Einstein Hat Awards (entries 241-215)
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.
241 Marc Geller 48 USA
"(pro)spectre"
240 Sophia Nichols 23 USA
For my submission, I have created a Chinese dumpling folded in the shape of an einstein hat. The shape is created from a single uncut sheet of homemade wonton dough folded around a traditional meat and vegetable filling, held together using only water.
I grew up folding wonton wrappers into dumplings. My mother, my sister, and I spent evenings tessellating many trays with squat dumplings ready to be cooked or frozen. The shapes my mother taught me to fold have no formal names. Like the hat and spectre tiles, the shapes of dumplings can only be described through squinting, head-tilting metaphors: a purse, a fish, a bishop’s mitre.
The kitchen table was where I first felt that math is the language of the universe. Shaping perfect, identical dumplings from scratch requires engineering, spatial thinking, and precise reasoning. As a dumpling takes shape under my hands, it’s easy for me to believe that this is what makes us human— not just to create but to engineer. Maybe that’s why every culture has a traditional dumpling. From the samosa to the pierogi, traditional dumpling folds demonstrate our species’ affinity for engineering, design and form.
But for all the dumplings across humankind, I don't think I'll ever have one the same as the ones my mother makes. The dumpling is a single food, multiplied infinitely, yet never identically repeated.
238 Tulio Koneçny 22 Brazil
What would it be like if instead of the great pyramid of Giza there was a great Einstein Mad Hat? In this work of reimagination, I sought to surpass the technology of the time, with representations of the wonderful world of tiling: Escher arts, Archimedean tiling and other forms.
Digital art made in Procreate
236 Julie-Anne Castonguay 32 Canada
This is a quilt top made of einstein tiles, or rather made out of simpler tiles, composed of two kites each, using color to make out each hat. The mirrored tiles are indicated by the purple pieces.
It is entirely pieced by hand using a technique called English Paper Piecing, or EPP, where each piece is attached around a piece of paper before being sewn by hand to the others, and the paper removed as the pieces are all enclosed. This ensures crisp and precise edges.
Quilting by hand is most often a matter of years of work, so this one lives as a wall hanging for now, but it still has the potential to become a full-size quilt.
235 Agnes Grinchenko 39 USA
There's no better way to celebrate The Hat Tile than with this delectable dessert! These sweet sugar cookies in the distinctive hat shape may be infinitely tiled to make an endlessly sharable tray of sweets. The hat tile-shaped cutter was 3D modeled, sliced, and 3D printed in an at-home workshop. Then the cookie cutter was polished up with sandpaper and cleaned for food safety. Using this self-made Hat cookie cutter, a typical batch of sugar cookies was turned into a beautiful mosaic of Einstein hat cookies, ready to be enjoyed and shared.
234 Skona Brittain 69 USA
Within a day of the arxiv announcement, Robert Fathauer had produced tessellation artwork, where he sketched the inverted hat as a shirt, which I actually thought the shape resembled more than a hat. See https://twitter.com/RobFathauerArt/status/1638244474349555714
Within another day, I'd decided to create a real shirt that looked like it.
Being lazy, I mean efficient, I preferred to modify an already-made shirt, so I scoured the thrift shops looking for a plain short-sleeved button-down, but after weeks of no success, resorted to a shirt I'd outgrown decades ago.
I've re-inverted the shirt image since women's shirts button in the opposite direction from men's.
The first step was to move the buttons to be on a slant. Then I hemmed the straight side, which was easy, and then the pointed side, which was the hardest step. Finally, I modified the collar to be asymmetric.
Although it is tight on me, I have since worn it to all of the many monotile talks I've attended, including Chaim Goodman-Strauss's at the MOVES conference in August.
However, for the the contest photographs, I asked one of my thinner students to model it.
Before the spectre was announced, people tried to order them from me but luckily I didn't take any orders.
233 Liam Thomas 19 UK
A computer program that takes in an image as an input (such as the first image) and produces a "hatified" version of the image (such as one half of the second image).
Firstly, it generates a large enough H-Supertile, from smaller supertiles and metatiles as described in the arXiv paper, until the patch of hat tiles covers the entire image (depending on the resolution chosen in the program settings). Then the hats are overlaid on top of the image, and the modal pixel colour of the image pixels within each individual hat is calculated, and finally the colour of all pixels within a hat is set to be the modal pixel colour for said hat, such that the new image constructed consists only of solid-coloured hat tiles. Optionally, it can also produce a version with borders around the hat tiles (as seen in the second half of the second image) so it is easier to identify the individual hat tiles. The size and resolution of the output image can also be configured.
232 Megan Mealey 24 USA
I made a set of crochet hexagons that can be joined together to make the hat tiles. There are 14 unique hexagons for the coloring I chose. The yellow/gold was chosen to make the reflected hats stand out. I crocheted all of these by hand with cotton yarn and a 1.5mm hook.
231 Lisa Peterson 53 Canada
A set of modular ceramic tile candle holders that can be arranged side-by-side or stacked to create a single candle holder
The temple of the spectracleer, where the oracles divine by proof what beautiful mathematical concept will come next/ what follows from what is known.
229 Chris Nho Public Math 33 USA
For our Einstein Mad Hat Competition submission, we wanted to prompt young learners to play with the hat tile, and to express their questions and ideas.
To that end, we mailed sets of laser-cut paper hat tiles to teachers and others who shared them with learners of many ages. We asked those recipients to collect their learners’ questions and observations about the hat tiles, and then we assembled the submissions into a book.
The first draft of this book contains the initial submissions we received. We will complete a final draft of the book, and share it for free on the Public Math website. We will also take pre-orders and have a number of these books printed and shipped to interested parties at-cost.
228 Katia Francisca Nellen Mondragón 23 Mexico
This is the hat made out of hats. It's a cross between a beanie and a beret, and it's 100% aperiodic!
The individual hats were hand knit based on a modified version of the pattern "Aperiodic 'Einstein' Monotile" by Brent Annable on ravelry.com.
The hat is made out of twenty-five hats, three of them are reflected hat-tiles, depicted in red.
It is a very close replica of a patch of the hat tiling, with slight distortions near the red edge, to fit the shape of a sphere.
Periodically thinking about your problems? This hat can help you do it aperiodically, too.
227 Mateo Cely 38 Colombia
These pictures show the result of a project developed during the architectural studio "Materialidades Emergentes" at Universidad de los Andes, in Bogotá – Colombia.
The project, which is called "Teselado Wo’ Usü" aims to upscale a traditional local material as clay, using technology and biodesign. The main focus, is to address environmental and local development issues.
In this case, the replacement vernacular architecture from the desertic area of la Guajira for generic materials imported from other regions, creates bigger problems such as the dependency of big companies that produce generic materials and its oil dependent transportation, the loss of local knowledge about raw materials, and the lack of bioclimatic design due to the restriction of mass fabricated materials such as concrete bricks.
The project avoids nostalgia form the past and looks for the exploration on the sustainable use of raw materials, and the creation of a new language adapted to the local needs. The result is a clay tile that uses The Einstein Mad Hat pattern, that can be adapted to be used in facades and exteriors, creating very different results in architecture. Plus, the inner pattern that was designed using indigenous patterns is covered in a bioluminescent bacteria obtained from a local jellyfish that adds a symbolic component to the whole composition.
The project is a collaboration of the professors Mateo Cely and Juana Leal, with the group of students Juan Camilo Chaparro, Valentina Manotas, Santiago Sierra and Sergio Sierra.
225 Rebecca Klemm 73 USA
Having created lessons in edible polygon puzzles, it seemed natural to design a Hat puzzle linked to the Four Color Theorem in a manner different from using each kite as the map pieces. The design consists of 12 hexagons deconstructed to create the Hat as an island inside a complete black frame created by kites of the nearby hexagons. The other two colors complete nine hexagons that surround the three used to form the Hat so that no two neighbor pieces are the same color. Since the submission is around Halloween, it made sense to use all four common colors of that spooky day.
The illustration shows the underlying kites to demonstrate the design and how they complete the 12 hexagons. As a map, those internal segments should disappear. In the picture of the cookie, frosting hides the underlying cookie cuts to create larger individual map pieces.
224 Jamie Mitchell 22 USA
When I first saw the Hat, I noticed the quadrilateral (kite) tiles, each as 1/6 of a hexagon. I immediately thought of bees as they live in honeycombs formed by hexagonal cells. Wouldn’t it be fun if the worker bees changed the walls to Hats and surprised the Queen Bee?!
223 Pierre Broca 32 France
As a hobby, I do paper artworks. Paper volumes, on canvas.
This is my take on the Einstein Hats, in gold, and the surroundings shirts.
I usually enjoy working with hexagons, and these shapes were more of a challenge. However I find this fascinating for the future, just as Penrose's tilings have been.
Einstein's Hat, 30×40 cm, paper and paint
222 Daniel Rollan 21 Spain
The ability to tessellate the plane but only in a non-repeating way presents a fantastic opportunity to help our planet and its great natural spaces. Environmental protection organizations can add special protection zones, to a map and offer donors to sponsor unique tiles over mountains, rainforests or coral reefs for example. These tiles can be added as needed thanks to the properties they posses and everyone can later see their unique tile on a map representing everyone's.
221 Neelkantha Mukherjee 16 UK
220 Robert Durant 26 Canada
In this tabletop game (titled “Einstein Mad Hat Chase”) players collect and place hat tiles on the board to form paths they can travel along to collect pebbles. Players aim to collect five pebbles before their opponent does. While both players are moving around the same space and are competing for the same pebbles, each player has different paths available to them (based on the unique tessellation in their colour). Additionally, despite the pieces players are able to play being open information, each player does not know exactly how the other player is able to play the pieces down!
The game uses the “hat tile’s” unique unpredictable tessellation pattern to give information about a game state, without making the game state completely solvable. Game states like this are used in bluffing games like Poker, where you may have an idea of what hands your opponent can create (based on the probability of certain hands and how they are betting), but you can not deduce with full confidence what their hand might be. This can create interesting decisions for players to make, where they must wager on what their best move is rather than being able to fully calculate it. Unlike Poker though, “Einstein Mad Hat Chase” allows each player to see each other’s hand of tiles, so it is the tiles unpredictable tessellation pattern that creates the uncertainty.
The game accomplishes this by overlaying two different hat tile tessellations in the same 2D space, but only allowing each player to see one of the tessellations (in the digital prototype (https://steamcommunity.com/workshop/filedetails/?id=3063197996) this is done digitally, but this theoretically could be accomplished in physical space using lenticular printing). In addition to the tessellations, there are certain points on the board that are selected that both players want to travel to. Players must use paths on hat tiles to move, and these tiles must be placed such that they align with the pattern the player can see on the board, which importantly, does not align with what their opponent sees. This means that while you have a rough idea of how far a particular point on the board is from your opponent, due to not seeing how they are able to place their tiles, and the pattern in which tiles can be placed being unpredictable, it is not possible to fully deduce how many tiles your opponent needs to get to any given point. This allows players to wager whether they want to collect pebbles as quickly as possible, or try to collect them in one large move (which is more efficient), but risk their opponent collecting them before them.
This game would not work with tiles that have a predictable tessellation pattern (triangles, squares, hexagons) as it would be possible to fully deduce all possible paths your opponent could create with their pieces using these shapes (removing the wagering aspect of the game). Something like this theoretically could be achieved with other unpredictable tessellations involving multiple different tiles, but there is an elegance in all tiles having the same shape. In physical space, all tiles having the same shape allows the order that you acquire them to be easily be randomize, as they can easily be shuffled together (which is not easy with different shaped objects), or they can be drawn from a bag (the way in which different shapes sit in a bag would influence what is drawn if they were different shapes).
The game is completely playable, but it should be understood that the game is fairly bare bones and serves as more of a “tech demo” of a particular game mechanism a hat tile can uniquely create.
219 Iris Hebbeker 22 Germany
Material: Pencil on toned paper
218 Liina Roots 36 Estonia
Project name - perfectly disorganized kitchen.
It all started when I needed a backsplash for my kitchen counter. I decided to make the tiles myself as I was fascinated with the newly discovered aperiodic monotile. No seeming order, yet gets the job done perfectly - just the vibe I wanted my personal space to convey. Then I got drawn into the fantastic local patterns these tiles can create, inspired by the swirly lines on @ateldsign on Printables. I used the spectre tile in order to optimize manufacturing ease. The design I chose has distant influences from prehistoric rock carvings around the Baltic sea region, reminiscent of the robust animals, boats and tools scattered at all angles. It has different themes depending on whether you look at a single tile or the surface. A single tile looks like a turtle, but zooming out you can see rather odd patterns of snakes and eggs or eyes. Even further you start to see purely geometric spirals. There is a special three-eyed egg from which a quasi-symmetry emanates - there is a threefold rotational symmetry around this point.
Unfortunately I became aware of this competition so late I was not able to manufacture and install the clay tiles yet - they are at the bisque firing stage as of now. I present the 3D-printed mock-up instead, which will be used by the undoubtedly amused and confounded tiler to test the pattern before setting it on the wall.
217 Sofia Dey Choudhury 11 UK
My complete submission for the competition is in the link below. The uploaded images are only screenshots from the video itself.
https://vimeo.com/878099048/efbf694f9d
216 Sophie Bean 25 USA
Cookies.
215 Manisha Sharma 13 UK
I have made a clock using clock face using aperiodic monotile design. The design incorporates the aperiodic monotile using regular hexagon and different patterns inside the monotiles. The design incorporated the shape of triangles with an optical illusion. Finally, I incorporated a clock to create the aperiodic monotile clock.
