Octagonal prism

Octagonal prism
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Op
Coxeter diagram	x x8o ()
Elements
Faces	8 squares, 2 octagons
Edges	8+16
Vertices	16
Vertex figure	Isosceles triangle, edge lengths √2+√2, √2, √2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{5+2\sqrt2}}2 ≈ 1.39897$
Volume	$2(1+\sqrt2) ≈ 4.82843$
Dihedral angles	4–4: 135°
	4–8: 90°
Height	1
Central density	1
Number of external pieces	10
Level of complexity	3
Related polytopes
Army	Op
Regiment	Op
Dual	Octagonal tegum
Conjugate	Octagrammic prism
Abstract & topological properties
Flag count	96
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	I2(8)×A1, order 32
Convex	Yes
Nature	Tame

The octagonal prism, or op, is a prismatic uniform polyhedron. It consists of 2 octagons and 8 squares. Each vertex joins one octagon and two squares. As the name suggests, it is a prism based on an octagon.

It can also be obtained from the small rhombicuboctahedron by removing two opposing square cupolas. It can therefore also be thought of as a bidiminished small rhombicuboctahedron.

Vertex coordinates

An octagonal prism of edge length 1 has vertex coordinates given by:

• $\left(±\frac12,\,±\frac{1+\sqrt2}2,\,±\frac12\right),$
• $\left(±\frac{1+\sqrt2}2,\,±\frac12,\,±\frac12\right).$

Representations

An octagonal prism has the following Coxeter diagrams:

• x x8o (full symmetry)
• x x4x (generally a ditetragonal prism)
• s2s8x (generally a ditetragonal trapezoprism)
• xx8oo&#x (octagonal frustum)
• xx4xx&#x (ditetragonal frustum)
• xxxx xwwx&#xt (A₁×A₁ axial)
• xx xw wx&#zx (A₁×A₁×A₁ symmetry)

Semi-uniform variant

The octagonal prism has a semi-uniform variant of the form x y8o that maintains its full symmetry. This variant uses rectangles as its sides.

With base edges of length a and side edges of length b, its circumradius is given by $\sqrt{a^2\frac{2+\sqrt2}{2}+\frac{b^2}{4}}$ and its volume is given by $2(1+\sqrt2)a^2b$.

An octagonal prism with base edges of length a and side edges of length b can be alternated to form a square antiprism with base edges of length $\sqrt{2+\sqrt2}a$ and side edges of lengths $\sqrt{a^2+b^2}$. In particular if the side edges are $\sqrt{1+\sqrt2}$ times the length of the base edges this gives a uniform square antiprism.

Variations

An octagonal prism has the following variations:

• Ditetragonal prism - prism with ditetragons as bases, and 2 types of rectangles
• Ditetragonal trapezoprism - isogonal with trapezoid sides
• Octagonal frustum
• Ditetragonal frustum

Related polyhedra

A square cupola can be attached to a base of the octagonal prism to form the elongated square cupola. If a second square cupola is attached to the other base in the same orientation, the result is the elongated square orthobicupola, better known as the small rhombicuboctahedron. If the second cupola is rotated 45º the result is the elongated square gyrobicupola.

The small rhombihexahedron can be obtained by blending three orthogonal octagonal prisms so that half of the square faces blend out.

External links

• Klitzing, Richard. "op".

• Quickfur. "The Octagonal Prism".

• Wikipedia Contributors. "Octagonal prism".
• McCooey, David. "Octagonal Prism"

• Hi.gher.Space Wiki Contributors. "Octagonal prism".