Hexagonal prism

Summary

Uniform hexagonal prism
Hexagonal prism.png
Type Prismatic uniform polyhedron
Elements F = 8, E = 18, V = 12 (χ = 2)
Faces by sides 6{4}+2{6}
Schläfli symbol t{2,6} or {6}×{}
Wythoff symbol 2 6 | 2
2 2 3 |
Coxeter diagrams CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 2.pngCDel node h.pngCDel 6.pngCDel node h.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node 1.png
Symmetry D6h, [6,2], (*622), order 24
Rotation group D6, [6,2]+, (622), order 12
References U76(d)
Dual Hexagonal dipyramid
Properties convex, zonohedron
Hexagonal prism vertfig.png
Vertex figure
4.4.6

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

3D model of a uniform hexagonal prism.

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 tilarity 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) polyhedronEdit

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.

VolumeEdit

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 it's surface area can be  .

SymmetryEdit

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

Name Regular-hexagonal prism Hexagonal frustum Ditrigonal prism Triambic prism Ditrigonal trapezoprism
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 tesselationsEdit

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
       
       

Related polyhedra and tilingsEdit

Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622) [6,2]+, (622) [6,2+], (2*3)
                 
                                                     
{6,2} t{6,2} r{6,2} t{2,6} {2,6} rr{6,2} tr{6,2} sr{6,2} s{2,6}
Duals to uniforms
                 
V62 V122 V62 V4.4.6 V26 V4.4.6 V4.4.12 V3.3.3.6 V3.3.3.3

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.

*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]
*∞32
[∞,3]
 
[12i,3]
 
[9i,3]
 
[6i,3]
 
[3i,3]
Figures                        
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals                        
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i

See alsoEdit

Family of uniform n-gonal prisms
Prism name Digonal prism (Trigonal)
Triangular prism
(Tetragonal)
Square prism
Pentagonal prism Hexagonal prism Heptagonal prism Octagonal prism Enneagonal prism Decagonal prism Hendecagonal prism Dodecagonal prism ... Apeirogonal prism
Polyhedron image                       ...
Spherical tiling image                 Plane tiling image  
Vertex config. 2.4.4 3.4.4 4.4.4 5.4.4 6.4.4 7.4.4 8.4.4 9.4.4 10.4.4 11.4.4 12.4.4 ... ∞.4.4
Coxeter diagram                                                                   ...      

ReferencesEdit

  1. ^ a b Pugh, Anthony (1976), Polyhedra: 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.

External linksEdit

  • Uniform Honeycombs in 3-Space VRML models
  • The Uniform Polyhedra
  • Virtual Reality Polyhedra The Encyclopedia of Polyhedra Prisms and antiprisms
  • Weisstein, Eric W. "Hexagonal prism". MathWorld.
  • Hexagonal Prism Interactive Model -- works in your web browser