Hexagonal prism

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Hexagonal prism
Hexagonal prism.png
Rank3
TypeUniform
SpaceSpherical
Bowers style acronymHip
Info
Coxeter diagramx x6o
SymmetryG2×A1, order 24
ArmyHip
RegimentHip
Elements
Vertex figureIsosceles triangle, edge lengths 3, 2, 2
Faces6 squares, 2 hexagons
Edges6+12
Vertices12
Measures (edge length 1)
Circumradius
Volume
Dihedral angles4–4: 120°
 4–6: 90°
Height1
Central density1
Euler characteristic2
Number of pieces8
Level of complexity3
Related polytopes
DualHexagonal tegum
ConjugateHexagonal prism
Properties
ConvexYes
OrientableYes
NatureTame

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[edit | edit source]

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

Representations[edit | edit source]

A hexagonal prism has the following Coxeter diagrams:

Semi-uniform variant[edit | edit source]

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 and its volume is given by .

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 and side edges of lengths . In particular if the side edges are times the length of the base edges this gives a regular octahedron.

Variations[edit | edit source]

A hexagonal prism has the following variations:

Related polyhedra[edit | edit source]

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:

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

External links[edit | edit source]

  • Klitzing, Richard. "Hip".