Truncated cube

Truncated cube
Type	Archimedean solid
Faces	14 (6 octagons and 8 triangles
Edges	36
Vertices	24
Symmetry group	octahedral symmetry ${\displaystyle \mathrm {O} _{h}}$
Dual polyhedron	triakis octahedron
Vertex figure
Net

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.

If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and δ_S +1, where δ_S is the silver ratio, √2 +1.

Construction

The truncated cube is constructed by cutting off all the vertices of a cube. ^[1] The resulting polyhedron has six octagons and eight triangles, having in total fourteen regular polygonal faces, thirty-six edges, and twenty-four vertices. ^[2]

Cartesian coordinates for the vertices of a truncated cube centered at the origin with edge length ${\textstyle 2{\frac {1}{\delta _{S}}}}$ are all the permutations of $${\displaystyle \left(\pm {\frac {1}{\delta _{S}}},\pm 1,\pm 1\right),}$$ where ${\displaystyle \delta _{S}=1+{\sqrt {2}}}$ is a silver ratio.^{[ citation needed ]}

Properties

3D model of a truncated cube

The truncated cube is an Archimedean solid, having a highly symmetric and semi-regular polyhedron with two or more different regular polygonal faces that meet in a vertex. ^[3] Every vertex is surrounded by two octagons and one triangle, thereby the vertex figure is ${\displaystyle 3\cdot 8^{2}}$. ^[4] The truncated octahedron has the same three-dimensional symmetry group as the regular octahedron does, the octahedral symmetry ${\displaystyle \mathrm {O} _{\mathrm {h} }}$. ^[5] The dual polyhedron of a truncated cube is a triakis octahedron, a Catalan solid obtained by gluing two short pyramids onto the faces of a regular octahedron. ^[4]

To find the surface area of a truncated cube, one may calculate the total area of all polygonal faces, namely six regular octagons and eight equilateral triangles, all of which have the same edge length. On the other hand, its volume can be calculated from the volume of a cube and the volume of the smaller pieces that have been truncated, and then subtracting them. Let ${\displaystyle a}$ be the edge length of a truncated cube. The formulation for its surface area ${\displaystyle A}$ and the volume ${\displaystyle V}$ are: ^[2] $${\displaystyle {\begin{aligned}A&=2\left(6+6{\sqrt {2}}+{\sqrt {3}}\right)a^{2}&&\approx 32.435a^{2}\\V&={\frac {21+14{\sqrt {2}}}{3}}a^{3}&&\approx 13.600a^{3}.\end{aligned}}}$$

A truncated cube has two different dihedral angles, an angle between two polygonal faces: An angle between a triangle and an octagon is 125.26°, whereas an angle between two octagons is a right angle, 90°. ^[4]

Dissection

Dissected truncated cube, with elements expanded apart

The truncated cube can be dissected into a central cube, with six square cupolae around each of the cube's faces, and 8 regular tetrahedra in the corners. This dissection can also be seen within the runcic cubic honeycomb, with cube, tetrahedron, and rhombicuboctahedron cells.

This dissection can be used to create a Stewart toroid with all regular faces by removing two square cupolae and the central cube. This excavated cube has 16 triangles, 12 squares, and 4 octagons. ^{[6] [7]}

Graph

Graph of a truncated cube

In the mathematical field of graph theory, a truncated cubical graph is the graph of vertices and edges of the truncated cube, one of the Archimedean solids. It has 24 vertices and 36 edges, and is a cubic Archimedean graph. ^[8] As a Hamiltonian cubic graph, it can be represented by LCF notation as LCF[2,-9,-2,2,9,-2]⁴.

Orthographic	LCF[2,-9,-2,2,9,-2]4
		Configuration \ v1 v2 e1 e2 e3 e4 v1 16 * 1 1 1 0 v2 * 8 2 0 0 1 e1 1 1 16 * * * e2 2 0 * 8 * * e3 2 0 * * 8 * e4 0 2 * * * 4

See also

• Cube-connected cycles, a family of graphs that includes the skeleton of the truncated cube
• Chamfered cube, obtained by replacing the edges of a cube with non-uniform hexagons

References

1. Cromwell, P. (1997). Polyhedra. pp. 81–86.
2. Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR   0290245.
3. Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. Vol. 10. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN   978-3-319-64123-2.
4. Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 76. ISBN   978-0-486-23729-9.
5. Koca, M.; Koca, N. O. (2013). "Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes". Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27–31 October 2010. World Scientific. p. 48.
6. B. M. Stewart, Adventures Among the Toroids (1970) ISBN   978-0-686-11936-4
7. "Adventures Among the Toroids - Chapter 5 - Simplest (R)(A)(Q)(T) Toroids of genus p=1".
8. Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN   0-486-23729-X. (Section 3-9)

External links

• Weisstein, Eric W., "Truncated cube" ("Archimedean solid") at MathWorld .
  • Weisstein, Eric W. "Truncated cubical graph". MathWorld .
• Klitzing, Richard. "3D convex uniform polyhedra o3x4x - tic".
• Editable printable net of a truncated cube with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
  • VRML model
  • Conway Notation for Polyhedra Try: "tC"