Einstein Hat Awards (entries 72-44)
Thank you to our sponsors, XTX Markets, UK Maths Trust, National Museum of Mathematics (MoMath.org), Amplify, G-Research, Jane Street and Dexter and Deborah Senft.
Contributors - Geoff Smith, Simon Coyle, Samuel Monnier, Dianne Flatt, Cindy Lawrence, Chaim Goodman-Strauss, Guillermo Acevedo, Kit Reagan, Hayley Richardson, Philipp Legner, Craig Kaplan, Robert Fathauer, Yoshiaki Araki, Dexter Senft, David Smith and Ewart Shaw.
72 Peter Larsson 47 USA
My submission is my living-room wall. This was done by hand and took a week to do from start to finish.
Philosophy:
I'm charmed by the Einstein's tile's simultaneous simplicity and complexity. By applying the shape in its purest form I can create a pattern that is inherently chaotic but patterns emerge naturally.
This makes for a cosy space.
The colours were chosen to reflect the purity of the pattern (red, green, blue) and the background and hues of the colours to make it look natural and earthy.
Process:
1. Use https://cs.uwaterloo.ca/~csk/hat/app.html to generate a correct pattern. One image for the left part and a separate image for the right part (while ensuring both combine correctly in the middle).
2. Take the two images and digitally re-colour the tiles to make the colour layout look random and evenly distributed.
3. Use a projector to project the image on the wall.
4. Buy a set of wooden Einstein tiles of the right size on Etsy.
5. Use the wooden tiles to make stencils out of Foam Sheets
6. Use the stencils to paint each Einstein tile on the wall in the colour according to the projection.
7. Repeat for the left and right part of the wall, aligning the middle to blend correctly.
71Emma Laughton 65 UK
The quilted patchwork wall hanging shows a tessellation of hats which are all hand sewn by the English paper piecing method. The image size is height 66cm x width 69cm while the height including the hanging pole is 76cm. The patchwork consists of approximately 1200 ‘mitre’-shaped individual pieces. Each mitre shape is a third of the area of a regular hexagon with sides 2cm, and each of the 300 hats is composed of four of these pieces. The colours I’ve included are shades of red, blue, yellow and green using 33 different cotton fabrics. The colours are largely allocated according to the underlying structure of supertiles, however there is an energetic burst of yellow and red near the centre; while four large triangular areas and a border are gently suggested by the use of different blues and yellows. The design aims to please the eye with a combination of elements of repetition and apparent (near) symmetry, contrasting with the unpredictable overall aperiodicity.
70 Alec Sparks 29 USA
I propose to develop a system of theater stage platforms based on this tile shape.
Most theaters keep a stock of square or rectangular wood platforms in storage to use to build platforms or risers out of. Since they're re-usable, they're cheap and environmentally friendly.
However, if a script specifies a setting that doesn't have right angles, like rocky cliffs or a dark forest, custom-shaped platforming may be constructed to emulate this terrain. After the show is done, these bespoke shapes are unable to be used for another purpose and may be thrown out.
The Einstein 'hat' tile would be a useful tool for stage designers to build organically-shaped sets at a lower monetary and environmental cost, as they can be stored and re-used in many configurations!
I would use the money to make a set of these platforms for The Voxel (voxel.org) theater in Baltimore, Maryland and study their use with the rotating theater companies it hosts. I would also share the design so other theaters can emulate it.
69 Lukas Kavaliauskas 11 UK
No description
67 Will Rocke 31 UK
This painting is my rendition of a spectre tile, where all thirteen edges are of the same length. There seems to still be no reason as to why this is the only known shape to tessellate aperiodically. The concept behind the painting was to evoke the mystery behind the shape.
I used compressed charcoal and acrylic paint on 50x70cm 400 gsm acrylic paper.
I found out about this competition through a colleague at school. I am a 31 year old Maths teacher in Central London who draws and paints as a hobby, hence my interest in the competition. Thinking about it, it could be interesting to see kids investigate this in school.
I found the project intriguing.
66 Ben Edwards 16 UK
See all the source code (written in Python) alongside further explanation of this entry here: https://github.com/Ben-Edwards44/Aperiodic-Monotile-Walk
This entry uses the positions of tiles from the weakly chiral aperiodic monotile Tile(1, 1), or 'spectre' tile, as a sort of pseudo-random number generator. The image shows a collection of lines, each of which are drawn one after the other and at an angle between 0 and 360 degrees. This angle is generated pseudo randomly from the positions of tiles from the spectre tiling. The colour of each line is also determined by the angle it is drawn at: lines with angles between 0 and 72 degrees are one colour, lines drawn between 72 and 144 degrees are another, and so on.
I believe the beauty in the resulting image lies in how, seemingly, random the produced pattern is - but it is, in fact, produced from positions of tiles that are far from random. Also, because the spectre tiling is aperiodic, the shown pattern will - like the original tiling - never repeat. I find this fascinating and believe it adds to the overall intricacy of the pattern.
This is a more detailed explanation of the underlying algorithm to help you better understand this entry:
1. The spectre tiling is produced using a system of tiles and meta tiles.
2. Each time a new tile is drawn, its (x, y) position is added to a list.
3. Once the desired number of tiles have been drawn, the list is iteratively looped over.
4. For each (x, y) position in the list, the product of the x and y positions is calculated modulo 360 to produce and angle between 0 and 360 degrees (angle = xy MOD 360).
5. A line of arbitrary length is then drawn at this angle, starting at the end of the previous line.
6. This is repeated until a line has been drawn for each tile in the tiling.
The shown image displays lines for 34649 tiles (generated using 5 iterations), but any arbitrary number of tiles can be used. Using a different number of tiles would completely change the produced pattern because it would change the order in which tiles are drawn and, thus, added to the list of tile positions.
65 Randhir Chavhan 46 India
Commonality maintained in the form of combinations digit 1 (0.11, 1.1 , 11 etc or multiples) in the creations of various components of the universe. Hence we can say that fate decided for structures ranging from microscopic (atom) to macroscopic (solar system) , aspects of evolution of life, interrelated universal physical constants, etc.
64 Owafa Owajiha Abedin 12 UK
I added embroidery to a fabric inspired by Einstein's hat. I made it by first creating 2 stencils of Einstein's hat out of cardboard. The big one is for the outer shape and the small one is for the tilling on the inside. Then I traced them on piece of fabric before embroidering the entire thing using back stitch.
62 Martin Windischer 39 Austria
This is my take on decorating the spectre tile.
I love mazes and I love tiles and I'm happy that the spectre tile is just perfect for this concept. In the past I tried something like this with the penrose tilings but couldn't get a satisfying result.
By omitting dead ends while allowing circles and by having the pathes cross each other the most obvious maze solution algorithms will often fail. It also creates a nice aesthetics that I'm very happy with. Obviously the aperiodicity adds a lot to the meaning of the maze.
I printed the tiles and glued them on thick cardboard so I got tiles which feel great to play with.
If you simply put the tiles together you will occassionally come to situations where nothing fits and you have to remove some tiles to be able to continue – sometimes you have to remove quite a number of them. Not an easy task.
For example there is an unfillable gap just in the middle of the top edge of my first picture – I realized it only after taking the image.
Having these tiles is a bit addictive. I started by making 60 tiles and after playing for a while I found this number too small. So I made an additional 36 tiles. But I'm again in a position where I want to have more. And I'm afraid this will not change no matter how many I have.
61 David Barros 34 Equador
What if instead of old square pixels we make the using hat pixels instead.
60 William Thuburn 81 Italy
An Art Deco style table with “hat” marquetry top
59 Mohammed Mudasserul Islam 17 UK
No description
58 Milena Hoehn 26 Germany
Einstein Cookie Cutters, so that you only need to roll out your cookie dough once (as long as you have perfectly made cutters and infinite cookie dough)!
57 Dominic Jenner 42 UK
What a wonderful competition it was an absolute delight to come up with ideas for this.
My entry is designed to celebrate the discovery of these monotiles and consists of two parts firstly an infinite zoom art work created by me the first frame of which is uploaded as the first file the complete works can be viewed here:
https://drive.google.com/file/d/1qPqsIG93L0xos_PrAS6IVIWeSVIiBw5b/view?usp=drive_link
Happy to provide the full video if you would like it.
The second image is the front page to a spectre tile
playground of 3D printable games, puzzles and office equipment that is all based around the spectre tile. These can be accessed here:
https://www.printables.com/model/576028-spectre-tile-playground
55 Andrew Trevorrow 67 Australia
I've written a Lua script called HatLife.lua that lets people explore various cellular automata rules on an aperiodic hat tiling. The script is run from Glu, a free application for Windows/macOS/Linux. More details, including links to download Glu and HatLife, are available here:
https://glu1.sourceforge.io/
https://conwaylife.com/forums/viewtopic.php?f=11&t=6015
The first uploaded image shows 6 gliders in a rule called Worms moving away from a small still-life. The gliders (called "worms") are extendable and travel within the wide gaps between two parallel lines of reflected hats.
54 Joshua Wilson 38 Taiwan
The tread pattern on a car tire is designed with several objectives in mind, one of which is to reduce road noise. It is generally undesirable to space grooves evenly in a tread pattern, because doing so would concentrate a great deal of acoustic energy in a narrow range of frequencies. Instead, by placing the grooves in a tread pattern so that there are many different distances between grooves, the resulting road noise is distributed between a broader range of frequencies, and the human ear generally perceives this as quieter.
The concept that I present here takes advantage of the aperiodicity of the tessellation of the hat tile to reduce recurring frequencies within the tread pattern. I have also applied some subtle warping to the tile pattern to reduce the impact of local scale repetition.
53 Jesse Louis-Rosenberg 37 USA
The Spectre Puzzle is a wooden jigsaw puzzle and mathematical toy consisting of identically shaped pieces which tile in a unique way that never repeats. It is based on a recent mathematical discovery, the Spectre tile which answered a long-standing mathematical mystery: does a so-called “einstein” exist? That is an aperiodic monotile, a single shape which tiles the plane but never makes the same pattern twice. Seems impossible? Now you can play with the mind bending shape and discover how it works.
My version of the Spectre tile has curvy, interlocking edges for better puzzling. The set comes with 111 wooden tiles with two different truchet-style patterns printed on them. You can assemble the puzzle as you like or try to figure out the rules to make an infinite tile. By having two different patterns on the tiles, it enables a little bit of creativity in the assembly. If you prefer your puzzle to be patternless, flip the pieces over to construct a plain wood version.
52 Patrick Liddell 44 US
I am a game and graphic designer in Oakland, CA. The very moment I read the monotile article back in March the idea for this game popped into my head!
The gameplay is similar to tabletop classic "Carcassonne", where players take turns laying tiles on a constantly expanding board, and adding a token if they choose. As opposed to Carcassonne, however, players are working cooperatively to win, not competitively.
The geometry of the Einstein hat allows for relatively simple placement rules: in order to legally place a hat tile, at least two sides of equal length must match. As long as players follow that main rule, hat tiling takes place.
Further, the requirement for every 9-or-so hats to be reversed allowed for a fascinating gameplay mechanism. Players attempt to build gaps in their pieces such that a reversed hat would perfectly fill the empty space; once that occurs a reversed hat automatically appears and provides bonuses. Players are flabbergasted by the idea of achieving this at first, but almost always succeed in it during their game as the shape becomes familiar to them.
The theme of the game is one of spreading peace. You play as one of the Cultural Ministers of a Kardeshev Scale Type-III Civilization. Your goal is to spread your art (music, dance, art, or poetry) as you explore the local superclusters. In the process you forge cosmic gardens (reversed hat tiles), birth star children (a result of geometry), and uncover void shards (gaps in the tiles). The players collectively win if they successfully spread their arts a prescribed amount before running out of Universe (tiles).
I have been playtesting (in the parlance of game designers, "protospieling") this game for several months now and everyone loves it! People who are into math really are surprised and thrilled by it. I would like to hope that I will be able to mass produce it in the next year or two! This version in the photos is made from laser-cut wood.
The two photos I included are from a protospiel recently at Pacificon convention in San Jose, CA. I also included a 'sell sheet' which is a rundown of the game that one might send to developers. I do have other photos, of other events, and earlier versions, if those would be of interest.
Thank you!
51 Yossi Elran 56 Israel
Inspired by the discovery of the "hat" mono tile and following the recent discovery of techniques that combine Kirigami with the construction of different kinds of flexagons, I was able to come up with the "hat" hexa-hexa-flexagon. The attached pdf file show snapshots of a portion of the Tuckerman traverse performed on the flexagon.
50 Kevin Horecka 35 USA
My mom is a quilter, and I've been trying to inspire her to use more mathematics in her quilts. While on a vacation 2 weeks after the tile discovery was published, I had the opportunity to take a jewelry making class. I decided to make my mom a replica of the tile as a necklace to hopefully be a nice inspirational reminder for any future quilts she makes that mathematical art is always a fun option.
49 Brian Ennis 43 Ireland
Presenting the Infinite Hat Dungeon... many enter... but none escape!
This is an old school dungeon for Dungeons & Dragons or other roleplaying games.
How I made this:
I constructed a Hat Tile in Adobe illustrator (after sketching it a few times on isometric paper). I decided to make the "exits" to the tile in the centre of each short and long edge. Afterwards, I added a hex grid (proper D&D old school style) and printed the template.
Next, I hand drew the dungeon (as seen bottom left in the second image). I used copic multiliner pens and a lightboard for this. I sketched a layout for the dungeon on a separate page and then inked as you see. After scanning to photoshop, I added the hex grid I had made previously and shaded the "walls" of the dungeon.
I hope you like it (especially if there are any gamer nerds among your midst). I plan to make more variations with different dungeon layouts to use with this.
About me:
I'm a fantasy map maker from Ireland (under the moniker "Jog Brogzin" and an avid fan of Numberphile and Standup Maths. Most of my maps are world maps for fantasy authors, or city / dungeon maps for players of Dungeons & Dragons. I have made the Penrose tiles into a dungeon tile previously and made strange globes from geometric shapes.
Thanks for reading my submission,
Brian Ennis
(AKA Jog Brogzin, intrepid explorer of fantasy worlds)
48 Albin Thorning 43 Sweden
When I first heard of the aperiodic monotile i was so excited :)
I work at a concrete plant, and decided to create something to celebrate the occasion.
I made a 150 lbs reinforced slab in the shape of the monotile, with the kites outlined on one side and the other side flat.
It's currently on display at a math/logic themed park a few towns over, along with a few sheets of information about the mathematical properties of the tile.
47 Edric Haleen 49 USA
Crossword puzzle
46 Theodore Wilson 51 Canada
Use the aperiodic tiles for earthquake, vibration or wave resistant walls or structures. stacked longer versions of these tiles like a pack of pencils or pile of logs may be very resistant to changing loads such as passing heavy trucks or breaking waves.
Blending long and short versions of the tiles may tie back a retaining or sea wall into the earth as is necessary.
45 Kiumars Sharifmoghaddam 35 Austria
The image shows a folding sequence of two rigid-foldable origami tubes with einstein tile cross-section and a zig-zag trajectory, to a flat state, while they are in contact. It is a digital image rendered by Rhino software and generated by a Grasshopper plugin I developed. It is based on a novel generalization of rigidly foldable origami tubes, I did as a researcher at Vienna University of Technology. The research has been published in the Journal of Mechanics Research Communications, very recently.
Link to the article:
https://www.researchgate.net/publication/371880874_Generalizing_rigid-foldable_tubular_structures_of_T-hedral_type
DOI: 10.1016/j.mechrescom.2023.104151
Einstein tile, as a very interesting example, also belongs to the class of valid cross-sections of T-hedral tubes. Specially, because a meta-material with einstein tilling could exhibit extra out-of-plane stiffness, due to its irregularity and asymmetry of the covering. Another observation is that the 2d tiling as a system of bars and joints (for edges and vertices of the tiles) can flex, at least with one degree of freedom, even for infinite tilling.
Note that, I have found this example, after publishing the article. Therefore this very example and its possible applications are not published yet.
Extra images, and animations can be provided.
44 Francesca Carter 32 UK
I'm a first-year PhD student working in the field of aperiodic tilings, and I also came out almost exactly when I started my PhD as a transwoman, and my studies have always deeply linked closely to my struggles with my gender identity. When I was invited to Hatfest earlier this year, I knew I wanted to wear something mathematics-themed, and this quickly evolved into the idea of making a custom dress - one with the Spectre Tiling pattern no less.
It was only about 1.5 months away from the date when the idea coalesced into what you see now, a fabric made of the spectre tiling, in the colour palette of the trans pride flag. We had to work absurdly quickly to get it finished in time, ordering samples, creating algorithms to correctly colour the tiling, deciding on a dress style, getting fitted, and so on. It was an absolute rush!
Literally the day I left for Hatfest, the dress was ready! With no time for a test fit, I just had to take the bundle with me and pray it looked good, and it looked *amazing* :) You can read the whole story of its creation here
https://mathstodon.xyz/@FrannaCotta/110774207144411188
