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.^[1]

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

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		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

Honeycomb

Ten-of-diamonds honeycomb
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

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.^[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]

Dissected models in symmetric projections

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

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 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.^[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)

Bow-tie model (two ten-of-diamonds)

Skew	Symmetric

See also

• Elongated gyrobifastigium