Triangular orthobicupola

Triangular orthobicupola
Type	Johnson J26 – J27 – J28
Faces	8 triangles 6 squares
Edges	24
Vertices	12
Vertex configuration	6(32.42) 6(3.4.3.4)
Symmetry group	D3h
Dihedral angle (degrees)	triangle-to-triangle:141° triangle-to-square:125.3° square-to-square:109.4°
Dual polyhedron	trapezo-rhombic dodecahedron
Properties	convex, composite
Net

In geometry, the triangular orthobicupola is one of the Johnson solids (J₂₇). As the name suggests, it can be constructed by attaching two triangular cupolas (J₃) along their bases. It has an equal number of squares and triangles at each vertex; however, it is not vertex-transitive. It is also called an anticuboctahedron, twisted cuboctahedron ^[1] or disheptahedron. It is also a canonical polyhedron.

The triangular orthobicupola is the first in an infinite set of orthobicupolae.

Construction

The triangular orthobicupola is composite, which can be constructed by attaching two triangular cupolas onto their bases. ^[2] Similar to the cuboctahedron, which would be known as the triangular gyrobicupola, the difference is that the two triangular cupolas that make up the triangular orthobicupola are joined so that pairs of matching sides abut (hence, "ortho"); the cuboctahedron is joined so that triangles abut squares and vice versa. Given a triangular orthobicupola, a 60-degree rotation of one cupola before the joining yields a cuboctahedron. ^[3] Hence, another name for the triangular orthobicupola is the anticuboctahedron. ^[4] Because the triangular orthobicupola has the property of convexity and its faces are regular polygons —eight equilateral triangles and six squares —it is categorized as a Johnson solid. Out of 92 solids, it is enumerated as the twenty-seventh Johnson solid ${\displaystyle J_{27}}$ ^{[5] [6]}

Properties

The surface area ${\displaystyle A}$ and the volume ${\displaystyle V}$ of a triangular orthobicupola are the same as those of a cuboctahedron. Its surface area is the sum of all of its polygonal faces, and its volume is obtained by slicing it off into two triangular cupolas and adding their volume. With edge length ${\displaystyle a}$, they are: ^[5] $${\displaystyle {\begin{aligned}A&=2\left(3+{\sqrt {3}}\right)a^{2}\approx 9.464a^{2},\\V&={\frac {5{\sqrt {2}}}{3}}a^{3}\approx 2.357a^{3}.\end{aligned}}}$$

A triangular orthobicupola has the same symmetry as a triangular prism, the dihedral group ${\displaystyle \mathrm {D} _{3\mathrm {h} }}$, which contains one three-fold axis and three two-fold axes. ^[1] There are three different dihedral angles, which can be ascertained by triangular cupolae adjoining together along their bases. These angles between two triangles: 70.5° + 70.5° = 141°; two squares 54.7° + 54.7° = 109.4°; as well as a square and a triangle, the same as the triangular cupola 125.3°. ^[7]

Dual polyhedron

The dual of a triangular orthobicupola can tessellate with its copy in three-dimensional space

The dual polyhedron of a triangular orthobicupola is a twelve-faced rhombic dodecahedron of six rhombi and six trapezoidal shapes, a trapezo-rhombic dodecahedron; ^[4] alternative names are rhombo-trapezoidal dodecahedron^{[ citation needed ]} and trapezoidal dodecahedron. ^[8] The trapezo-rhombic dodecahedron is a plesiohedron, a special kind of space-filling polyhedron that can be defined as the Voronoi cell of a symmetric Delone set, generated by hexagonal close-packing. ^[8] Alongside its copy, the polyhedron forms a honeycomb. ^[1]

References

1. Pearce, Peter (1978). Structure in Nature Is a Strategy for Design. MIT Preszs. p. 132. ISBN   978-0-262-66045-7.
2. Timofeenko, A. V. (2009). "Convex Polyhedra with Parquet Faces" (PDF). Doklady Mathematics. 80 (2): 720–723. doi:10.1134/S1064562409050238.
3. Ogievetsky, O.; Shlosman, S. (2021). "Platonic compounds and cylinders". In Novikov, S.; Krichever, I.; Ogievetsky, O.; Shlosman, S. (eds.). Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry. American Mathematical Society. p. 477. ISBN   978-1-4704-5592-7.
4. Becker, David A. (2012). "A Peculiarly Cerebroid Convex Zygo-Dodecahedron is an Axiomatically Balanced "House of Blues": The Circle of Fifths to the Circle of Willis to Cadherin Cadenzas". Symmetry. 4 (4): 644–666. Bibcode:2012Symm....4..644B. doi: 10.3390/sym4040644 .
5. Berman, M. (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR   0290245.
6. Francis, D. (2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
7. Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics . 18: 169–200. doi: 10.4153/cjm-1966-021-8 . MR   0185507. S2CID   122006114. Zbl   0132.14603.
8. Lagarias, Jeffrey C. (2011). "The Kepler conjecture and its proof". The Kepler Conjecture: The Hales–Ferguson proof. Springer, New York. pp. 3–26. doi:10.1007/978-1-4614-1129-1_1. ISBN   978-1-4614-1128-4. MR   3050907.; see especially p. 11

External links

• Weisstein, Eric W., "Johnson solid" ("Triangular orthobicupola") at MathWorld .