icositetrahedron#Other icositetrahedron

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

Four Catalan solids, convex:

• Triakis octahedron - isosceles triangles
• Tetrakis hexahedron - isosceles triangles
• Deltoidal icositetrahedron - kites
• Pentagonal icositetrahedron - pentagons

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

• Small rhombihexacron, Great rhombihexacron
• Small hexacronic icositetrahedron, Great hexacronic icositetrahedron
• Great deltoidal icositetrahedron
• Great triakis octahedron

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

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

Icositrigonal pyramids are a type of cone with an icositrigon as a base, with 24 faces, 46 edges, and 24 vertices.{{Cite web |title=Icositrigonal pyramid - Wolfram{{!}}Alpha |url=https://www.wolframalpha.com/input/?i=Icositrigonal+pyramid |archive-url=http://web.archive.org/web/20241130162738/https://www.wolframalpha.com/input/?i=Icositrigonal+pyramid |archive-date=2024-11-30 |access-date=2025-02-02 |website=www.wolframalpha.com |language=en}} Regular icositrigonal pyramids have a regular icositrigon as a base, and its Schläfli symbol is {}∨{23}. The surface area $S$ and volume $V$ with side length $s$ and height $h$ can be calculated as follows:

: $V=\backslash frac\{23\; h\; s^2\; \backslash cot\{\backslash frac\{\backslash pi\}\{23\}\}\}\{12\}\backslash approx\; 13.9448\; h\; s^2$

: $S=\backslash frac\{23s\backslash left(\backslash sqrt\{4h^2+s^2\backslash cot^\{2\}\{\backslash frac\{\backslash pi\}\{23\}\}\}+s\backslash cot\{\backslash frac\{\backslash pi\}\{23\}\}\backslash right)\}\{4\}\backslash approx\; 5.75s\backslash left(\backslash sqrt\{4h^2+52.9335s^2\}+7.27554s\backslash right)$

Icosidigonal prisms are a type of cylinder with an icosidigon as a base, with 24 faces, 66 edges, and 44 vertices.{{Cite web |title=Icosidigonal prism |url=https://www.wolframalpha.com/input/?i=Icosidigonal+prism |access-date=2025-02-02 |website=Wolfram Alpha}} 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 $4\{.\}4\{.\}22$, its Schläfli symbol is {22}×{} or t{2,22}, its Coxeter diagram is {{CDD|node_1|2x|2x|node|2|node_1}}, and its Conway polyhedron notation is P22. The surface area $S$ and volume $V$ with side length $s$ and height $h$ can be calculated as follows:

: $V=\backslash frac\{11\; h\; s^2\; \backslash cot\{\backslash frac\{\backslash pi\}\{22\}\}\}\{2\}\backslash approx\; 38.2533\; h\; s^2$

: $S=11s\backslash left(\; 2h+s\backslash cot\{\backslash frac\{\backslash pi\}\{22\}\}\backslash right)\backslash approx\; 11s\backslash left(2h+6.95515s\backslash right)$

File:Hendecagonal_antiprism.png

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 $11\{.\}3\{.\}3\{.\}3$.

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.{{Cite web |title=12-trapezohedron |url=https://www.wolframalpha.com/input/?i=12-trapezohedron |access-date=2025-02-02 |website=Wolfram Alpha}} Its Schläfli symbol is { }⨁{12}, its Coxeter diagram is {{CDD||node_fh|2|node_fh|2x|4|node}} or {{CDD||node_fh|2|node_fh|12|node_fh}}, and its Conway polyhedron notation is dA12.

Dodecagonal trapezohedra are isohedral figures.

{{main|Johnson solid}}

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

{{main|Catalan solid}}

There are 5 types of icositetrahedra with different topologies.{{Citation |title=Sur la théorie des quantités positives et négatives |date=2009-07-20 |work=Cours d'analyse de l'École Royale Polytechnique |pages=403–437 |url=https://doi.org/10.1017/cbo9780511693328.016 |access-date=2025-02-02 |publisher=Cambridge University Press}} The pentagonal icositetetrahedron has two mirror images (enantiomorphs), so geometrically there are 4 distinct Catalan icositetetrahedra.

{{main|Uniform star polyhedron}}

Some uniform star polyhedra also have 24 faces:

• Icositetragon

• {{Mathworld |urlname=Icositetrahedron |title=Icositetrahedron}}

{{Polyhedra}}

Category:Polyhedra

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