Hexagonal prism

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Hexagon prism
Hexagonal Prism.svg
Type prism,
parallelohedron
Symmetry group prismatic symmetry of order 24
Dual polyhedron hexagonal bipyramid
3D model of a uniform hexagonal prism. Prisma hexagonal 3D.stl
3D model of a uniform 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]

Contents

As a semiregular 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 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

Hexagonal prismatic honeycomb Hexagonal prismatic honeycomb.png
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
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
Snub triangular-hexagonal prismatic honeycomb
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
Rhombitriangular-hexagonal prismatic honeycomb
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
Triangular-hexagonal prismatic honeycomb.png Snub triangular-hexagonal prismatic honeycomb.png Rhombitriangular-hexagonal prismatic honeycomb.png

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

truncated tetrahedral prism
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
truncated octahedral prism
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
Truncated cuboctahedral prism
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Truncated icosahedral prism
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.png
Truncated icosidodecahedral prism
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Truncated tetrahedral prism.png Truncated octahedral prism.png Truncated cuboctahedral prism.png Truncated icosahedral prism.png Truncated icosidodecahedral prism.png
runcitruncated 5-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
omnitruncated 5-cell
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
runcitruncated 16-cell
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
omnitruncated tesseract
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
4-simplex t013.svg 4-simplex t0123.svg 4-cube t023.svg 4-cube t0123.svg
runcitruncated 24-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
omnitruncated 24-cell
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
runcitruncated 600-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
omnitruncated 120-cell
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
24-cell t0123 F4.svg 24-cell t013 F4.svg 120-cell t023 H3.png 120-cell t0123 H3.png

References

  1. Pugh, Anthony (1976), Polyhedra: A Visual Approach, University of California Press, pp. 21, 27, 62, ISBN   9780520030565 .
  2. Flusser, J.; Suk, T.; Zitofa, B. (2017), 2D and 3D Image Analysis by Moments, John Wiley & Sons, p. 126, ISBN   978-1-119-03935-8
  3. Wheater, Carolyn C. (2007), Geometry, Career Press, pp. 236–237, ISBN   9781564149367
  4. Alexandrov, A. D. (2005), "8.1 Parallelohedra", Convex Polyhedra, Springer, pp. 349–359
  5. Delaney, Gary W.; Khoury, David (February 2013), "Onset of rigidity in 3D stretched string networks", The European Physical Journal B, 86 (2): 44, Bibcode:2013EPJB...86...44D, doi:10.1140/epjb/e2012-30445-y