Ten-of-diamonds decahedron
# Ten-of-diamonds decahedron

In geometry, the ten-of-diamonds decahedron is a space-filling polyhedron with 10 faces, 2 opposite rhombi with orthogonal major axes, connected by 8 identical isosceles triangle faces. Although it is convex, it is not a Johnson solid because its faces are not composed entirely of regular polygons. Michael Goldberg named it after a playing card, as a 10-faced polyhedron with two opposite rhombic (diamond-shaped) faces. He catalogued it in a 1982 paper as 10-II, the second in a list of 26 known space-filling decahedra.

## Coordinates

If the space-filling polyhedron is placed in a 3-D coordinate grid, the coordinates for the 8 vertices can be given as: (0, ±2, −1), (±2, 0, 1), (±1, 0, −1), (0, ±1, 1).

## Symmetry

The ten-of-diamonds has $D_{2d}$ symmetry, which projects as order-4 dihedral (square) symmetry in two dimensions. It can be seen as a triakis tetrahedron, with two pairs of coplanar triangles merged into rhombic faces. The dual is similar to a truncated tetrahedron, except two edges from the original tetrahedron are reduced to zero length making pentagonal faces. The dual polyhedra can be called a skew-truncated tetragonal disphenoid, where 2 edges along the symmetry axis completely truncated down to the edge midpoint.

Symmetric projection :

| Ten of diamonds | Ten of diamonds | Related | Dual | Dual | Related |
| - | - | - | - | - | - |
| Solid faces | Edges | triakis tetrahedron | Solid faces | Edges | Truncated tetrahedron |
| v=8, e=16, f=10 | v=8, e=16, f=10 | v=8, e=18, f=12 | v=10, e=16, f=8 | v=10, e=16, f=8 | v=12, e=18, f=8 |

## Honeycomb

Ten-of-diamonds honeycomb :

| Schläfli symbol | $dht_{1,2}{4,3,4}$ |
| - | - |
| Coxeter diagram | |
| Cell | Ten-of-diamonds |
| Vertex figures | dodecahedron tetrahedron |
| Space Fibrifold Coxeter | I3 (204) $8^{−o}$ $[[4,3^+,4]]$ |
| Dual | Alternated bitruncated cubic honeycomb |
| Properties | Cell-transitive |

The ten-of-diamonds is used in the honeycomb with Coxeter diagram , being the dual of an alternated bitruncated cubic honeycomb, . Since the alternated bitruncated cubic honeycomb fills space by pyritohedral icosahedra, , and tetragonal disphenoidal tetrahedra, vertex figures of this honeycomb are their duals – pyritohedra, and tetragonal disphenoids.

Cells can be seen as the cells of the tetragonal disphenoid honeycomb, , with alternate cells removed and augmented into neighboring cells by a center vertex. The rhombic faces in the honeycomb are aligned along 3 orthogonal planes.

| Uniform | Dual | Alternated | Dual alternated | Dual alternated |
| - | - | - | - | - |
| $t_{1,2}{4,3,4}$ | $dt_{1,2}{4,3,4}$ | $ht_{1,2}{4,3,4}$ | $dht_{1,2}{4,3,4}$ | $dht_{1,2}{4,3,4}$ |
| Bitruncated cubic honeycomb of truncated octahedral cells | tetragonal disphenoid honeycomb | Dual honeycomb of icosahedra and tetrahedra | Ten-of-diamonds honeycomb | Honeycomb structure orthogonally viewed along cubic plane |

## Related space-filling polyhedra

The ten-of-diamonds can be dissected in an octagonal cross-section between the two rhombic faces. It is a decahedron with 12 vertices, 20 edges, and 10 faces (4 triangles, 4 trapezoids, 1 rhombus, and 1 isotoxal octagon). Michael Goldberg labels this polyhedron 10-XXV, the 25th in a list of space-filling decahedra.

The ten-of-diamonds can be dissected as a half-model on a symmetry plane into a space-filling heptahedron with 6 vertices, 11 edges, and 7 faces (6 triangles and 1 trapezoid). Michael Goldberg identifies this polyhedron as a triply truncated quadrilateral prism, type 7-XXIV, the 24th in a list of heptagonal space-fillers.

It can be further dissected as a quarter-model by another symmetry plane into a space-filling hexahedron with 6 vertices, 10 edges, and 6 faces (4 triangles, 2 right trapezoids). Michael Goldberg identifies this polyhedron as an ungulated quadrilateral pyramid, type 6-X, the 10th in a list of space-filling hexahedron.

Dissected models in symmetric projections :

| Relation | Decahedral half model | Heptahedral half model | Hexahedral quarter model |
| - | - | - | - |
| Symmetry | $C_{2v},$ order 4 | $C_s,$ order 2 | $C_2,$ order 2 |
| Edges | | | |
| Net | | | |
| Elements | v=12, e=20, f=10 | v=6, e=11, f=7 | v=6, e=10, f=6 |

### Rhombic bowtie

Pairs of ten-of-diamonds can be attached as a nonconvex bow-tie space-filler, called a rhombic bowtie for its cross-sectional appearance. The two right-most symmetric projections below show the rhombi edge-on on the top, bottom and a middle neck where the two halves are connected. The 2D projections can look convex or concave.

It has 12 vertices, 28 edges, and 18 faces (16 triangles and 2 rhombi) within $D_{2h}$ symmetry. These paired-cells stack more easily as inter-locking elements. Long sequences of these can be stacked together in 3 axes to fill space.

The 12 vertex coordinates in a 2-unit cube. (further augmentations on the rhombi can be done with 2 unit translation in z.)

    (0, ±1, −1), (±1, 0, 0), (0, ±1, 1),
    (±1/2, 0, −1), (0, ±1/2, 0), (±1/2, 0, 1)

Bow-tie model (two ten-of-diamonds) :

| Skew | Symmetric | Symmetric | Symmetric | Symmetric |
| - | - | - | - | - |
