A Cute Triangle

Friend of the Blog, Max Forester, told us about an interesting math problem which is both ancient and modern.  It’s ancient because Pythagoras himself probably wondered about the question.  It’s modern because it’s still an active area of research.  What is it?

Well, first of all, remember that an acute triangle is a triangle where all three angles are strictly less than 90 degrees.  For example, an equilateral triangle is an acute triangle.   The question is:

Can you tile a square with acute triangles?

The answer is well known, and it’s yes!  Here’s an example which uses 60 triangles:

An acute triangulation with 60 triangles.

60 triangles is a lot of triangles, can it be done with less?  Yes:

An acute triangulation with 14 triangles.

It turns out that you can prove that the best possible is an acute triangulation with 8 triangles.  A good puzzle is to try to do better than 14 triangles and, ideally, find one with 8 triangles.

But squares are rather boring.  What about an octagon?:

It was proven by Burago and Zalgaller in 1960 that in fact any polygon in the plane has an acute triangulation!

Let’s take up a dimension?  What about polytopes in 3 dimensions?:

Well, we need to agree on what an acute triangulation should mean in 3D.  A triangle is what you get in 2D when you take the convex hull of three points (not all in a line, of course!).  An acute triangle is a triangle where the angle between any two adjacent sides is strictly less than 90 degrees.

What about 3D?  If you take the convex hull of four points in 3-space (not all in a plane, of course!), then the result is called a 3-simplex or tetrahedron.   (By the way, another name for a triangle is a 2-simplex.)  An acute tetrahedron is a tetrahedron where the angle between any two adjacent sides is strictly less than 90 degrees.  Here’s an example of a tetrahedron:

So a natural 3D version of our ancient problem is to find a filling of the cube with acute tetrahedra.

Heck, your little cousin can sit down with a square and find an acute triangulation in less than an hour.  How hard can it be to find a filling of the cube?

Very, very hard, it turns out! Two research papers have just come out showing a yes answer.

One is by Evan VanderZee, Anil N. Hirani, Vadim Zharnitsky, and Damrong Guoy called “A Dihedral Acute Triangulation of the Cube”.  You can find it here.

Another paper is by Eryk Kopczynski, Igor Pak, and Piotr Przytycki and called “Acute triangulations of polyhedra and Rn“.  You can find it here.

In both papers they prove that there is a filling of the cube.  They use a mixture of theoretical and computer-based techniques.  Here is a picture of the filling:

From the Kopczyński, Pak, and Przytycki paper.

In case you have trouble counting, there are 2715 tetrahedra!  VanderZee, Hirani, Zharnitsky, and Guoy do better by using a mere 1370 tetrahedra.

On the other hand, Kopczyński, Pak, and Przytycki have a more theoretical approach which lets them prove that you can fill any of the platonic solids with acute tetrahedra.

We should mention two other amazing facts:

First, it is an open problem whether all convex polygons in 3D space have a filling by acute tetrahedra!  That’s a good summer project to think about.

Second, amazingly, it can be proven that for any n bigger than 3, there is no filling of the n-dimensional cube with acute n-simplicies!   Of course, if you went to Dr. Goodey’s Math Club talk last fall then you aren’t so surprised…