One-sided Rhombsliced Heptiamonds
Since there are no symmetric pieces in the set of rhombsliced heptiamonds we have 54 pairs of mirror pieces in the one-sided set. Putting them together we get a set of strange symmetric 'animals' as shown in the picture below. I cut the pieces from wood for parquet floor and glued magnets on the back to display the constructions on iron surfaces.
For two pieces and their mirror pieces it is possible to make three 6-fold replicas or one large replica using the whole set. Therefore the ratio of corresponding edges must be sqrt(3) which is also the ratio of basic and sliced edges in the grid. For the other pieces you can't get the large replicas, because they have more than one 60° angle.
But for all pieces without 30° angles three 6-fold replicas are possible. Some constructions are shown on
Andrew Clarke's site the missing solutions are
I looked for some other figures to make three small congruent copies or one large replica. The heart is a nice example I found.
Hexagons of size 6 or 6sqrt(3) respectively are shown on
Andrew Clarke's site. With six congruent rectangles of size 9x3sqrt(3) you can make three 9x6sqrt(3) rectangles or one 9sqrt(3)x18 rectangle. Three 18x3sqrt(3) rectangles or one 18sqrt(3)x9 rectangles makes another set of similar figures.
If you want to make symmetric constructions, you should try to find a solution for half of it with the two-sided set. Then you can reflect this part to get the whole solution. This was made for the star.
Unfortunately this method failed for the wheel, because each axis of symmetry leaves halfs of sliced edges. So the solution was much harder to find.