Hexagonal prism

Prism with a 6-sided base From Wikipedia, the free encyclopedia

Hexagonal prism

In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.[1]

Quick Facts Hexagon prism, Type ...
Hexagon prism
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Typeprism
Symmetry groupprismatic symmetry of order 24
Dual polyhedronhexagonal bipyramid
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3D model of a uniform hexagonal prism.

As a semiregular polyhedron

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Perspective

If faces are all regular, the hexagonal prism is a semiregular polyhedronmore generally, a uniform polyhedronand 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 prismatic symmetry of order 24, consisting of rotation around an axis passing through the regular hexagon bases' center, and reflection across a horizontal plane.[2]

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] and its surface area is by summing the area of two regular hexagonal bases and the lateral faces of six squares:

As a parallelohedron

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Hexagonal prismatic honeycomb

The hexagonal prism is one of the parallelohedron, a polyhedral class that can be translated without rotations in Euclidean space, producing honeycombs; this class was discovered by Evgraf Fedorov in accordance with his studies of crystallography systems. The hexagonal prism is generated from four line segments, three of them parallel to a common plane and the fourth not.[4] Its most symmetric form is the right prism over a regular hexagon, forming the hexagonal prismatic honeycomb.[5]

The hexagonal prism also exists as cells of four prismatic uniform convex honeycombs in 3 dimensions:

Triangular-hexagonal prismatic honeycomb
Snub triangular-hexagonal prismatic honeycomb
Rhombitriangular-hexagonal prismatic honeycomb
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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
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runcitruncated 5-cell
omnitruncated 5-cell
runcitruncated 16-cell
omnitruncated tesseract
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runcitruncated 24-cell
omnitruncated 24-cell
runcitruncated 600-cell
omnitruncated 120-cell
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References

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