I set out to blend the mathematics that solved the Einstein Problem with the artistic style of M.C. Escher. Follow along as I recount my creative process and display my art.

The Einstein Problem

The Einstein Problem is solved by a tile. The shape of that tile, or if it even existed, was a longstanding problem in mathematics. Then in 2022 a solution was discovered, and in 2023, published.

The Einstein Problem asks if there exists an “aperiodic monotile”.

1. Tile: It covers a plane without gaps.
2. Mono: The tiles are all the same shape.
3. Aperiodic: It can’t have translational symmetry.

When the solution was published, the mathematics community was ecstatic. However, detractors were quick to point out that the solution was technically incomplete. This controversy ended when the same researchers discovered and published a second solution.

The announcement of the first solution spurred on a wave of tessellation art. News of the discovery entered the mainstream and tasteful mosaics abound. The second solution, mired in controversy, and arriving well after the hype, earned little fan art. It’s sad. The second solution is more elegant, storied, emotional, than the first.

Specter

That second solution was named “Specter” by its discoverers. Specter isn’t Just a single tile thought, it’s a whole family of tiles. There’s an infinite number of Specter tiles, and this makes it flexible enough to stylize. I’ve created a few examples to clarify what I mean.

A tessellation of Specters. The animation fades between differently stylized edges.

Getting Hands On

The initial discovery was made by David Smith, a retired print technician and self-identified “shape enthusiast.” In his pragmatic methodology he used software to design tiles and then tested them in the real world by cutting them out of paper. Smith has talked about the moment of discovery. He describes laying down the paper tiles, one after another. As he laid down the tiles, he recounts the strange feeling, as each fit into the next without any pattern. I wanted to understand that feeling.

Instead of cutting the tiles from paper, I would use my own medium, 3D printing. There are countless decorative variations published online, so I browsed for a bit and picked my favorites. Here are the mosaics I made with those tiles:

Mosaics made with 3D printed monotiles.¹

Outside of a computer, I was free to place tiles however felt natural. As I placed tiles, I kept trending into repeating motifs. The aperiodic tile resists this. The only way to make a repeating motif with an aperiodic tile is to leave occasional gaps between tiles. Only an aperiodic mosaic will close the gaps.

Making an aperiodic mosaic feels strange. Without a visual pattern to guide tile placement, it’s unclear where the next tile belongs. The tiles can’t be placed randomly however, otherwise you’ll end up with unfillable gaps. There’s a single pattern that all Specter tessellations follow. Each additional tile has a place.

Controversy revisted

Leaving gaps between tiles is an ‘illegal’ move in the world of mathematical tessellation. I’m reminded of another move that’s natural in the real world, but significant in math: flipping a tile over. I mentioned before that there was some controversy surrounding the first published monotile solution. When David Smith was testing his tile design with paper tile cutouts, a few of his paper tiles were upside down. When a tile is flipped over, is it the same shape? That was the subject of the controversy surrounding the first publication. The second publication, Specter, doesn’t require any flipped tiles.

The first monotile publication had found a spectrum of solutions. At the center of that spectrum is Tile (1,1). That central tile, coincidentally, is a Specter tile. It simply took some time for the research team to realize the significance of Tile (1,1). In the end, the partial solution led to the complete one. If Smith hadn’t flipped over that paper tile, would there be a Specter?

Making Specters

After playing with tiles designed by other people, I wanted to make some of my own. Perhaps because I have a soft spot for the strange, I wanted to do it with the lesser-known Specter.

Unfortunately, there’s few options for working with Specter. Nearly all of the tools and art I found build off of the first published tile, not the second. After a long search, I found this forum thread. It’s a veritable gold mine of Specter art and software. I modified another user’s script to allow custom tile designs and started to play around. The way a design flows between Specters is hard to describe. There’s a lot of order and chaos to it. This would inspire the decorations that I designed.

(First) A single decorated tile that’s used to tessellate the plane. (Second) A glimpse of the grasshopper script used.

Art Gallery

The following mosaics are made by decorating a single Specter tile, then tessellating it. The aperiodic tessellation of Specter it may appear chaotic, but it is ordered. In this artwork I attempted to capture that feeling of order from chaos. Individual tile decorations are abstract, and only when tessellated dose a structure emerge. What structures do you see?

Untitled #1

Untitled #2

Untitled #3

Untitled #4

Untitled #5

Untitled #6

Repetition Emerges

As the tessellation expands, repeating patterns start to appear. This is strange because the tessellation lacks translational symmetry. A repeating motif shouldn’t be possible. What’s especially strange is that these patterns are imperfect — ghost like. For an example of what I’m describing, look for triangles in the large version of untitled #4.

Untitled #4, Large

It took some research, but I think I know what’s going on. When Specters are tessellated, they organize into small groups. These small groups tile in the same way as hexagons.

Animated fade between tessellated Specters² and tessellated hexagons.

According to the seminal paper on Specter, there is always an equivalent tessellation of hexagons. However, the hexagon tessellation and the Specter tessellation never perfectly align. This is both the reason for orderly patterns that emerge, and also why they’re imperfect.

I had designed the tile decorations to create meandering structures. I imagined the tiles would tesselate into an endless maze. What I had not anticipated was this emergent symmetry; that it looks orderly as the tessellation expands.

Something new, something old

A public sculpture inspired by tessellation art of M.C. Escher.³

For 800 years the Alhambra has stood atop the hill in Granada, Spain. At one time it was a simple stronghold, then an Islamic palace. After the Reconquista, the Spanish king took it for himself. It was in the Alhambra that Cristopher Columbus received a royal endorsement for his expedition to the new world. By the 1500s the Renaissance was in Spain, and the Alhambra was not a Renaissance palace. Construction of a new palace began.

The kings that occupied the Alhambra added to it. A fountain here, a mosaic there and over time the walls became dense with ornamentation. The style might have been old, but the beauty was undeniable. When the new Renaissance-style palace was completed, the old one was not demolished, it was simply abandoned. The immaculately decorated halls of the Alhambra went quiet.

1936 M.C. Escher began to explore the empty Alhambra. Escher was a little-known artist at the time. His work was unremarkable, capturing the world as it appeared to the human eye, a style popularized in the Renaissance. He took a special interest in the ancient building, especially its mosaics. For a time, he was obsessed. He worked feverishly to document and study its tessellations.

After this study, Escher’s work changed. He blended what he saw in the Alhambra with contemporary techniques. He found his style, his legacy. To this day, he is known for his mathematical and tessellating art.

When I set out to make art with the aperiodic monotile, I did so with good company. My journey was made possible by those who came before me. Like them, I made something new with something old.

Citations

1. 3D modeling authorship for these 3D printed tiles are @bengineering on Thingaverse, @ateldsign on Printables, @ateldsign on Printables, @CarlosLuna on Printables. ↩︎
2. This diagram comes from the paper “A chiral aperiodic monotile” by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Straussis. It’s released under the CC BY license (International 4.0). My modifications to the diagram are shared alike. ↩︎
3. Photograph by Bouwe Brouwer. It’s released under the Creative Commons Attribution-Share Alike 3.0 Unported license. ↩︎