Just 185 781 (1,34%) of all tetrakaidecahexes are theoretically able to form pairs of identical pieces, which can be tiled with heptiamonds. This is so because such piece need to have no horns. Restrictions on perimeter certainly diminish this percentage in a considerable proportion, bringing the actual count of 14-hex with such property to a few tens of thousands. All these, however, are particularly interesting, because they are a key to discovering symmetric bi-tilable shapes.
George Sicherman has already proved that there are only 11 bi-tilable shapes with ternary symmetry. All other symmetric shapes would be divisible into two identical 14-hexes. There is no reason to assume that these 14-hex would be tilable separately, but still a great number of them should be.
It is convenient to explore such twin pairs of tilable 14-hexes in increasing order of their perimeter. For p = 26 there is no tilable combination at all. For p = 28, however, there are 200 symmetric tilable combinations, obtained from the 16 tetrakaidecahexes with this perimeter, which turn out to be all tilable separately in pairs.
| Number | Tiling |
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Beauty cannot be defined in mathematical terms nor calculated but still, here are a few examples of the symmetric shapes that we can obtain from this twin pairs:
When we look at 14-hexes with p = 30, things look as follows: all 104 pieces with no horns form tilable pairs ot twins and these combine to 829 symmetric shapes which can be tiled with heptiamonds. Some of these shapes overlap with the 200 obtained from the p = 28 14-hexes. The table contains only the symmetric 14-hexes.
| Number | Tiling |
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A few of the symmetric bi-tilable shapes: