Ten-of-diamonds decahedron

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 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

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

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 heptagonal space-fillers.[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

• Elongated gyrobifastigium

• Koch 1972 Koch, Elke, Wirkungsbereichspolyeder und Wirkungsbereichsteilunger zukubischen Gitterkomplexen mit weniger als drei Freiheitsgraden (Efficiency Polyhedra, and Efficiency Dividers, cubic lattice complexes with less than three degrees of freedom) Dissertation, University Marburg/Lahn 1972 - Model 10/8–1, 28–404.