Mosaic on plywood, 50 cm x 80 cm (20” x 32”), portrait.

## Introduction

Essentially, this piece of art has the humble goal of to capturing the mystery of mathematics itself. In that respect it may need some explanation.

I find the phoenix itself to be a powerful symbol of mathematics. The mythical firebird periodically sets itself in fire, just to born again from it’s own ashes. Like mathematics, its radiant beauty is only visible to those who choose to seek it. And like the phoenix, mathematics has been born again twice – after burning itself with the infinity (with the Zeno’s and Russell’s paradoxes).

As an artist I set myself a challenge was to say all that in just one single mosaic opus.

Mathematics has been aptly said to be 50 % theorems, 50 % proofs and 50 % sheer imagination. On one hand it is extremely strict and definite, on the other hand many new fascinating and useful things are literally product playful imagination of the mathematicians. I’ve tried to somehow present some of the most intriguing paradoxes of the nature of mathematics in this piece:

- Strictly formal yet highly creative – the phoenix is a symbol of creation, but has been embedded in a square grid.
- Intertwining the finite and the infinite – The aperiodic tilings this work is based on contain a finite tileset that can cover the infinite plane with no finite period.
- The organised and the organic – this manifests in the contrast between the regular hexagon versus the three loose tiles in the lower part.
- The dynamic relationship of the absolute and the personal – this is visible in the decision to place uncut standard-sized mosaic pieces in a regular square grid by hand, without any supports to enforce even-sized and -shaped gaps.

## Visual representation

The proportions of the work were chosen to reflect the golden ratio. The area is split between a massive regular hexagon in the upper part of the work filled with a regular square pattern tiling, tiles laid down in freehand, and three irregularly placed tiles, like sparks flying from the tip of the bird’s wing, in the lower left part of the frame balancing the hexagon.

The phoenix shape is drawn by a set of iridescent glass tiles embedded in a field of ceramic tiles with a matte tone. The colors of the iridescent and ceramic tiles match closely, so while there are eight kinds of tiles, there are only four different colors, two types of tiles for each color.

The colors of the tiling form strips in the SW-NE direction, which is also the direction the phoenix is flying, giving it an air of ascent. The colors symbolize ash, fire, air and life.

## Mathematics behind the piece

The pattern in the hexagon features two recent mathematical results from the field of aperiodic tilings:

- Emmanuel Jeandel (CARTE), Michael Rao (LIP): An aperiodic set of 11 Wang tiles (arXiv:1506.06492).
- Joshua E. S. Socolar, Joan M. Taylor: An aperiodic hexagonal tile (arXiv:1003.4279).

A tileset is a finite set of prototiles, each of which can be copied infinitely many times. There is some local matching rule telling which tiles may be placed in each position. If the copies can cover seamlessly the entire infinite Euclidean plane, the tileset is called valid.

### Jeandel-Rao Wang tile set

So called Wang tiles are a common example. They are unit squares with colored edges. The local rule is that tiles may not be mirrored or rotated, and two tiles may only be placed next to each other if their adjacent edges have matching colors. The usual way to visualize valid patches of Wang tiles is to divide each tile into four right-angled triangles, with edges as bases. The triangles are given the color of the edge, and the patch is presented as squares of the edge colors.

It’s common to visualize the tiles by drawing lines from each corner to the opposite corner and coloring the resulting triangles with the corresponding edge colors. Then a patch tiled with them looks like a grid of squares with the edge colors, tile centers and corners located where the squares meet. As it is customary to present the tiles as horizontal rows, the edge squares become angled by 45 degrees.

A tile set is periodic, if there exists a finite area that can be copied and placed seamlessly next to itself in two directions to produce a valid tiling. In terms of symmetry, periodic tilings have a P1 -symmetry group. Valid tilesets without such finite areas are called aperiodic. A related open question is the einstein -problem: Is there a valid tileset with just one tile that only admits aperiodic tilings?

Hao Wang, who invented the Wang tiles, conjenctured that all valid tilesets are periodic. He was proven wrong by his student Berger, who constructed a tileset with 20426 tiles. There’s a connection between Turing machines, undecidability problems and aperiodic tilings: It is undecidable if an arbitrary Wang tile set is invalid, periodic or aperiodic. Arbitrary Turing machine can be converted into a set of Wang tiles, and if an algorithm to solve the tiling problem for any tile set would exist, it could be applied to the Turing machine tileset to solve the halting problem. As the halting problem is otherwise known to be undecidable, we can conclude that the supposed algorithm to solve the tiling problem does not exist either.

Since Berger’s dissertation several aperiodic Wang tilesets have been constructed, culminating to a result presented by Emmanuel Jeandel and Michaël Rao in 2015. They used an exhaustive computer search to determine that there are no aperiodic tileset with less than 11 tiles, and show that there exists an aperiodic set of Wang tiles with 11 tiles over 4 colors:

The phoenix features a small patch of this Jeandel-Rao -tiling: each mosaic square represents a pair of edge colors, like in the picture above. In the Phoenix the tiling has been rotated by 45 degrees, and colors reassigned to fit the theme. The ceramic and iridescent tiles with corresponding colors are interpreted to have the same color (see the leftmost picture at the top of this post.)

### Socolar – Taylor monotile

The second aperiodic tileset featured is the Socolar–Taylor monotile, presented by Joshua Socolar and Joan Taylor in 2011. This monotile is the first known solution to the einstein -problem, but it is only so and so: it’s not possible to enforce the local rule with the tile shape only. The following images are from their paper:

This monotile is a regular decorated hexagon with a matching rule for the decorations. Rotations and reflections of the tile are allowed. For the usual representation the matching rule is:

- The black bars must be continuous across the egdes.
- The magenta arrows In the opposite ends of seams must point to opposite directions.

While this representation presents the symmetries of an individual tile very nicely, it becomes a bit messy once you try to tile larges patches. Fortunately there is an alternative representatin of the monotile I could use as an starting point:

Here their symmetry is not as apparent, but the second matching rule becomes more clear:

This yields a very nice tiling:

The matching rules are strictly determined by certain points on the edge of the tiles. As long as they are obeyed, the interior decoration of the tile has no effect on the aperiodicity properties. Essentially all the signals are binary, so one of the states can be replaced with a blanc. Thus the following hexagon is a simplified representation of the Socolar-Taylor tile, already fitted into square grid:

Based on this simplified form, I implemented the monotile in the phoenix shape by coding the appropriate locations with iridiscent tiles, and by applying a two-part local rule to the fenix shape:

- If a hexagon has an iridescent tile at an edge, there must be a matching iridescent tile next to it in the adjacent hexagon.
- If a hexagon has an iridescent tile at a corner, then in the corner at the other end of the seam departing from the tile must have a ceramic tile – and vice versa.

## Further considerations

This project was also an investigation to how large-scale mosaics with square tiles in a square grid could be designed from a smaller patches. Here I took it one step forward, and decided to create batch based aperiodic pattern based on single design. In reality I had to design the details of all six rotated copies of the the design to make them fit in the square grid, after which I could mirror the other six copies.

This design process opens possibilities for architectural application, as it allows designing large-scale mosaic murals to be implemented at a relatively low cost. The manufacturing process might even be automated to an extent. In any case, producing such mosaics is considerably faster than those with irregular tiles in irregular grids.

On the other hand, such mosaics require larger areas to compensate for the lost resolution. For this particular monotile the smallest aperiodic area with any visual interest is a hexagon with a bounding box closer to 8 m x 5 m, with the standard 20 mm mosaic tile and the standard 3 mm gap:

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