Where mathematics meets visual art.

6 tiles · Periodic

The Mosaic set uses six tile shapes — equilateral triangle, square, hexagon, trapezoid, parallelogram, and narrow rhombus — all built on a shared 60° angle grid. The first three are regular polygons; the other three are complementary shapes whose angles are multiples of 30°, keeping every edge and corner on the same underlying grid.

This shared angle system is what allows the tiles to fit together cleanly, forming patterns that repeat with exact translational symmetry in two directions.

These shapes have appeared across centuries of decorative tilework — in Islamic geometric patterns, Roman floor mosaics, and architectural facades worldwide. The same underlying geometry: simple angle rules, endlessly extendable patterns.

3 tiles · Symmetric

The Rhombus family uses three tile shapes — thin, medium, and wide — all built on the same base angle step: 180° ÷ 7 (≈ 25.714°). Each tile angle is an integer multiple of that step:

• Thin: acute 180/7°, obtuse 1080/7°
• Medium: acute 360/7°, obtuse 900/7°
• Wide: acute 540/7°, obtuse 720/7°

This 7-fold angle grid gives the tiling its rotational symmetry. Rotating any tile in increments of 180/7° keeps all edges and corners on the same angular grid, so they always align.

The Rhombus family is not aperiodic. The tiles form symmetric, repeating patterns — a distinct mathematical world with its own visual logic.

2 tiles · Aperiodic

The Penrose tiling uses two golden rhombuses — a thin rhombus and a thick rhombus — each constructed from two golden triangles, whose sides are in ratio φ. Held together by edge-matching rules, they force an aperiodic arrangement: the pattern never repeats, no matter how far it extends.

Both tiles share the same edge length. Their angles are built on a single base step of 36° (= 180° ÷ 5). Every angle in both tiles is an integer multiple of that step:

• Thin: acute 36°, obtuse 144°
• Thick: acute 72°, obtuse 108°

This 5-fold angle grid gives Penrose tilings their rotational symmetry and keeps edges and corners aligned across the whole plane. The result looks structured but is provably non-repeating — order without periodicity.

1 tile · Monotile

The Spectre is a 14-sided polygon that tiles the plane using a single shape. Like Penrose, the pattern is aperiodic — it never repeats. Unlike Penrose, it needs no second tile type and no edge-matching rules. One shape does everything.

It was discovered in 2023 by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss — one of the most significant results in tiling theory in decades. An earlier finding from the same research, the Hat, required both a tile and its mirror image to tile aperiodically. The Spectre removes that requirement: a single shape, in a single orientation, with no reflections needed.

There is no translational symmetry, no rotational repeat. The Spectre is aperiodic at the most fundamental level — and it does it alone.