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 features games and puzzles focusing on tilings.

The Penrose Tiles (January 1977)

The Hat (announced 20 March 2023)

https://arxiv.org/abs/2303.10798
https://cs.uwaterloo.ca/~csk/hat/
It’s a rather complicated tile.

What is periodicity? Triangles

Out[]=

What is periodicity? Quadrilaterals.

Out[]=

What is periodicity? Pentagons.

Out[]=

Ponting Square Packing

Melting tile

It’s possible to remove a tile.

Freese’s Dissection of a Regular Dodecagon into a Square

What is nonperiodicity? Irregularity.

What is nonperiodicity? Substitution Tilings

What is nonperiodicity? Pinwheel tiling

What is nonperiodicity? Isopenta

Anyone recognize this?

Penrose Tilings and Wieringa Roofs
Traditionally, the P3 Penrose tiling is made of thin and thick rhombi. However, by raising the vertices in three dimensions, it is possible to force all of the rhombi to be congruent. The resulting surface is known as a Wieringa roof. Due to similarities with three-dimensional quasicrystals, you can see rhombic triacontahedra and hexecontahedra hidden in the tiling.

Prior shapes are made from triangles and quadrilaterals.
Can we get non-periodic from pentagons?

What is non-periodicity? Rep-Tiles

What is non-periodicity? More Reptiles

What is non-periodicity? Even More Reptiles (order 5)

https://demonstrations.wolfram.com/RepTilesAndFractalsOfOrderFive/

Substitution Tilings

https://demonstrations.wolfram.com/SubstitutionTilings/

Aperiodic Tilings

A tiling set is aperiodic if
1. A non-periodic tiling is possible.
2. Arbitrarily large periodic regions or patches are impossible.

Aperiodic Tilings -- Robinson Tilings

Aperiodic tiling showing hierarchical structure, published by Raphael M. Robinson in 1971.

Code 686 Builds the Chair Tiling

https://demonstrations.wolfram.com/Code686BuildsTheChairTiling/

Let’s get back to the Hat (or ein stein)

It’s a single tile with a non-periodic tiling.

Enter David Smith (Nov 2022))

A retired print technician and shape enthusiast. He’s the one that identified the Hat.

Jaap’s PolyForm Puzzle Solver

“I think I found one that won’t tile.”
https://www.jaapsch.net/puzzles/polysolver.htm

The Good Fences App (Craig Kaplan)

https://play.google.com/store/apps/details?id=net.isohedral.goodfences&hl=en_US&gl=US

First pictures

At this point, they were still thinking of it as a Heesch problem.

IT’S A SHAPE JIM, BUT NOT AS WE KNOW IT (Dave Smith’s Site)

https://hedraweb.wordpress.com/2023/03/23/its-a-shape-jim-but-not-as-we-know-it/

Dave Smith - “I found another one. Look familiar?”

An aperiodic monotile (20 March 2023)

https://arxiv.org/abs/2303.10798
David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss
A longstanding open problem asks for an aperiodic monotile, also known as an “einstein”: a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the “hat” polykite, can form clusters called “metatiles”, for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

CITE THIS NOTEBOOK

Mathematical Games: The Hat and Other Tilings​
by Ed Pegg
Wolfram Community, STAFF PICKS, May 8 2023
​https://community.wolfram.com/groups/-/m/t/2916247