere, they’ll continue to get under your feet at midnight.

When everyone’s special, no one is.

Can I have a sphere… but only for me?

The best things in life make us feel special compared with everyone else. What if we had objects that behaved like balls in specific cases, but did not act like balls in all the others?

Such objects are called solids of constant width (in two dimensions, shapes of constant width).

We recognize a few shapes in the animation above. The sphere, obviously, but also the Meissner tetrahedron (looks like a pyramid) and a revolved Reuleaux triangle (looks like an ice-cream cone).

Here are a few others:

This time, no sphere at all. A Meissner tetrahedron, and revolved Reuleaux pentagon and heptagon. The book doesn’t know the difference.

But these objects are boring?—?algorithmic and predictable. Is there a way we can get exotic shapes? I recently found a paper titled On Curves and Surfaces of Constant Width by Howard L. Resnikoff, which presents a few new shapes never-before seen and not “revolved polygons.” I rendered these in 3D printable models, and this is how they function:

This time, there are two exotic-looking blobs and a revolved Reuleaux pentagon. The book spins exactly the same as before.

Semantic Identity for You

Semantically, each solid is considered “identical for the purpose of rotating a book.” What this means is for that particular purpose, all objects might as well be the same.

This is a fundamental concept we are deeply in tune with at Polyverse. A semantically identical text-editor could be anything that, under the load of editing text, you wouldn’t know the difference.

Now You Are More Special Than Everyone Else.

Polymorphism for Everyone

For every purpose other than spinning a book on top of them, such as rolling helter-skelter, the physical morphology of our objects differs from a sphere and they quickly come to a halt. Moreover, once word spreads that your rolling objects can’t be used for anything but spinning books in animated GIFs, it deters attempts to steal them.

Consider a Polymorphic Text Editor that under the wrong load, such as impersonating a kernel device driver, simply comes to a halt. This unusual behavior at once Detects something is amiss, while at the same time Defends against misuse, and Deters people trying. It simply isn’t shaped like a kernel device driver. It’s useless.

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