Hendecagrammic prism

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The four regular hendecagrams
{11/2}, {11/3}, {11/4}, and {11/5} HendecagramTypes.png
The four regular hendecagrams
{11/2}, {11/3}, {11/4}, and {11/5}

In geometry, a hendecagrammic prism is a star polyhedron made from two identical regular hendecagrams connected by squares. The related hendecagrammic antiprisms are made from two identical regular hendecagrams connected by equilateral triangles.

Contents

Hendecagrammic prisms and bipyramids

There are 4 hendecagrammic uniform prisms, and 6 hendecagrammic uniform antiprisms. The prisms are constructed by 4.4.11/q vertex figures, CDel node 1.pngCDel 2.pngCDel node.pngCDel 11.pngCDel rat.pngCDel q.pngCDel node 1.png Coxeter diagram. The hendecagrammic bipyramids, duals to the hendecagrammic prisms are also given.

SymmetryPrisms
D11h
[2,11]
(*2.2.11)
Prism 11-2.png
4.4.11/2
CDel node 1.pngCDel 2.pngCDel node.pngCDel 11.pngCDel rat.pngCDel d2.pngCDel node 1.png
Prism 11-3.png
4.4.11/3
CDel node 1.pngCDel 2.pngCDel node.pngCDel 11.pngCDel rat.pngCDel d3.pngCDel node 1.png
Prism 11-4.png
4.4.11/4
CDel node 1.pngCDel 2.pngCDel node.pngCDel 11.pngCDel rat.pngCDel d4.pngCDel node 1.png
Prism 11-5.png
4.4.11/5
CDel node 1.pngCDel 2.pngCDel node.pngCDel 11.pngCDel rat.pngCDel d5.pngCDel node 1.png
D11h
[2,11]
(*2.2.11)
11-2 dipyramid.png
CDel node f1.pngCDel 2.pngCDel node.pngCDel 11.pngCDel rat.pngCDel 2x.pngCDel node f1.png
11-3 dipyramid.png
CDel node f1.pngCDel 2.pngCDel node.pngCDel 11.pngCDel rat.pngCDel 3x.pngCDel node f1.png
11-4 dipyramid.png
CDel node f1.pngCDel 2.pngCDel node.pngCDel 11.pngCDel rat.pngCDel 4.pngCDel node f1.png
11-5 dipyramid.png
CDel node f1.pngCDel 2.pngCDel node.pngCDel 11.pngCDel rat.pngCDel 5.pngCDel node f1.png

Hendecagrammic antiprisms

The antiprisms with 3.3.3.3.11/q vertex figures, CDel node h.pngCDel 2.pngCDel node h.pngCDel 11.pngCDel rat.pngCDel q.pngCDel node h.png. Uniform antiprisms exist for p/q>3/2, [1] and are called crossed for p/q<2. For hendecagonal antiprism, two crossed antiprisms can not be constructed as uniform (with equilateral triangles): 11/8, and 11/9.

SymmetryAntiprismsCrossed- antiprisms
D11h
[2,11]
(*2.2.11)
Antiprism 11-2.png
3.3.3.11/2
 
CDel node h.pngCDel 2.pngCDel node h.pngCDel 11.pngCDel rat.pngCDel d2.pngCDel node h.png
Antiprism 11-4.png
3.3.3.11/4
 
CDel node h.pngCDel 2.pngCDel node h.pngCDel 11.pngCDel rat.pngCDel d4.pngCDel node h.png
Antiprism 11-6.png
3.3.3.11/6
3.3.3.-11/5
CDel node h.pngCDel 2.pngCDel node h.pngCDel 11.pngCDel rat.pngCDel 6.pngCDel node h.png
Nonuniform
3.3.3.11/8
3.3.3.-11/3
D11d
[2+,11]
(2*11)
Antiprism 11-3.png
3.3.3.11/3
 
CDel node h.pngCDel 2.pngCDel node h.pngCDel 11.pngCDel rat.pngCDel d3.pngCDel node h.png
Antiprism 11-5.png
3.3.3.11/5
 
CDel node h.pngCDel 2.pngCDel node h.pngCDel 11.pngCDel rat.pngCDel d5.pngCDel node h.png
Antiprism 11-7.png
3.3.3.11/7
3.3.3.-11/4
CDel node h.pngCDel 2.pngCDel node h.pngCDel 11.pngCDel rat.pngCDel d7.pngCDel node h.png
Nonuniform
3.3.3.11/9
3.3.3.-11/2

Hendecagrammic trapezohedra

The hendecagrammic trapezohedra are duals to the hendecagrammic antiprisms.

SymmetryTrapezohedra
D11h
[2,11]
(*2.2.11)
11-2 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 2x.pngCDel node fh.png
11-4 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 4.pngCDel node fh.png
11-6 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 6.pngCDel node fh.png
D11d
[2+,11]
(2*11)
11-3 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 3x.pngCDel node fh.png
11-5 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 5.pngCDel node fh.png
11-7 deltohedron.png
CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 11.pngCDel rat.pngCDel 7.pngCDel node fh.png

See also

Related Research Articles

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<span class="mw-page-title-main">Uniform 4-polytope</span> Class of 4-dimensional polytopes

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<span class="mw-page-title-main">Uniform polyhedron</span> Isogonal polyhedron with regular faces

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References

  1. Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, Bibcode:1976MPCPS..79..447S, doi:10.1017/S0305004100052440, MR   0397554 .