Ten-of-diamonds decahedron
| Ten-of-diamonds | |
|---|---|
| Faces | 8 triangles 2 rhombi |
| Edges | 16 |
| Vertices | 8 |
| 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. 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]
Related 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-filling 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 hexahedra.[4][4]
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. 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]
See also
References
- ^ Goldberg, Michael (1982). "On the Space-filling Decahedra". Structural Topology. Type 10-II.
- ^ Goldberg (1982), Type 10-XXV.
- ^ Goldberg, Michael (June 1978). "On the space-filling heptahedra". Geometriae Dedicata. 7 (2): 175–184. doi:10.1007/BF00181630. Type 7-XXIV.
- ^ a b Goldberg, Michael (June 1977). "On the space-filling hexahedra". Geometriae Dedicata. 6 (1): 99–108. doi:10.1007/BF00181585. Type 6-X.
- ^ Robert Reid, Anthony Steed Bowties: A Novel Class of Space Filling Polyhedron 2003
- Goldberg, Michael (November 1976). "Several new space-filling polyhedra". Geometriae Dedicata. 5: 517–523. doi:10.1007/BF00150781.
{{cite journal}}: CS1 maint: date and year (link)
- Koch, Elke (1972). 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] (Thesis). University Marburg/Lahn. Model 10/8–1, 28–404.