Catalan solid

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Set of Catalan solids Catalan-18.jpg
Set of Catalan solids
The rhombic dodecahedron's construction, the dual polyhedron of a cuboctahedron, by Dorman Luke construction DormanLuke.svg
The rhombic dodecahedron's construction, the dual polyhedron of a cuboctahedron, by Dorman Luke construction

The Catalan solids are the dual polyhedron of Archimedean solids, a set of thirteen polyhedrons with highly symmetric forms semiregular polyhedrons in which two or more polygonal of their faces are met at a vertex. [1] A polyhedron can have a dual by corresponding vertices to the faces of the other polyhedron, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. [2] One way to construct the Catalan solids is by using the method of Dorman Luke construction. [3]

Contents

These solids are face-transitive or isohedral because their faces are transitive to one another, but they are not vertex-transitive because their vertices are not transitive to one another. Their dual, the Archimedean solids, are vertex-transitive but not face-transitive. Each has constant dihedral angles, meaning the angle between any two adjacent faces is the same. [1] Additionally, both Catalan solids rhombic dodecahedron and rhombic triacontahedron are edge-transitive, meaning there is an isometry between any two edges preserving the symmetry of the whole.[ citation needed ] These solids were also already discovered by Johannes Kepler during the study of zonohedrons, until Eugene Catalan first completed the list of the thirteen solids in 1865. [4]

The pentagonal icositetrahedron and the pentagonal hexecontahedron are chiral because they are dual to the snub cube and snub dodecahedron respectively, which are chiral; that is, these two solids are not their own mirror images.

Eleven of the thirteen Catalan solids are known to have the Rupert property (a copy of the same solid can be passed through a hole in the solid). [5]

The thirteen Catalan solids
NameImageFacesEdgesVerticesDihedral angle [6] Point group
triakis tetrahedron Triakistetrahedron.jpg 12 isosceles triangles 188129.521°Td
rhombic dodecahedron Rhombicdodecahedron.jpg 12 rhombi 2414120°Oh
triakis octahedron Triakisoctahedron.jpg 24 isosceles triangles3614147.350°Oh
tetrakis hexahedron Tetrakishexahedron.jpg 24 isosceles triangles3614143.130°Oh
deltoidal icositetrahedron Deltoidalicositetrahedron.jpg 24 kites 4826138.118°Oh
disdyakis dodecahedron Disdyakisdodecahedron.jpg 48 scalene triangles 7226155.082°Oh
pentagonal icositetrahedron Pentagonalicositetrahedronccw.jpg 24 pentagons 6038136.309°O
rhombic triacontahedron Rhombictriacontahedron.svg 30 rhombi 6032144°Ih
triakis icosahedron Triakisicosahedron.jpg 60 isosceles triangles9032160.613°Ih
pentakis dodecahedron Pentakisdodecahedron.jpg 60 isosceles triangles9032156.719°Ih
deltoidal hexecontahedron Deltoidalhexecontahedron.jpg 60 kites12062154.121°Ih
disdyakis triacontahedron Disdyakistriacontahedron.jpg 120 scalene triangles18062164.888°Ih
pentagonal hexecontahedron Pentagonalhexecontahedronccw.jpg 60 pentagons15092153.179°I

Related Research Articles

<span class="mw-page-title-main">Archimedean solid</span> Polyhedra in which all vertices are the same

The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons, but not all alike, and whose vertices are all symmetric to each other. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric vertices. Some Archimedean solids were portrayed in the works of artists and mathematicians during the Renaissance.

<span class="mw-page-title-main">Cuboctahedron</span> Polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron.

<span class="mw-page-title-main">Icosidodecahedron</span> Archimedean solid with 32 faces

In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

<span class="mw-page-title-main">Rhombicuboctahedron</span> Archimedean solid with 26 faces

In geometry, the rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named by Johannes Kepler in his 1618 Harmonices Mundi, being short for truncated cuboctahedral rhombus, with cuboctahedral rhombus being his name for a rhombic dodecahedron.

<span class="mw-page-title-main">Truncated icosahedron</span> A polyhedron resembling a soccerball

In geometry, the truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. Intuitively, it may be regarded as footballs that are typically patterned with white hexagons and black pentagons. It can be found in the application of geodesic dome structures such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It is an example of an Archimedean solid, as well as a Goldberg polyhedron.

<span class="mw-page-title-main">Stellation</span> Extending the elements of a polytope to form a new figure

In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from the Latin stella, "star". Stellation is the reciprocal or dual process to faceting.

<span class="mw-page-title-main">Truncated tetrahedron</span> Archimedean solid with 8 faces

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges. It can be constructed by truncating all 4 vertices of a regular tetrahedron.

<span class="mw-page-title-main">Truncated icosidodecahedron</span> Archimedean solid

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

<span class="mw-page-title-main">Truncated dodecahedron</span> Archimedean solid with 32 faces

In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.

<span class="mw-page-title-main">Rhombic dodecahedron</span> Catalan solid with 12 faces

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb. There are some variations of the rhombic dodecahedron, one of which is the Bilinski dodecahedron. There are some stellations of the rhombic dodecahedron, one of which is the Escher's solid. The rhombic dodecahedron may also appearances in the garnet crystal, the architectural philosophies, practical usages, and toys.

<span class="mw-page-title-main">Rhombic triacontahedron</span> Catalan solid with 30 faces

The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

<span class="mw-page-title-main">Disdyakis dodecahedron</span> Geometric shape with 48 faces

In geometry, a disdyakis dodecahedron,, is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons. It resembles an augmented rhombic dodecahedron. Replacing each face of the rhombic dodecahedron with a flat pyramid creates a polyhedron that looks almost like the disdyakis dodecahedron, and is topologically equivalent to it.

<span class="mw-page-title-main">Disdyakis triacontahedron</span> Catalan solid with 120 faces

In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.

In geometry, the term semiregular polyhedron is used variously by different authors.

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

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular, quasi-regular, or semi-regular. The faces and vertices don't need to be convex, so many of the uniform polyhedra are also star polyhedra.

<span class="mw-page-title-main">Vertex configuration</span> Notation for a polyhedrons vertex figure

In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron.

<span class="mw-page-title-main">Conway polyhedron notation</span> Method of describing higher-order polyhedra

In geometry and topology, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

A dual uniform polyhedron is the dual of a uniform polyhedron. Where a uniform polyhedron is vertex-transitive, a dual uniform polyhedron is face-transitive.

References

Footnotes

  1. 1 2 Diudea (2018), p.  39.
  2. Wenninger (1983), p. 1, Basic notions about stellation and duality.
  3. Fredriksson (2024).
  4. Williams (1979).

Works cited

  • Cundy, H. Martyn; Rollett, A. P. (1961), Mathematical Models (2nd ed.), Oxford: Clarendon Press, MR   0124167 .
  • Diudea, M. V. (2018), Multi-shell Polyhedral Clusters, Springer, doi:10.1007/978-3-319-64123-2, ISBN   978-3-319-64123-2 .
  • Fredriksson, Albin (2024), "Optimizing for the Rupert property", The American Mathematical Monthly , 131 (3): 255–261, arXiv: 2210.00601 , doi:10.1080/00029890.2023.2285200 .
  • Gailiunas, P.; Sharp, J. (2005), "Duality of polyhedra", International Journal of Mathematical Education in Science and Technology, 36 (6): 617–642, doi:10.1080/00207390500064049, S2CID   120818796 .
  • Heil, E.; Martini, H. (1993), "Special convex bodies", in Gruber, P. M.; Wills, J. M. (eds.), Handbook of Convex Geometry, North Holland
  • Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN   978-0-521-54325-5, MR   0730208
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN   0-486-23729-X. (Section 3-9)