Two years ago, in March of 2023, an unlikely team of mathematical hobbyists announced the discovery of a remarkable 13-sided polygon that they nick-named the "Hat". Today, a Google search for the Hat's more formal name, "Aperiodic Monotile", yields dozens of links.

This blog post is about the Hat and the resulting polyshape object.

Contents

An Aperiodic Monotile

The authors of the paper that announced the discovery of the Hat are David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss. The announcement was made in an archive>math preprint with the title "An Aperiodic Monotile".

Within days of the announcement, articles like this one in Science News appeared. Two months later, Florentin Waligorski created an origami Hat.

The official paper was published in June of 2024 in the journal Combinatorial Theory. It was also titled "An Aperiodic Monotile".

I first heard about the Hat is an article by Erica Klarreich in the magazine Quanta, Hobbyist Finds Math’s Elusive ‘Einstein’ Tile.

Tilings

Quoting Wikipedia:

  A "tessellation" or "tiling" is the covering of a surface,
  often a plane, using one or more geometric shapes, called "tiles"
  with no overlaps and no gaps.

  A "periodic tiling" has a repeating pattern. Some special kinds
  include regular tilings with regular polygonal tiles all of the
  same shape, and semiregular tilings with regular tiles of more
  than one shape and with every corner identically arranged.

  A tiling that lacks a repeating pattern is called "aperiodic".
  An aperiodic tiling uses a small set of tile shapes that
  cannot form a repeating pattern.

Tridecagons

A tridecagon is a polygon with 13 sides, like this gold coin from the Czech Republic. The sides of a regular tridecagon are all the same length. Any attempt to tile your floor with these coins inevitably has gaps. Regular tridecagons cannot tile the plane. The Hat is an irregular tridecagon that can tile the plane.

Penrose Tiling

Roger Penrose is a Nobel prize-winning mathematician and physicist. Among his many achievements are the Moore-Penrose pseudoinverse and the Penrose tiling. The Penrose tiling uses these regular quadrilateral tiles, the "kite" and the "dart", to produce an aperiodic tiling with two tiles. The Hat generates an aperiodic tiling with a single tile.

Steve Eddins is a FOC-winning French horn player. Among his many achievements is a Cleve's Corner about this Penrose tiling.

Reflections

We say that the Hat on the left is reflected and the one on the right is not reflected.

Monotile Tiling

Here is (a finite piece of) an infinite, aperiodic tiling of the plane using only the Hat and its reflection. This figure is one half of figure 2.12 from the original preprint, "An Aperiodic Monotile". Each dark blue tile is surrounded by three light blue tiles. The white tiles appear alone or in pairs. If you look carefully, you can see the grey tiles form filaments. The filaments are more apparent in the other half of figure 2.12 on page 20 of the preprint.

The dark tiles are not reflected; all of the other tiles are reflected.

Polyshapes

Quoting the MATLAB documentation for the polyshape object.

  The polyshape function creates polygon-like shapes from 2-D vertices.
  However, unlike polygons, a polyshape can have discontiguous regions
  and holes. The properties of a polyshape object describe its vertices,
  solid regions, and holes.

All of the figures after this point in the blog post were made with polyshape/plot.

We begin with a pentagonal polyshape made from a portion of a regular hexagon.

The Hat itself is a polyshape formed from the union of four rotated and translated copies of the pentagon.

Level 0

We have experimented with a tiling created by expanding rings of unreflected hats centered around a single reflected hat. We stop after three rings because additional reflected hats are needed to continue.

The zeroth level is a single reflected hat.

Level 1

The first level adds a ring of three hats.

Level 2

The second ring has nine more hats.

Level 3

The level 3 ring has 18 hats.

Numbers

Our Hats program allows you to move a hat around the screen with your mouse. When you get close to another hat, numbers appear to guide your final approach. Here are vertices 8, 9 and 10 on hat number 6 near vertices 12, 11 and 10 on hat number 2.

This crowded figure shows all the hat numbers and all the vertex indices at level 2.

Level Color

Each level has a single color.

Compare this with the detail in figure 2.12 of "An Aperiodic Monotile".

Convex Hull

Convex hull is one of many other methods available for polyshape objects.

Get the MATLAB code

Published with MATLAB® R2024b