Wolfram Cloud Document

This is part of live presentation series called Mathematical Games in which we explore a variety of games and puzzles using Wolfram Language. In this episode, we explore games that involve space groups and filling space.

Watch on YouTube: https://www.youtube.com/watch?v=YQRHsnd9yCA

demonstrations.wolfram.com

Many Demonstrations deal with crystals, tiling and space groups.

We just did a relaunch to improve some features.

230 Space Groups

There’s a nice free poster of the space groups available. MaXrd is good package of programs for space groups.

Terrence Tao and SMBC

Also, a few days ago, Terrence Tao teamed up with SMBC to give a 5-part exploration of how math is perceived, before an extended discussion of sphere packing. It’s always great when you pick a topic for a talk... and one of the most famous mathematicians in the world (Terrence Tao) has been looking at the same item. The 5-part comic is worth a read.

Lattices can be tricky

For example, a 4×4×4 lattice.

In[]:=

cube=Tuples[{-3,-1,1,3},{3}]

Out[]=

{{-3,-3,-3},{-3,-3,-1},{-3,-3,1},{-3,-3,3},{-3,-1,-3},{-3,-1,-1},{-3,-1,1},{-3,-1,3},{-3,1,-3},{-3,1,-1},{-3,1,1},{-3,1,3},{-3,3,-3},{-3,3,-1},{-3,3,1},{-3,3,3},{-1,-3,-3},{-1,-3,-1},{-1,-3,1},{-1,-3,3},{-1,-1,-3},{-1,-1,-1},{-1,-1,1},{-1,-1,3},{-1,1,-3},{-1,1,-1},{-1,1,1},{-1,1,3},{-1,3,-3},{-1,3,-1},{-1,3,1},{-1,3,3},{1,-3,-3},{1,-3,-1},{1,-3,1},{1,-3,3},{1,-1,-3},{1,-1,-1},{1,-1,1},{1,-1,3},{1,1,-3},{1,1,-1},{1,1,1},{1,1,3},{1,3,-3},{1,3,-1},{1,3,1},{1,3,3},{3,-3,-3},{3,-3,-1},{3,-3,1},{3,-3,3},{3,-1,-3},{3,-1,-1},{3,-1,1},{3,-1,3},{3,1,-3},{3,1,-1},{3,1,1},{3,1,3},{3,3,-3},{3,3,-1},{3,3,1},{3,3,3}}

In[]:=

Graphics3D[Point[cube]]

Serhiy Grabarchuk noticed that each of these had six points at distance 6.
This is now called the Grabarchuk graph.

But what does it look like in 2D? Up until a week ago, no symmetric embedding was known.
Then I noticed the “face” vertices (in blue) made the Nauru graph.

And with that, I found an order-4 embedding. It’s always great when you pick a topic for a talk... and one of the most famous mathematicians in the world (Donald Knuth) has been looking at the same item.

Space-filling Polyhedra

Years ago, I knew there were space-filling polyhedra, such as cubes.
The Kepler packing of spheres relates to both the rhombic dodecahedron and truncated octahedron.

There one more parallelohedron as identified by Fedorov, the elongated dodecahedron. Here’s code I’ve rewritten many times.

The Engel-38, Peter Engel’s Geometric Crystallography

Starting about 30 years ago, I knew there were weirder polyhedra.
The record setter had 38-faces. Here was the best available picture:

Later, Branko Grünbaum and G. C. Shephard made a slightly better picture in Tilings with congruent tiles.
Still not enough for me to understand it.

I had no idea how to make a 3D picture of it.

Moritz W. Schmitt, On Space Groups and Dirichlet–Voronoi Stereohedra

A new paper by Moritz W. Schmitt, On Space Groups and Dirichlet–Voronoi Stereohedra, seemed to have most of an answer on page 135.
Within space group IT(214) generating grid point (427/6984, 761/6984, 1421/6984), gives the 38-sided space-filling polyhedron can be generated. The author also supplies the program https://github.com/moritzschmitt/plesiohedron.

I couldn’t get the program to run. So, faced with a tough problem, back in 2017, I asked other people to solve it for me.

In 2022, Anders Kaseorg posted an answer for the vertices.

Plesiohedra Code

So, I wanted code for generating all the plesiohedra in the Schmitt paper. I figured maybe I could browbeat an AI into writing the code for me. And since I also had some finalized answers, I could check for correctness. First off, I got data for 3903 plesiohedra from the Schmitt paper. ( See https://community.wolfram.com/groups/-/m/t/3688732 . )

I could open up that compressed form. Imagine 50 pages of code.
MaXrd is good package of programs for space groups, as I mentioned.

How many rotations are in each space group? I needed to know that later on.

JUST IN CASE

A lot of this data is large. Just in case I need to shrink this notebook, I put code and data at Wolfram Community,
Spacefilling Polyhedra, https://community.wolfram.com/groups/-/m/t/3688732 .

Back to talking

A lot of this data is large. Just in case I need to shrink this notebook, I put code and data at Wolfram Community,
Spacefilling Polyhedra, https://community.wolfram.com/groups/-/m/t/3688732 .

The following means that I found 16 4-sided space-fillers and 1 38-sided space-fillers:

Engel 38-sided space-filling polyhedron

The last one is Engel-38

This is the same one I made a post about. https://community.wolfram.com/groups/-/m/t/2617634

So, to solve this, I used the tried and true method of giving up and asking for help: 38-sided space-filling polyhedron. After five years, Anders Kaseorg posted an answer with approximate coordinates. Turns out the Moritz Schmitt method worked. With those approximate answers, I was able to use various WL tricks to get exact coordinates. The resource functions IntegerChop (multiply by 6984) and ReflectPoints helped to create 38 midplanes.

And we’ll get back to that. Now that I had this one, I’ve wanted to use the Schmitt paper to generate ALL of the plesiohedra.

Tetrahedra in Schmitt

The tetrahedra found in the Schmitt paper.

We can find their tiling cells.

Space-Filling Tetrahedra

Already, we seem to be missing some.

There are five known space-filling tetrahedra when mirror images are not allowed. Hill (1896), Baumgartner (1968), and Sommerville (1923) each enumerated four of the five.

Goldberg (1972) was the first to list all five. He also studied dissections of triangular prisms and found three basic ways in which a prism could be split into identical tetrahedra, if mirror images are allowed. Each of these three methods leads to infinite families of tetrahedra. All three methods are represented in the five tetrahedra listed here.

Whether other space-filling tetrahedra exist is a long unsolved question that dates back to Aristotle.

Five sided space-fillers

There are a lot of repeats in the data

I need to do another overnight run to add transition cells and max dihedral angles.

Triangles and Quadrilaterals

Any triangle can tile the plane. Drag the blue points to make a new triangle.

Six-sided space-filling polyhedra

We get a lot of cuboids in the full list

The 4th one is wrong.

The Michael Goldberg Hexahedra

In the 1970’s, Michael Goldberg tried to enumerate the polyhedra of various types. Here are his hexahedra.

Pentagons

On July 29, 2015, a 15th type was announced by Casey Mann, Jennifer McLoud, and David Von Derau. This Demonstration gives exact solutions for all 15 families.

Seven-sided space-filling polyhedra

Two of the tiling pentagons are sporadic,

Because of similarity problems, we only select those with a different graph structure. I should have kept track of dihedral angles.
I could have shown a much longer list to illustrate how many of the pentagonal prisms are missing.

I also need to keep track of the number of group elements and favor the smaller cells.

The Goldberg Heptahedra

From 1976, the Goldberg list of heptahedra.

Eight-sided space-filling polyhedra

Because of similarity problems, we only select those with a different graph structure. I should have kept track of dihedral angles.

Goldberg’s space-filling octahedra

Goldberg’s 1979 listing of octahedra.

Dihedral Angle

Since I’ve mentioned dihedral angle a few times.

Those faces look pretty flat.

But apparently not quite flat.

Nine-sided space-filling polyhedra

Because of similarity problems, we only select those with a different graph structure. I should have kept track of dihedral angles.

Hilbert’s 18th: Building up of space from congruent polyhedra

1. How many space groups are there? Answer: 230
2. Are there tilings where two of the tiles cannot be related by the underlying group?
3. Densest sphere packing.

A few answers are known for #2, but there is barely a start on a complete solution.

There are two solutions for #2 just in the pentagons. Here’s the image from Forbes ... by me.