The area of all heptiamonds is 168 triangles which corresponds to 28 hexagons, each comprising six triangles. It is possible to tile with the heptiamonds two heptahexes and one tetrakaidecahex (14-hex) at the same time. There are nearly 1000 triplets of 7-7-14-hexes (details in the tables below). The minimal cumulative perimeter of such a triplet is 64 and the known highest is 82, well under the theoretical maximum of 95 that a hexagon built heptiamond shape can have.
Since perimeter is a key feature, it is handy to categorize the triplets according to the perimeter of the 14-hex that they contain, noted p. I have thoroughly checked all possibilities for 26 <= p <= 34; results for p = 36 and p = 38 are partial; for higher values of p, solutions have been obtained by modifying solutions with lower p.
Most 14-hexes are labelled with numbers but some have been assigned names. When a 14-hex has is a unicorn (has one horn), it is much easier to check whether it can be combined with two heptahexes: there are 15 times less configurations to explore. That is why I indicate how many of the 14-hexes are unicorns.
Triplets by value of p: 26, 28, 30, 32, 34, 36, 38, 40.
Due to lack of space, I haven't yet published the actual pairs of 7-hexes nor actual tilings with heptiamonds. However, pairs of 7-hexes have one notable feature which should be mentioned here: they all contain at least one of a set of six 7-hexes that I call key 7-hexes. These are as follows:
|
Flower |
Turtle |
Comet |
Saucer |
Weight |
Canary |
Detailed results can be seen in tabular form.
There is only one 14-hex with a perimeter of 26 edges. It can be combined with 135 pairs of heptahexes to obtain a heptiamond tilable configuration:
| 14-hex | Pairs of 7-hexes |
|---|---|
000000 | 135 |
There are 16 14-hexes with a perimeter of 28 edges and one or no horns. All of them can be tiled together with a pair of heptahexes. However, the number of actual distinct pairs combinable with each p28 14-hex varies greatly from just 3 to 106, as can be seen from the table below. Symmetric 14-hexes combine with significantly fewer pairs. All in all, there are 690 distinct triplets in which the 14-hex has a perimeter of 28:
| 14-hex | Pairs of 7-hexes | Note |
|---|
There are 120 14-hexes with a perimeter of 30 edges and one or no horns. 104 pieces have no horns and only 17 are unicorns. 38 of them can combine with pairs of 7-hexes to form triplets tilable with heptiamonds. Only one unicorn (#53) forms a tilable triplet but it has the property to lead to a triplet for p = 32.
| 14-hex | Pairs of 7-hexes | Note |
|---|
There are 529 14-hexes with perimeter of 32 edges and one or no horns. Slightly more than half of them are unicorns -- 268, whilst no horns are 261. Only 17 of these pieces can combine with pairs of 7-hexes to form 50 triplets tilable with heptiamonds.
A curious feature of the no-horn pieces with p = 32 is that they are often composed of two of the key 7-hexes. This is noted where pertinent.
| 14-hex | Pairs of 7-hexes | Note |
|---|
There are 1 799 14-hexes with perimeter of 34 edges and one or no horns. Two thirds of them are unicorns -- 1158, whilst no horns are 641. 15 pieces have a hole of a single hexagon and 12 of them can combine with pairs of 7-hexes and form triplets tilable with heptiamonds. Besides these, there are just 4 pieces which have no horns but can be combined.
| 14-hex | Pairs of 7-hexes | Note |
|---|
There are 5 220 14-hexes with perimeter of 36 edges and one or no horns: 1 660 have no horns and 3 560 are unicorns. 194 of the 14-hexes have a hole of a single hexagon. 40 pieces with perimeter of 36 can be tiled with the heptiamonds, together with a pair of heptahexes.
| 14-hex | Pairs of 7-hexes | Note |
|---|
There are 13 769 14-hexes with perimeter of 38 edges and one or no horns. 801 pieces have one hole -- for 797 of them hole is the size of a single hexagon and for the remain 4 it is the size of two hexagons. 20 of all p = 38 pieces form a triplet tilable with heptiamonds.
| 14-hex | Pairs of 7-hexes | Note |
|---|
There are 33 989 14-hexes with perimeter of 40 edges and one or no horns. 2 643 pieces have one hole, 2 have two holes. Unicorns are 25 901. Only five pieces are co-tilable with two heptahexes: two of them are unicorns and the other three have a hole comprising of two hexes.
| 14-hex | Pairs of 7-hexes | Note |
|---|