Prisms
If you stack equal figures, you get a prisms, which are symmetric with respect to the center plane. I used figures
with rotational symmetry or symmetry with respect to one or two lines. This way I got constructions with up
to three planes of symmetry and rotational symmetry.
Unfortunately it's impossible to construct a prism using all pieces, because the number of layers is even or the
area of the cross-section is even, contrary to the conditions for constructions..
For some cross-sections I tested, whether a prism of given height is possible to construct with the stacked tetrahexes.
If a solution was found, it's indicated by a +. If a solution is impossible, the prism got a -. The prisms with a ?
seem to be impossible, too, but I haven't got a proof.
 | Triangular Cross-section |
| Shape of Cross-section |
Area of Cross-section |
Hight / Number of Hexagons / Solvable |
| T2 | 3 |
4/12/+, 8/24/+ , 12/36/- |
|---|
| T3 | 6 |
2/12/+, 4/24/+, 6/36/+, 8/48/+, 10/60/+, 12/72/+ , 14/84/- |
|---|
| T4 | 10 |
2/20/+, 4/40+, 6/60+, 8/80/+ |
|---|
| T5 | 15 |
4/60/+ |
|---|
| T6 | 21 |
4/84/+ |
|---|
| T7 | 28 |
1/28/-, 2/54/+, 3/72/+ |
|---|
| T8 | 36 |
1/36/-, 2/72/+ |
|---|
| |
|---|
 | Hexagonal Cross-section |
| Shape of Cross-section |
Area of Cross-section |
Hight / Number of Hexagons / Solvable |
| H2 | 7 |
4/28+, 8/56/+ , 12/84/- |
|---|
| H3 | 19 |
4/76/+ |
|---|
| |
|---|
 | Rhombic Cross-section |
| Shape of Cross-section |
Area of Cross-section |
Hight / Number of Hexagons / Solvable |
| RH2 | 4 |
1/4/+, 2/8/+, 3/12/+, 4/16/+, 5/20/+, 6/24/+, 7/28/+,
8/32/+, 9/36/+, 10/40/+ , 11/44/?, 15/60/- |
|---|
| RH3 | 9 |
4/36/+, 8/72/+ |
|---|
| RH4 | 16 |
1/16/+, 2/32/+, 3/48/+, 4/80/+ |
|---|
| RH6 | 36 |
1/36/-, 2/72/+ |
|---|
| |
|---|
 | Rectangular Cross-section |
| Shape of Cross-section |
Area of Cross-section |
Hight / Number of Hexagons / Solvable |
| R 3 4 3 | 10 |
2/20/+, 4/40/+, 6/60/+, 8/80/+ |
|---|
| R 7 8 7 | 22 |
2/44/+, 4/88/+ |
|---|
| R 9 10 9 | 28 |
1/28/+, 2/56/+, 3/84/+ |
|---|
| Z 11 12 11 | 34 |
1/34/-, 2/68/+ |
|---|
| R 3 2 3 | 8 |
1/8/-, 2/16/+, 3/24/+,...,10/80/+ , 11/88/- |
|---|
| R 4 3 4 | 11 |
4/44/+, 8/88+ |
|---|
| R 11 10 11 | 32 |
1/32/-, 2/64/+ |
|---|
| R 13 12 13 | 38 |
2/76/+ |
|---|
| R 2 3 2 3 2 | 12 |
1/12/+, 2/24/+,...,7/84/+ |
|---|
| R 3 4 3 4 3 | 17 |
4/68/+ |
|---|
| R 4 5 4 5 4 | 22 |
2/44/+, 4/88/+ |
|---|
| R 2 1 2 1 2 | 8 |
1/8/-, 2/16/+, 3/16/+,...8/64/+ ,
9/72/?, 10/80/- |
|---|
| r 6 5 6 5 6 | 28 |
1/28/+, 2/56/+, 3/84/+ |
|---|
| R 8 7 8 7 8 | 38 |
1/38/-, 2/76/+ |
|---|
The maximum height of the prisms in the table is 12. If you allow an asymmetric cross-section even a height of 14
can be achieved.
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