Hexagonal prism

In geometry, the hexagonal prism is a prism with hexagonal base. This polyhedron has 8 faces, 18 edges, and 12 vertices.[1]

Uniform Hexagonal prism
TypePrismatic uniform polyhedron
ElementsF = 8, E = 18, V = 12 (χ = 2)
Faces by sides6{4}+2{6}
Schläfli symbolt{2,6} or {6}x{}
Wythoff symbol2 6 | 2
2 2 3 |
Coxeter diagrams


SymmetryD6h, [6,2], (*622), order 24
Rotation groupD6, [6,2]+, (622), order 12
ReferencesU76(d)
DualHexagonal dipyramid
Propertiesconvex, zonohedron

Vertex figure
4.4.6

Since it has 8 faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces. Because of the ambiguity of the term octahedron and the dissimilarity of the various eight-sided figures, the term is rarely used without clarification.

Before sharpening, many pencils take the shape of a long hexagonal prism.[2]

As a semiregular (or uniform) polyhedron

If faces are all regular, the hexagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the fourth in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a truncated hexagonal hosohedron, represented by Schläfli symbol t{2,6}. Alternately it can be seen as the Cartesian product of a regular hexagon and a line segment, and represented by the product {6}×{}. The dual of a hexagonal prism is a hexagonal bipyramid.

The symmetry group of a right hexagonal prism is D6h of order 24. The rotation group is D6 of order 12.

Volume

As in most prisms, the volume is found by taking the area of the base, with a side length of , and multiplying it by the height , giving the formula:[3]

Symmetry

The topology of a uniform hexagonal prism can have geometric variations of lower symmetry, including:

Symmetry D6h, [2,6], (*622) C6v, [6], (*66) D3h, [2,3], (*322) D3d, [2+,6], (2*3)
Construction {6}×{}, t{3}×{}, s2{2,6},
Image
Distortion

As part of spatial tesselations

It exists as cells of four prismatic uniform convex honeycombs in 3 dimensions:

Hexagonal prismatic honeycomb[1]
Triangular-hexagonal prismatic honeycomb
Snub triangular-hexagonal prismatic honeycomb
Rhombitriangular-hexagonal prismatic honeycomb

It also exists as cells of a number of four-dimensional uniform 4-polytopes, including:

truncated tetrahedral prism
truncated octahedral prism
Truncated cuboctahedral prism
Truncated icosahedral prism
Truncated icosidodecahedral prism
runcitruncated 5-cell
omnitruncated 5-cell
runcitruncated 16-cell
omnitruncated tesseract
runcitruncated 24-cell
omnitruncated 24-cell
runcitruncated 600-cell
omnitruncated 120-cell

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

See also

References

  1. Pugh, Anthony (1976), Polyheda: A Visual Approach, University of California Press, pp. 21, 27, 62, ISBN 9780520030565.
  2. Simpson, Audrey (2011), Core Mathematics for Cambridge IGCSE, Cambridge University Press, pp. 266–267, ISBN 9780521727921.
  3. Wheater, Carolyn C. (2007), Geometry, Career Press, pp. 236–237, ISBN 9781564149367.
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