Pentiamond Shapes
When five triangles are joined in all possible ways via their edges, only four different pieces are obtained: the pentiamonds. They have a total area of 20 triangular cells and can be combined in 491 030 different shapes. I'm greatly indebted to George Sicherman, who double checked my results and inspired me how to correct them.
I have ordered these shapes in a rather obscure way. Below are shown, as samples, shapes with indexes 0, 70,000, 140,000 and so on, every 70,000. Because the patterns rely on triangles, many of the shapes look spiky and scattered:
Solutions per shape
An overwhelming majority of the shapes have single solutions. However, there are 12 495 shapes with two solutions, 265 shapes with three solutions. There are even 5 shapes with four solutions! Two of the five champions shapes have holes.
Symmetric shapes
Out of more than half a million shapes, only 66 are symmetric: 51 with axial symmetry and 15 with rotational. Here are a few examples:
George Sicherman has observed that there is only one way to simultaneously form two identical shapes with the pentiamonds. All possible combinations of these two are the source of many of the symmetric full shapes mentionned above:
Convexity
Out of all the 491 030 pentiamond shapes there is exactly one which is convex. This remarkable result has been obtained by George Sicherman.
Shapes with holes
90,702 shapes have at least one hole, which corresponds to nearly 18,5% of all combinations of pentiamonds. Among those, 86,590 have just one hole, 4,104 have two of them and only 8 have three holes. Here I provide two visual examples:
One might want to know what is the biggest hole size. The answer is 8 and one single shape fulfills this condition. On the other hand, there are 8 shapes with a hole of 7 triangles. All but one of them have a hole in the form of the D heptiamond; the exceptional hole has the form of the V heptiamonds. Here are a few examples:
Icosahedron nets
The shape below can be folded into a regular icosahedron. It could be interesting to count all such shapes, knowing that there are 43 380 distinct nets for this Platonic solid.
[Added on 29 November 2020]Rick Mabry has a keen interest in nets of 3D solids and he was able to find all icosahedron nets that can be tiled with the Pentiamonds. He provided me with the list of all 43 380 icosahedron nets and I could confirm his result: only 2850 of them are tilable with the pentiamonds. Here are a bunch of random examples:
