Hexagonal prism

Hexagonal prism
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Hip
Coxeter diagram	x x6o ()
Stewart notation	P6
Elements
Faces	6 squares, 2 hexagons
Edges	6+12
Vertices	12
Vertex figure	Isosceles triangle, edge lengths √3, √2, √2
Measures (edge length 1)
Circumradius	$\frac{\sqrt5}{2} ≈ 1.11803$
Volume	$\frac{3\sqrt3}{2} ≈ 2.59808$
Dihedral angles	4–4: 120°
	4–6: 90°
Height	1
Central density	1
Number of external pieces	8
Level of complexity	3
Related polytopes
Army	Hip
Regiment	Hip
Dual	Hexagonal tegum
Conjugate	None
Abstract & topological properties
Flag count	72
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	G2×A1, order 24
Convex	Yes
Nature	Tame

The hexagonal prism or hip is a prismatic uniform polyhedron. It consists of 2 hexagons and 6 squares. Each vertex joins one hexagon and two squares. As the name suggests, it is a prism based on a hexagon.

It can tile 3D space to form the hexagonal prismatic honeycomb.

Vertex coordinates

A hexagonal prism of edge length 1 has vertex coordinates given by:

• $\left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac12\right),$
• $\left(±1,\,0,\,±\frac12\right).$

Representations

A hexagonal prism has the following Coxeter diagrams:

• x x6o (full symmetry)
• x x3x (A2×A1 symmetry, generally a ditrigonal prism)
• s2s6x (generally a ditrigonal trapezoprism)
• xx6oo&#x (hexagonal frustum)
• xx3xx&#x (ditrigonal frustum)
• xxx xux&#xt (A1×A1 axial, square-first)
• xxxx ohho&#xt (A1×A1, vertex-first)
• xx xu ho&#zx (A1×A1×A1 symmetry)

Semi-uniform variant

The hexagonal prism has a semi-uniform variant of the form x y6o 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 $\sqrt{a^2+\frac{b^2}{4}}$ and its volume is given by $\frac{3\sqrt3}{2}a^2b$.

A hexagonal prism with base edges of length a and side edges of length b can be atlernated to form a triangular antiprism with base edges of length $\sqrt3a$ and side edges of lengths $\sqrt{a^2+b^2}$. In particular if the side edges are $\sqrt2$ times the length of the base edges this gives a regular octahedron.

Variations

A hexagonal prism has the following variations:

• Ditrigonal prism - prism with ditrigons as bases, and 2 types of rectangles
• Ditrigonal trapezoprism - isogonal with trapezoid sides
• Hexagonal frustum
• Ditrigonal frustum

Related polyhedra

A triangular cupola can be attached to a base of the hexagonal prism to form the elongated triangular cupola. If a second triangular cupola is attached to the other base in the same orientation, the result is the elongated triangular orthobicupola. If the second cupola is rotated 60º from the first cupola the result is the elongated triangular gyrobicupola.

It is also possible to augment square faces of the hexagonal prism with square pyramids. If one square is augmented the result is the augmented hexagonal prism. If a second square, opposite to the first,, is augmented the result is the parabiaugmented hexagonal prism. If two non-opposite, non-adjacent squares are augmented the result is the metabiaugmented hexagonal prism. If three mutually non-adjacent squares are augmented the result is the triaugmented hexagonal prism.

Two non-prismatic uniform polyhedron compounds are composed of hexagonal prisms:

• Great rhombisnub octahedron (4)
• Rhombisnub icosahedron (10)

There are also an infinite amount of prismatic uniform compounds that are the prisms of compounds of hexagons.

External links

• Klitzing, Richard. "hip".

• Quickfur. "The Hexagonal Prism".

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

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