Octahedron Explained

In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular octahedra also exist, including both convex and non-convex shapes.

Regular octahedron

See main article: Regular octahedron. The regular octahedron has eight equilateral triangle sides, six vertices at which four sides meet, and twelve edges. Its dual polyhedron is a cube.^[1] It can be formed as the convex hull of the six axis-parallel unit vectors in three-dimensional Euclidean space. It is one of the five Platonic solids,^[2] and the three-dimensional case of an infinite family of regular polytopes, the cross polytopes.^[3] Although it does not tile space by itself, it can tile space together with the regular tetrahedron to form the tetrahedral-octahedral honeycomb.^[4]

Combinatorially equivalent to the regular octahedron

The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it:

• Triangular antiprisms: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral.^[5]
• Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all three quadrilaterals are planar squares.^[6]
• Schönhardt polyhedron, a non-convex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices.^[7]
• Bricard octahedron, a non-convex self-crossing flexible polyhedron.^[8]

Other convex polyhedra

The regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges.^[9] There are 257 topologically distinct convex octahedra, excluding mirror images. More specifically there are 2, 11, 42, 74, 76, 38, 14 for octahedra with 6 to 12 vertices respectively.^{[10] [11]} (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)

Notable eight-sided convex polyhedra include:

References

1. Book: Erickson, Martin . 2011 . Beautiful Mathematics . . 62 . 978-1-61444-509-8.
2. Book: Herrmann . Diane L. . Sally . Paul J. . 2013 . Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory . Taylor & Francis . 978-1-4665-5464-1 . 252.
3. Book: Coxeter, H. S. M. . Harold Scott MacDonald Coxeter . Regular Polytopes . Regular Polytopes (book) . Methuen and Co. . 1948 . 121–122.
4. Book: Posamentier . Alfred S. . Maresch . Guenter . Thaller . Bernd . Spreitzer . Christian . Geretschlager . Robert . Stuhlpfarrer . David . Dorner . Christian . 2022 . Geometry In Our Three-dimensional World . . 9789811237126 . 233 - 234.
5. Book: O'Keeffe . Michael . Hyde . Bruce G. . Crystal Structures: Patterns and Symmetry . 2020 . 141 . . 978-0-486-83654-6.
6. Trigg . Charles W. . Charles W. Trigg . 1 . Mathematics Magazine . 2689647 . 55–57 . An Infinite Class of Deltahedra . 51 . 1978 . 10.1080/0025570X.1978.11976675.
7. Schönhardt . E. . Erich Schönhardt . . 309–312 . Über die Zerlegung von Dreieckspolyedern in Tetraeder . 1928 . 98 . 10.1007/BF01451597.
8. Book: Connelly, Robert . Robert Connelly . . Flexing surfaces . 10.1007/978-1-4684-6686-7_10 . 978-1-4684-6688-1 . 79–89 . Springer . The Mathematical Gardner . 1981. .
9. Web site: Enumeration of Polyhedra . 2 May 2006 . https://web.archive.org/web/20111010185122/http://www.uwgb.edu/dutchs/symmetry/polynum0.htm . 10 October 2011 . dead .
10. Web site: Counting polyhedra.
11. Web site: Polyhedra with 8 Faces and 6-8 Vertices . 14 August 2016 . dead . https://web.archive.org/web/20141117072140/http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm . 17 November 2014 .

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