Polyiamonds, polyhexes and other polyforms
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.
Mathematical Problems
- 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.
Artistic Creations
- 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.
JavaScript solvers
You can use my JavaScript solvers for easy puzzles. The user interface is a bit confusing: click on a cell to fill or unfill it. Click on the right-most coloured cell to solve the puzzle. Use the middle yellow cell to visualize a particular solution. To discover the use of the other colours cells, move your mouse over them (works only on a computer, not a smartphone).
Triangle based puzzles
Square based puzzles
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).
