This article may be too technical for most readers to understand.(April 2017) |
Ten-of-diamonds decahedron | |
---|---|
Faces | 8
triangles 2 rhombi |
Edges | 16 |
Vertices | 8 |
Symmetry group | D2d, order 8 |
Dual polyhedron | 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.
Ten of diamonds | Related | Dual | Related | ||
---|---|---|---|---|---|
Solid faces |
Edges |
triakis tetrahedron |
Solid faces |
Edges |
Truncated tetrahedron |
v=8, e=16, f=10 | v=8, e=18, f=12 | v=10, e=16, f=8 | v=12, e=18, f=8 |
Ten-of-diamonds honeycomb | |
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Schläfli symbol | dht1,2{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 | |
---|---|---|---|---|
t1,2{4,3,4} |
dt1,2{4,3,4} |
ht1,2{4,3,4} |
dht1,2{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 |
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]
Rhombic bowtie | |
---|---|
Faces | 16
triangles 2 rhombi |
Edges | 28 |
Vertices | 12 |
Symmetry group | D2h, order 8 |
Properties | space-filling |
Net | |
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.)
Skew | Symmetric | |||
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