I've built minesweeper over different tilings. Posting it here because it is on topic and crowd should understand this well.

Flat, by family:

• Regular — squares, hexagons, a triangle grid, plus four boards whose outline is a polygon of the tiling's own symmetry, exactly filled: triangles on a triangular board, triangles on a hexagonal board, hexagons on a hexagonal board, hexagons on a triangular board.
• Uniform — the eight non-regular Archimedeans: 3.3.3.3.6, 3.3.3.4.4, 3.3.4.3.4, 3.4.6.4, 3.6.3.6, 3.12.12, 4.6.12, 4.8.8.
• Laves — their eight duals, built mechanically from the templates rather than drawn by hand: prismatic pentagonal, Cairo pentagonal, rhombille, floret pentagonal, tetrakis square, triakis triangular, deltoidal trihexagonal, kisrhombille.
• Isogonal — six tilings by regular polygons that are not edge-to-edge, where a tile's corner lands in the middle of its neighbour's edge: offset square, staggered triangular, Pythagorean, rotated hexagonal, rotated triangular, three-scale triangular. The T-vertices have to be recorded so adjacency still works, then dropped again before any tile is measured.
• Congruent rectangles — five brick bonds, where all the interest is in how the courses stagger: stacked, running, basket weave, basket weave 3×3, herringbone.
• Aperiodic — Penrose P3 rhombi; the Spectre, Tile(1,1), whose tiling uses rotations only and never mirrors a tile; a phyllotactic spiral of one equilateral hexagon with angles 72°/144° in five arms, nonperiodic because a five-fold centre forbids any translation; and brick rings, 2×1 bricks in concentric square rings about a 2×2 core, nonperiodic by symmetry rather than by substitution.
• Fractals — sphinx, chair, Sierpiński carpet, pentaflake, Gosper island.

Each periodic family also wraps a cylinder and a torus, and — unless the tiling is chiral — a Möbius strip and a Klein bottle too. That comes to 180 boards.

Three details worth pulling out:

Adjacency never rounds. Vertices are exact hashable ids and two cells are neighbours exactly when they share one, with no floating-point tolerance anywhere. The arithmetic follows the tiling: ℤ[ζ5] for Penrose, ℤ[ζ12] for the Spectre, ℤ[ζ10] for the pentaflake, since five-fold symmetry needs rank 4. The Spectre's rotation ring is dense in the plane, so there's no lattice to snap a float back to — its placements are carried as exact (rotation, mirror, translation) triples and no floating point enters the substitution at all.

The Spectre is stored as a 14-gon and drawn as a 13-gon. The 14th corner is collinear and exists only so that shared-vertex adjacency finds the right neighbour; it's dropped before the shape is measured, so the tile measures as the 13-gon it looks like.

Chirality decides which boards exist. Both the Möbius and the Klein seam reverse orientation, so the snub hexagonal 3.3.3.3.6 and the floret pentagonal wrap a cylinder and a torus and cannot wrap a Möbius strip or a Klein bottle — the menu just doesn't offer those. Same for the chiral Spectre. And one tiling, the three-scale triangular (p3), reverses y at no height in any orientation, so it's the single tiling with no cylinder either: it ships flat and on the torus only.

Playable free in the browser, no account.

Penrose: https://hypersweeper.pages.dev/?mode=penrose&difficulty=medium

The Spectre: https://hypersweeper.pages.dev/?mode=spectre&difficulty=medium

I noticed a cloudy edge to three tiles that my tiler replaced this afternoon — he had laid the others yesterday but had to replace three. Tomorrow morning he’s laying the last eight and then is due to apply the sealant but everything I’m reading online suggests that he should be waiting at least 24 hours between laying the tiles and applying the sealant to avoid trapping the moisture. The cloudy edge to the tiles indicates that the edges are drying but the rest of the tile has absorbed moisture from the adhesive, and if he applies the sealant to them before they finish drying they’ll trap that moisture. Is that correct? Should I ask him to lay the tiles tomorrow and then come back the next day to seal them and grout..?

The sealant is either Lithofin stain stop (solvent based) or Lithofin stain stop eco (water based).

so I started by creating a mathematically perfect Einstein tile in a vector program. I had found an image that had guide lines of triangles and mid points along those triangles that just made it make sense to me visually. Is there anything that exists for the spectre tile? I believe some of the points along it don't follow the same guidelines as the einstein. Any help is appreciated however if it's in math language please do not use big formulas and break it down to basic language please. Or using geometry terms is ok. Thank you in advance.

Hello, Alessandro Longa has made (with my input) a huge repository of patterns and tilings at https://tiling-atlas.vercel.app/

We have Euclidean, spherical, hyperbolic, and we've come up with several new varieties like the "freedraw" patterns. Have a look!

The grout in my shower floor has worn away down to the spacers in many areas, can I dye Davco Ready made Rejuvenation Grout to Havana and use it over the existing grout?

Hi,

My draftsman thinks have the bathroom floor wastes should be to the side of bath. I think in the middle (green). It's a large bathroom and the fall needs to be 1:100 to be code. It's quite a long bathroom; 5 meters.

What do you think?

I made this by decorating a substitution tiling from The Tilings Encyclopedia.

Can I link to my blog article about tiling 7 sided figures here?

https://midlarts.wordpress.com/2026/02/07/heptagonal-tessellation/

Hi All,

I based this tiling on one of the substitution rules from The Tilings Encyclopedia by adding some decorations to the prototile(s). Can you guess which one it is?

(colors are based on the type of corner in the original tiling)

Chaim Goodman-Strauss included them in his book "The Magic Theorem" -- I just got my copy!

Imagine an infinite grid of white square tiles. I arbitrarily pick one tile and call it (0,0). The process to make the pattern is as follows. Find the closest tile to (0,0). Check if it shares a relative relationship to any of the other tiles. If it doesn’t, color it black. If it does, find the next closest tile to (0,0) and check again. Now to describe what a relative relationship is. Imagine 2 tiles. A at (0,0) and B at (0,1). The relationship B has to A is the tile directly above another tile, therefore no other tiles can be directly above any other tile. The relationship A has to B is directly below another tile, so no other tiles can be directly below any other tile. So when looking to place the next tile, the “illegal” placements of tile C are (0,2) and (0,-1). It is important to note that the “relative relationships” between two tiles does NOT exclude rotationally similar moves. This means that the relationships “tile directly to the right or left” and “the tile directly above or below” are NOT the same, and can be used once each. Because of this, (-1,0) and (1,0) are both acceptable tile placements. Let’s say we pick (-1,0) to place tile C. Now, because of C and A, tiles cannot be directly left or right of any other tiles, and because of C and B, tiles cannot be directly diagonal in the (+,+) or (-,-) direction. This means for the next tile, the illegal placements are (-2,0), (-1,1), (0,2), (1,2), (1,1), (1,0), (0,-1), (-1,-1), and (-2,-1). Therefore the next closest tile to (0,0) is (1,-1). This continues on indefinitely. So far, whenever there have been two points that are the closest, as was the case for the placement of tile C, it has worked out so the pattern has rotational or mirrored symmetry. Due to the exponential nature of this pattern, and the fact I do not know how to code, I have made limited progress manually mapping this pattern. I believe I have made it to the ninth tile in the pattern, but I’m human so I may make mistakes. The reason I’m posting this here is to ask 1. if anyone knows a way to automate the creation of this pattern, 2. Does this pattern eventually not have mirrored or rotational symmetry with equidistant tiles, and if there is anywhere I can go to see more research on this very niche topic. Attached is a photo of my best attempt at making this pattern, with the fully colored tiles being the black tiles and the x’s notating “illegal” moves.

Imagine an infinite grid of white square tiles. I arbitrarily pick one tile and call it (0,0). The process to make the pattern is as follows. Find the closest tile to (0,0). Check if it shares a relative relationship to any of the other tiles. If it doesn’t, color it black. If it does, find the next closest tile to (0,0) and check again. Now to describe what a relative relationship is. Imagine 2 tiles. A at (0,0) and B at (0,1). The relationship B has to A is the tile directly above another tile, therefore no other tiles can be directly above any other tile. The relationship A has to B is directly below another tile, so no other tiles can be directly below any other tile. So when looking to place the next tile, the “illegal” placements of tile C are (0,2) and (0,-1). It is important to note that the “relative relationships” between two tiles does NOT exclude rotationally similar moves. This means that the relationships “tile directly to the right or left” and “the tile directly above or below” are NOT the same, and can be used once each. Because of this, (-1,0) and (1,0) are both acceptable tile placements. Let’s say we pick (-1,0) to place tile C. Now, because of C and A, tiles cannot be directly left or right of any other tiles, and because of C and B, tiles cannot be directly diagonal in the (+,+) or (-,-) direction. This means for the next tile, the illegal placements are (-2,0), (-1,1), (0,2), (1,2), (1,1), (1,0), (0,-1), (-1,-1), and (-2,-1). Therefore the next closest tile to (0,0) is (1,-1). This continues on indefinitely. So far, whenever there have been two points that are the closest, as was the case for the placement of tile C, it has worked out so the pattern has rotational or mirrored symmetry. Due to the exponential nature of this pattern, and the fact I do not know how to code, I have made limited progress manually mapping this pattern. I believe I have made it to the ninth tile in the pattern, but I’m human so I may make mistakes. The reason I’m posting this here is to ask 1. if anyone knows a way to automate the creation of this pattern, 2. Does this pattern eventually not have mirrored or rotational symmetry with equidistant tiles, and if there is anywhere I can go to see more research on this very niche topic. Attached is a photo of my best attempt at making this pattern, with the fully colored tiles being the black tiles and the x’s notating “illegal” moves.