Icositetrahedron

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Triakisoctahedron.jpg
Triakis octahedron
Tetrakishexahedron.jpg
Tetrakis hexahedron
Deltoidalicositetrahedron.jpg
Deltoidal icositetrahedron
Pentagonalicositetrahedroncw.jpg
Pentagonal icositetrahedron

In geometry, an icositetrahedron [1] refers to a polyhedron with 24 faces, none of which are regular polyhedra. However, many are composed of regular polygons, such as the triaugmented dodecahedron and the disphenocingulum. Some icositetrahedra are near-spherical, but are not composed of regular polygons. A minimum of 14 vertices is required to form a icositetahedron. [2]

Contents

Symmetry

There are many symmetric forms, and the ones with highest symmetry have chiral icosahedral symmetry:

Four Catalan solids, convex:

27 uniform star-polyhedral duals: (self-intersecting)

Examples with lower symmetry include certain dual polyhedra of Johnson solids, such as the gyroelongated square bicupola and the elongated square gyrobicupola.

Common examples

Common examples include prisms and pyramids, and include certain Johnson solids and Catalan solids.

Icositrigonal pyramids

Icositrigonal pyramids are a type of cone with an icositrigon as a base, with 24 faces, 46 edges, and 24 vertices. [3] Regular icositrigonal pyramids have a regular icositrigon as a base, and its Schläfli symbol is {}∨{23}. The surface area and volume with side length and height can be calculated as follows: [3]

Icosidigonal prism

Icosidigonal prisms are a type of cylinder with an icosidigon as a base, with 24 faces, 66 edges, and 44 vertices. [4] Regular icosidigonal prisms have a regular icosidigon as a base, with each face a rectangle. Every vertex borders 2 squares and an icosidigon base. Its vertex configuration is , its Schläfli symbol is {22}×{} or t{2,22}, its Coxeter diagram is CDel node 1.pngCDel 2x.pngCDel 2x.pngCDel node.pngCDel 2.pngCDel node 1.png, and its Conway polyhedron notation is P22. The surface area and volume with side length and height can be calculated as follows:

Hendecagonal antiprism

A hendecagonal antiprism Hendecagonal antiprism.png
A hendecagonal antiprism

Hendecagonal antiprisms are antiprisms with a hendecagon as a base, with 24 faces, 44 edges, and 22 vertices. Regular hendecagonal antiprisms have a regular hendecagon as a base, with each face an equilateral triangle. Every vertex borders 2 triangles and a hendecagon base. Its vertex configuration is .

Dodecagonal trapezohedron

Dodecagonal trapezohedra are the tenth member of the trapezohedra family, made of 24 congruent kites arranged radially. Every dodecagonal trapezohedron has 24 faces, 28 edges, and 26 vertices. There are two types of vertices, ones bordering 12 kits and ones bordering 3. Its dual polyhedron is the Hendecagonal antiprism. [5] Its Schläfli symbol is { }⨁{12}, its Coxeter diagram is CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 2x.pngCDel 4.pngCDel node.png or CDel node fh.pngCDel 2.pngCDel node fh.pngCDel 12.pngCDel node fh.png, and its Conway polyhedron notation is dA12.

Dodecagonal trapezohedra are isohedral figures.

Johnson solids

There are two examples of Johnson solids which are icositetrahedra. They are listed as follows:

NameImageDesignationVerticesEdgesFacesTypes of facesSymmetry groupNet
Disphenocingulum Disphenocingulum.png J9016382420 equilateral triangle,

4 squares

D2d Johnson solid 90 net.png
Triaugmented dodecahedron Triaugmented dodecahedron.png J6123452415 equilateral triangles,

9 pentagons

C3v Johnson solid 61 net.png

Catalan Solids

There are 5 types of icositetrahedra with different topologies. [6] The pentagonal icositetetrahedron has two mirror images (enantiomorphs), so geometrically there are 4 distinct Catalan icositetetrahedra.

NameImageNetDualFacesEdgesVertices Face Configuration Point Group
Triakis octahedron Triakisoctahedron.jpg

(animation)

Triakisoctahedron net.png Truncated cube 243614Isosceles triangle

V3.8.8

Oh
Tetrakis hexahedron Tetrakishexahedron.jpg

(animation)

Tetrakishexahedron net.png Truncated octahedron 243614Isosceles triangle

V4.6.6

Oh
Deltoidal icositetrahedron Deltoidalicositetrahedron.jpg

(animation)

Deltoidalicositetrahedron net.png Rhombicuboctahedron 244826Kite

V3.4.4.4

Oh
Pentagonal icositetrahedron Pentagonalicositetrahedronccw.jpg

(animation) Pentagonalicositetrahedroncw.jpg (animation)

Pentagonalicositetrahedron net.svg Snub cube 246038irregular pentagon

V3.3.3.3.4

O

Uniform star polyhedra

Some uniform star polyhedra also have 24 faces:

NameImage Wythoff symbol Vertex figure Symmetry groupFacesEdgesVertices Euler characteristic Density Faces by sides
Ditrigonal dodecadodecahedron Ditrigonal dodecadodecahedron.png 3 | 5/3 5 Ditrigonal dodecadodecahedron vertfig.svg

(5.5/3)3

Ih246020-16412{5}+12{5/2}
Dodecadodecahedron Dodecadodecahedron.png 5 5/2 Dodecadodecahedron vertfig.png

5.5/2.5.5/2

Ih246020-16412{5}+12{5/2}
Truncated great dodecahedron Great truncated dodecahedron.png 2 5/2 | 5 Truncated great dodecahedron vertfig.png

10.10.5/2

Ih249060-6312{5/2}+12{10}
Small stellated truncated dodecahedron Small stellated truncated dodecahedron.png 2 5 | 5/3 Small stellated truncated dodecahedron vertfig.png

10/3.10/3.5

Ih249060-6912{5}+12{10/3}

Types of icositetrahedra

NameTypeImageIdentifierFacesEdgesVerticesEuler characteristicTypes of facesSymmetryNet
Icosidigonal prismPrismt{2,22}

{22}x{} CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2x.pngCDel 2x.pngCDel node.png

24664422 icosidigons,

22 squares

D22h, [22,2], (*22 2 2), order 88
Icositrigonal pyramidPyramid( )∨{23}24462421 icositrigon,

23 triangles

C23v, [23], (*23 23)
Icosidigonal frustum Frustum 24664422 icosidigons,

22 trapezoids

D22h, [22,2], (*22 2 2), order 88
Dodecagonal bipyramid Bipyramid { } + {12}

CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 12.pngCDel node.png

243614212 trianglesD12h, [12,2], (*2 2 12), order 48
Dodecagonal trapezohedronTrapezohedron Dodecagonal trapezohedron.png { }⨁{12} [7] 244826224 kitesD12d, [2+,12], (2*12)
Hendecagonal antiprismAntiprism Hendecagonal antiprism.png s{2,22}

sr{2,11} CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel 2x.pngCDel node.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 11.pngCDel node h.png

24442222 hendecagons,

22 triangles

D11d, [2+,22], (2*11), order 44
Hendecagonal cupola Cupola 245533211 equilateral triangles,

11 squares, 1 regular hendecagon, 1 regular icosidigon

D11d, [2+,22], (2*11), order 44
Deltoidal icositetrahedronJohnson solid Deltoidalicositetrahedron.jpg 244826224 kitesD4d Pseudo-strombic icositetrahedron flat.png

See also

References

  1. "Greek numerical prefixes". www.georgehart.com. Retrieved 2025-02-02.
  2. "Enumeration of Polyhedra - Numericana". www.numericana.com. Retrieved 2025-02-02.
  3. 1 2 "Icositrigonal pyramid - Wolfram|Alpha". www.wolframalpha.com. Archived from the original on 2024-11-30. Retrieved 2025-02-02.
  4. "Icosidigonal prism". Wolfram Alpha. Retrieved 2025-02-02.
  5. "12-trapezohedron". Wolfram Alpha. Retrieved 2025-02-02.
  6. "Sur la théorie des quantités positives et négatives", Cours d'analyse de l'École Royale Polytechnique, Cambridge University Press, pp. 403–437, 2009-07-20, retrieved 2025-02-02
  7. Johnson, N.W. (2018). "Chapter 11: Finite symmetry groups". Geometries and Transformations. p. 235. ISBN   978-1-107-10340-5.