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# Regular Octahedron

## Make Your Own Regular Octahedron

The regular octahedron, often simply called "the" octahedron, is the Platonic solid with six polyhedron vertices, 12 polyhedron edges, and eight equivalent equilateral triangular faces, denoted . It is illustrated above together with a wireframe version and a net that can be used for its construction.

The regular octahedron is also the uniform polyhedron with Maeder index 5 (Maeder 1997), Wenninger index 2 (Wenninger 1989), Coxeter index 17 (Coxeter et al. 1954), and Har'El index 10 (Har'El 1993). It is given by the Schläfli symbol and Wythoff symbol . The octahedron of unit side length is the antiprism of sides with height . The octahedron is also a square dipyramid with equal edge lengths.

A number of symmetric projections of the regular octahedron are illustrated above.

The regular octahedron is implemented in the Wolfram Language as Octahedron[] or UniformPolyhedron["Octahedron"]. Precomputed properties are available as PolyhedronData["Octahedron", prop].

There are 11 distinct nets for the octahedron, the same as for the cube (Buekenhout and Parker 1998). Questions of polyhedron coloring of the octahedron can be addressed using the Pólya enumeration theorem.

The octahedron is the convex hull of the tetrahemihexahedron.

The dual polyhedron of an octahedron with unit edge lengths is a cube with edge lengths .

The illustration above shows an origami octahedron constructed from a single sheet of paper (Kasahara and Takahama 1987, pp. 60-61).

Like the cube, the regular octahedron has the octahedral group of symmetries.

The connectivity of the vertices is given by the octahedral graph.

The regular octahedron has a single stellation: the stella octangula. The solid bounded by the two tetrahedra of the stella octangula (left figure) is a regular octahedron (right figure; Ball and Coxeter 1987).

S. Wagon (pers. comm., Oct. 30, 2013) has constructed a closed loop of eight regular octahedra.

The following table gives polyhedra which can be constructed by augmentation of a regular octahedron by pyramids of given heights .

Three orientations of a regular octahedron are illustrated above. The left one has vertices , , , (for edge lengths ), the middle one has vertices , , (for edge lengths ), and the right one has vertices and (for edge lengths ).

In the former case, the face planes are , so a solid octahedron is given by the equation

 (1)

If the edges of a regular octahedron are divided in the golden ratio such that the points of division for any face form an equilateral triangle, then the twelve points of division form an icosahedron (Wells 1991). In fact, there are two ways in which the edges can be internally divided in the golden ratio and two ways in which they can be externally divided, resulting in four possible icosahedra. Keeping the same connectivity, but reversing the long and short ends of the division gives Jessen's orthogonal icosahedron.

A plane perpendicular to a axis of an octahedron cuts the solid in a regular hexagonal cross section (Holden 1991, pp. 22-23). Since there are four such axes, there are four possible hexagonal cross sections.

The centers of the faces of a regular octahedron form a cube, and the centers of the faces of a cube form an octahedron (Steinhaus 1999, pp. 194-195). Faceted forms of the regular octahedron include the cubitruncated cuboctahedron and tetrahemihexahedron.

Let a regular octahedron be length on a side. The height of the top polyhedron vertex from the square plane is also the circumradius

 (2)

where

 (3)

is the diagonal length, so

 (4)

 (5) (6) (7)

so

 (8)

Use similar triangles to obtain

 (9) (10) (11)

 (12)

and twice the inradius gives the height of the octahedron viewed as a 3-sided antiprism. The midradius of the octahedron is

 (13)

The area of one face of a regular octahedron is the area of an equilateral triangle

 (14)

The volume is two times the volume of a square-base pyramid,

 (15)

The dihedral angle is

 (16)

and the Dehn invariant of a unit regular octahedron is

 (17) (18)

where the first expression uses the basis of Conway et al. (1999).

The regular octahedron can be built using a Haűy construction. The Haűy octahedral numbers

 (19)

give another method for calculating the volume of the octahedron,

 (20)

in agreement with the result derived above.

Antiprism, Dürer's Solid, Escher's Solid, Haűy Construction, Icosahedron, Jumping Octahedron, Octahedral Graph, Octahedral Group, Octahedron, Octahedron 2-Compound, Octahedron 3-Compound, Octahedron 4-Compound, Octahedron 5-Compound, Octahedron 6-Compound, Octahedron 10-Compound, Platonic Solid, Polyhedron Coloring, Stella Octangula, Tritetrahedron, Truncated Octahedron

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## References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 1987.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 228, 1987.Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension ." Disc. Math. 186, 69-94, 1998.Conway, J. H.; Radin, C.; and Sadun, L. "On Angles Whose Squared Trigonometric Functions Are Rational." Discr. Computat. Geom. 22, 321-332, 1999.Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.Cundy, H. and Rollett, A. "Octahedron. ." §3.5.3 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 64, 1989.Davie, T. "The Octahedron." http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/octahedron.html.Geometry Technologies. "Octahedron." http://www.scienceu.com/geometry/facts/solids/octa.html.Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Harris, J. W. and Stocker, H. "Octahedron." §4.4.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 100, 1998.Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.Kasahara, K. Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, p. 204, 1988.Kasahara, K. and Takahama, T. Origami for the Connoisseur. Tokyo: Japan Publications, 1987.Maeder, R. E. "05: Octahedron." 1997. https://www.mathconsult.ch/static/unipoly/05.html.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 193-195, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 163, 1991.Wenninger, M. J. "The Octahedron." Model 2 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 15, 1989.

## Cite this as:

Weisstein, Eric W. "Regular Octahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RegularOctahedron.html