There are 24 heptiamonds, which is 5² - 1 and means that the whole set can tile a 5-plicated individual piece, provided that there are seven holes in it. Miroslav Vicher has shown that the holes can be situated in the center of each 5-plicated triangle; Kate Jones calls this "lace pattern"; David Goodger has shown that the holes can form a single hole in the shape of the 5-plicated piece; a solution for each piece in this case is shown on the Poly Pages.
Since 5² = 4² + 3², I considered simultaneously tiling a 3- and 4-plicated heptiamonds. Both cases -- a double piece and holes in the centers of the 4-plicated triangles -- are shown in the table below. In both case the operation fails for a single piece: C piece for double piece case and M piece for the lace pattern.
When I communicated these results to George Sicherman, he replied that it is possible to simultaneously produce two duplicated and one quadruplicated heptiamond for all pieces in the set, except for two. George let me rediscover these two: you will notice the holes in the corresponding column below.
| Double piece | Lace pattern | Two 2- and one 4-plicated |
|---|