Triakis octahedronTetrakis hexahedronDeltoidal icositetrahedronPentagonal icositetrahedron In geometry, an icositetrahedron 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.
Symmetry 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 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. 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:
Icosidigonal prism Icosidigonal prisms are a type of cylinder with an icosidigon as a base, with 24 faces, 66 edges, and 44 vertices. 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 , and its Conway polyhedron notation is P22. The surface area and volume with side length and height can be calculated as follows:
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. Its Schläfli symbol is { }⨁{12}, its Coxeter diagram is or , 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 groupNetDisphenocingulumJ9016382420 equilateral triangle, 4 squaresD2dTriaugmented dodecahedronJ6123452415 equilateral triangles, 9 pentagonsC3v
Catalan Solids
There are 5 types of icositetrahedra with different topologies. The pentagonal icositetetrahedron has two mirror images (enantiomorphs), so geometrically there are 4 distinct Catalan icositetetrahedra. NameImageNetDualFacesEdgesVerticesFace ConfigurationPoint GroupTriakis octahedron(animation)Truncated cube243614Isosceles triangle V3.8.8OhTetrakis hexahedron(animation)Truncated octahedron243614Isosceles triangle V4.6.6OhDeltoidal icositetrahedron(animation)Rhombicuboctahedron244826Kite V3.4.4.4OhPentagonal icositetrahedron(animation)
(animation)Snub cube246038irregular pentagon V3.3.3.3.4O
Uniform star polyhedra
Some uniform star polyhedra also have 24 faces: NameImageWythoff symbolVertex figureSymmetry groupFacesEdgesVerticesEuler characteristicDensityFaces by sidesDitrigonal dodecadodecahedron3 5/3 5(5.5/3)3Ih246020 -16412{5}+12{5/2}Dodecadodecahedron5 5/25.5/2.5.5/2Ih246020 -16412{5}+12{5/2}Truncated great dodecahedron2 5/2 510.10.5/2Ih249060 -6312{5/2}+12{10}Small stellated truncated dodecahedron2 5 5/310/3.10/3.5Ih249060 -6912{5}+12{10/3}
Types of icositetrahedra NameTypeImageIdentifierFacesEdgesVerticesEuler characteristicTypes of facesSymmetryNetIcosidigonal prismPrismt{2,22} {22}x{}24664422 icosidigons, 22 squaresD22h, [22,2], (*22 2 2), order 88Icositrigonal pyramidPyramid( )∨{23}24462421 icositrigon, 23 trianglesC23v, , (*23 23)Icosidigonal frustumFrustum24664422 icosidigons, 22 trapezoidsD22h, [22,2], (*22 2 2), order 88Dodecagonal bipyramidBipyramid{ } + {12}243614212 trianglesD12h, [12,2], (*2 2 12), order 48Dodecagonal trapezohedronTrapezohedron{ }⨁244826224 kitesD12d, [2+,12], (2*12)Hendecagonal antiprismAntiprisms{2,22} sr{2,11}24442222 hendecagons, 22 trianglesD11d, [2+,22], (2*11), order 44Hendecagonal cupolaCupola245533211 equilateral triangles, 11 squares, 1 regular hendecagon, 1 regular icosidigonD11d, [2+,22], (2*11), order 44Deltoidal icositetrahedronJohnson solid244826224 kitesD4d
See also
Icositetragon
References