This site was initially intended as a mean of sharing the dodecahexes that can be tiled with hexiamonds. The hexiamonds themselves are geometrical shapes formed of exactly **six** equilateral triangles. There exist **12** hexiamonds, if we do not consider rotated and mirrored shapes as different. They cover a total of **72** equilateral triangles. As for the dodecahexes, these consist of **twelve** hexagons. However, once this problem was extensively solved by George Sicherman, I started publishing various other ideas and results about polyforms, especially **polyiamonds** and **polyhexes**. My own partial results for the dodecahexes problem can still be seen here.

Unless explicitly stated, all problems presented here use **exactly once each piece of a given set** of polyforms, i. e. I don't analyse problems with repeating or missing pieces.

- Bi-tilability is the property of some shapes to be tilable with
**two different sets**of polyforms.

A proof that all**heptahexes**can be tiled with**heptiamonds**if an individual hex is side 2, i. e. 24 triangles.

Tiling simultaneously multiple side 1 polyhexes with the heptiamonds.

Tiling two heptahexes and one tetrakaidecahex with the heptiamonds.

Tiling two identical tetrakaidecahexes with the heptiamonds.

Tiling symmetric shapes with six holes with the heptiamonds and**tetrahexes**. - After constructing all possible shapes tilable with the
**pentiamonds**, I searched them for shapes with special properties. - Quarantined
**hexiamonds and heptiamonds**.

Fried eggs with hexiamonds: make two identical shapes, each of which contains a yolk, i.e. a piece that doesn't touch the edge of the shape.

In how many ways can two identical shapes of 36 triangles be tiled with**hexiamonds**?

Here is the only example of such shapes with**ternary**symmetry that I found manually.

An example with**two axis**of symmetry.

In how many ways can three identical shapes of 24 triangles be tiled with the**hexiamonds**? -- The answer is**131**and can be seen here (George Sicherman). **Heptiamond**shapes are seldom fully solved. I compiled a catalog of shapes for which the exact number of solutions is known.

An exhaustive database of all heptiamond shapes with maximal symmetry

Simultaneously tiling 3- and 4-plicated heptiamonds with the whole set.**Octiamonds**are to polyiamonds what heptominoes are to polyominoes.

A very partial collection of octiamond stars with full symmetry.- There is much variety in convex shapes on a
**triangular grid**.

Convex shapes with all hexiamonds (free and one-sided).

All possible combinations of convex shapes that can be tiled simultaneously with the hexiamonds.

Convex shapes with all heptiamonds.

Convex shapes with all octiamonds (only free).

- Tiling a shape with a
**hole similar to a piece**of the set is a classic problem with pentominoes.

Tiling a side 4 hexagon with the hexiamonds, leaving a hole in the shape of a doubled piece.

Tiling a 8 x 14 parallelogram with the heptiamonds, leaving two symmetrically placed holes in the shape of the same doubled piece. - I call the shape that has most solutions for a given set of polyforms
**a champion shape**. I have counted all possible shapes formed with the**trihexes**and determined the champions. - A few properties of a
**set of polyforms**to study with examples for**trihexes**and**pentiamonds**. - Some very simple results with the polycairos.

- Tilings of geographical objects

Map of Italy, tiled with the one-sided heptiamonds; one triangle corresponds to 1 000 kmē.

In a similar fashion, Bulgaria's 111 000 kmē can be tiled with the pentahexes plus a monohex for the capital. - The numbers of pentiamonds, hexiamonds and heptiamonds may be related to the calendar.
- A classification of the highest symmetry heptiamond shapes, inspired by Carl Linnaeus' nomenclature of living organisms.

A demonstration of the order in which Algorithm X places the pieces in an octiamond shape. This algorithm chooses always the place where the least number of pieces can be placed. This makes it faster than choosing places in a predefined order (for instance filling line after line top to bottom).

Developed by Todor Tchervenkov: tchervenkov@gmail.com