| Faces: | 8 triangles 2 rhombi |
| Edges: | 16 |
| Vertices: | 8 |
| Symmetry: | D2d, order 8 |
| Dual: | Skew-truncated tetragonal disphenoid |
| Properties: | space-filling |
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.[1]
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).
The ten-of-diamonds has D2d 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.
| bgcolor=#e7dcc3 colspan=2 | Ten-of-diamonds honeycomb | |
|---|---|---|
| Schläfli symbol | dht1,2 | |
| Coxeter diagram | ||
| Cell | Ten-of-diamonds | |
| Vertex figures | dodecahedron tetrahedron | |
| Space Fibrifold Coxeter | I (204) 8−o [[4,3+,4]] | |
| Dual | Alternated bitruncated cubic honeycomb | |
| Properties | Cell-transitive |
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.
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.[2]
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 space-fillering heptahedra.[3]
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.[4]
| Relation | Decahedral half model | Heptahedral half model | Hexahedral quarter model | |
|---|---|---|---|---|
| Symmetry | C2v, order 4 | Cs, order 2 | C2, 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 | |
| Faces: | 16 triangles 2 rhombi |
| Edges: | 28 |
| Vertices: | 12 |
| Symmetry: | D2h, order 8 |
| Properties: | space-filling |
| Net: | File:Double-ten-of-diamonds-net.png |
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 D2h 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.[5]
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)