Exploring The Einstein Problem Thesis submitted at the University of Leicester in partial fulllment of the requirements for the degree of Master of Mathematics by Zoe C. Allwood Department of Mathematics University of Leicester May 2014 Contents Declaration iii Abstract iv Introduction 1 1 Describing the Tile 4 1.1 Lines and Flags . 4 1.2 Lines and Coloured Diagonals . 5 1.3 Other Representations . 6 2 Proof of Aperiodicity 7 3 The Substitution Tiling 12 3.1 Simplifying the Tile Set . 13 3.2 The Substitution . 17 3.3 The Substitution on the Simplied Tile Sets . 19 3.4 Amalgamation . 22 3.5 Proof of Aperiodicity . 22 4 A Better Einstein 23 4.1 Unconnected Tile . 23 4.2 3D Tile . 24 4.2.1 Construction . 24 4.2.2 Issues . 26 i 4.3 Schmitt, Conway and Danzer Tile . 26 4.4 Issues . 27 4.4.1 Height . 27 4.4.2 Tile Orientation . 27 4.4.3 Aperiodic Layers . 28 4.5 Danzer Implementation Attempt . 28 4.5.1 Rotation of 2π .............................. 28 3 4.5.2 Irrational Rotation . 30 4.6 Aperiodic Stack . 31 4.6.1 Thue Morse exploration . 32 4.6.2 General explanation . 33 4.6.3 Two dimensional exploration . 33 5 Conclusions 34 5.1 Socolar and Taylor Tile . 34 5.2 Einsteins and Dimension . 35 Bibliography 36 ii Declaration All sentences or passages quoted in this project dissertation from other people's work have been specically acknowledged by clear cross referencing to author, work and page(s). I understand that failure to do this amounts to plagiarism and will be considered grounds for failure in this module and the degree examination as a whole. Name: Signed: Date: iii Abstract The Einstein problem is an attempt to nd an aperiodic prototile set that consists of a single tile. After some basic denitions and a more detailed explanation of the problem we look at the tile that ts this denition the best, created by Socolar and Taylor. The rst chapter is an explanation of two of the ways this tile can be dened. The second chapter is a proof of the non-periodicity of all tilings created by this prototile based on the hierarchical structure of them. The third chapter is looking at the tile set as a substitution tiling creating an easier way to make tilings in a computer and also show all tilings are non-periodic more easily. The fourth shows other constructions of the tile, including a three dimensional one, and explorations into trying to improve the tile by trying to remove some of the issues with the tile in its current forms. iv Introduction The Einstein problem is very simple to explain but very hard to nd a solution to, like Fermat's last theorem. It is the problem of whether a single aperiodic prototile exists. The name comes from a play on words as Einstein in German can be translated roughly as "one stone". As Fermat hinted in the margin of a book regarding his problem, there is quite possibly a simple and rigorous way to nd solutions to this type of problem, but at the moment we are working with trial and error. If a solution, or groups of solutions, can be found it will have a profound eect on the understanding of certain crystalline structures which are aperiodic and can be understood better in terms of these tilings. The current solutions to this problem are not complete; there are many ways that the current tiles can be improved. In this work we explore a tile that is a good Einstein, found by Socolar and Taylor [2, 3], show it is aperiodic in a way that is complementary to the original proof. We will then discuss the issues with it and try and look at ways to improve it. Basic Notions These are some denitions that we are going to be using throughout this work. The general details of tilings are explored thoroughly by Grunbaum and Shephard[1]. n Denition 0.1. A tiling is a subdivision of R into pieces called tiles. These tiles only n intersect at their boundaries and their union is R . Tiling is most commonly done in 2 dimensions, tiling the plane, but an equivalent is possible in any dimension. In one dimension line segments can be used to `tile' the line. A tile can be given a label, letting you distinguish between tiles of the same shape. 1 Thinking of these labels as colour would dierentiate between a simple square tiling and a checkerboard tiling. Denition 0.2. Prototiles are the set of tiles that make up a tiling ignoring position. For the checkerboard tiling the prototile set would be a black square and a white square. Another way to think of a tiling is as a collection of translates, translations and/ or rotations, of a prototile set so that each translate only intersects with others at its n boundary and the union of the collection is R . Denition 0.3. Matching rules are the guidelines for how tiles are allowed to be placed in regards to each other. The rule that tiles must meet full edge to full edge is an example of a matching rule that is generally applied to tilings. The two ways to force a matching rule are by markings on the tile or by the shape of the tile. The relationship between markings on the tiles can be dened in any way that creates the tilings wanted and most of these relationships can be dened instead using shape, though this can create very complicated tiles when dealing with complex matching rules. Denition 0.4. A patch is a nite section of a tiling. If you have a nite set of protiles and a set of matching rules, for example having to meet full edge to full edge, then you can only create a nite number of patches. This number generally increases very quickly as you increase the possible size of the patch but at any nite size there are only a nite number of possible patches. This is called having nite local complexity. Denition 0.5. A tiling is called periodic if the whole tiling is created by regular repetition of a nite patch of the tiling. This means the placement of one version of this patch can completely dene the position of all other tiles in the tiling. It is called non-periodic if this is not the case. Using the example of the checkerboard tiling, you can determine what colour of tile will be a certain distance from a known tile in the tiling without having to work out all the tiles 2 in between because of the regularity. If you took a standard square tiling and randomly split each square into two triangles by adding in one of the diagonals there is no way to deduce which diagonal was added using already known ones as it is completely random. This created tiling is therefore non-periodic. Though there is not any regularly repeating pattern, because the tiling has nite local complexity there are only nite patches of every size. Because of this there will be innitely many copies of every patch, but they will not appear with any regularity. There are many prototiles sets that can produce some periodic tilings and some non- periodic tilings as well. If you add the same diagonal to every square, instead of a random choice, you would get a periodic tiling from the same triangular prototile as the non-periodic tiling already dened. Denition 0.6. A prototile set is called aperiodic if every tiling it can create is non- periodic. These are the prototile sets we are interested in. A famous example of these are the Penrose tilings but with these there are always at least two dierent tiles in the prototile set, so they are not an Einstein. Using the denitions above an Einstein is an aperiodic prototile set consisting of a single tile. This means that using a single tile you can cover the entire plane without ever having a regularly repeating pattern. Because of the nite local complexity every patch of the tiling will repeat but this will never happen regularly. The tile we will describe below ts this denition. 3 Chapter 1 Describing the Tile Though Socolar and Taylor have dened a single tile, created from a hexagon, it can be useful to describe it in dierent ways. The matching rules cannot be dened simply by shape, unless the tile becomes unconnected or you use a substitution with multiple tiles. Also you need to use the fundamentally dierent mirror image to successfully tile the plane; you do not need this if you make the tile three dimensional. All these interpretations are explained in this work. Any pictures that do not have a reference were created for this work. 1.1 Lines and Flags The prototile is a hexagon with markings to enforce local matching rules. There are black lines, two short lines around the top and bottom vertexes and a long one connecting the two remaining edges o center. The distance between the end of the short black line and nearby top or bottom vertex must be the same distance as the end of the long black line and the top vertex of the edge it is touching. At every vertex there is a ag. The top and bottom ags always point the same way and the other four ags point away from the long black line. As the long black line is o-center of the edges it touches this prototile is chiral, its mirror image is fundamentally dierent to itself.