View source for Catalan solid

{{short description|13 polyhedra; duals of the Archimedean solids}}
[[File:Catalan-18.jpg|thumb|Set of Catalan solids]]

The '''Catalan solids''' are the [[dual polyhedron|dual polyhedra]] of [[Archimedean solid]]s. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices.{{sfnp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}} The faces of the Catalan solids correspond by duality to the vertices of Archimedean solids, and vice versa.{{sfnp|Wenninger|1983|p=1|loc=Basic notions about stellation and duality}}

== The solids ==

The Catalan solids are [[face-transitive]] or ''isohedral'' meaning that their faces are symmetric to one another, but they are not [[vertex-transitive]] because their vertices are not symmetric. Their duals, the Archimedean solids, are vertex-transitive but not face-transitive. Each Catalan solid has constant [[dihedral angle]]s, meaning the angle between any two adjacent faces is the same.{{sfnp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}} Additionally, two Catalan solids, the [[rhombic dodecahedron]] and [[rhombic triacontahedron]], are [[edge-transitive]], meaning their edges are symmetric to each other.{{cn|date=October 2024}} Some Catalan solids were discovered by [[Johannes Kepler]] during his study of [[zonohedron|zonohedra]], and [[Eugene Catalan]] completed the list of the thirteen solids in 1865.<ref>{{multiref
|{{harvp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}}
|{{harvp|Heil|Martini|1993|p=[https://books.google.com/books?id=M2viBQAAQBAJ&pg=PA352 352]}}
}}</ref>

[[File:DormanLuke.svg|thumb|upright|The [[rhombic dodecahedron]]'s construction, the dual polyhedron of a [[cuboctahedron]], by [[Dorman Luke construction]]]]
In general, each face of a dual uniform polyhedron (including the Catalan solid) can be constructed by using the [[Dorman Luke construction]].<ref>{{multiref
|{{harvp|Cundy|Rollett|1961|p=117}}
|{{harvp|Wenninger|1983|p=30}}
}}</ref> Some of the Catalan solids can be constructed by adding pyramids to the faces of Platonic solids. These examples are [[Kleetope]]s of Platonic solids: [[triakis tetrahedron]], [[tetrakis hexahedron]], [[triakis octahedron]], [[triakis icosahedron]], and [[pentakis dodecahedron]].<ref>{{multiref
 |{{harvp|Brigaglia|Palladino|Vaccaro|2018}}
 |{{harvp|Çolak|Gelişgen|2015}}
}}</ref>

Two Catalan solids, the [[pentagonal icositetrahedron]] and the [[pentagonal hexecontahedron]], are [[Chirality (mathematics)|chiral]], meaning that these two solids are not their own mirror images. They are dual to the [[snub cube]] and [[snub dodecahedron]] respectively, which are also chiral.

Eleven of the thirteen Catalan solids are known to have the [[Rupert property]] that a copy of the same solid can be passed through a hole in the solid.{{sfnp|Fredriksson|2024}}

{| class="wikitable sortable"
|+ The thirteen Catalan solids
! Name
! Image
! Faces
! Edges
! Vertices
! Dihedral angle{{sfnp|Williams|1979}}
! [[List of spherical symmetry groups#Polyhedral symmetry|Point group]]
|- align="center"
| [[triakis tetrahedron]]
| [[Image:triakistetrahedron.jpg|70px|Triakis tetrahedron]]
|  12 [[Isosceles triangle|isosceles triangles]]
|  18
|  8
| 129.521°
|  T<sub>d</sub>
|- align="center"
| [[rhombic dodecahedron]]
| [[Image:rhombicdodecahedron.jpg|70px|Rhombic dodecahedron]]
|  12 [[Rhombus|rhombi]]
|  24
|  14
| 120°
|  O<sub>h</sub>
|- align="center"
| [[triakis octahedron]]
| [[Image:triakisoctahedron.jpg|70px|Triakis octahedron]]
|  24 isosceles triangles
|  36
|  14
| 147.350°
|  O<sub>h</sub>
|- align="center"
| [[tetrakis hexahedron]]
| [[Image:tetrakishexahedron.jpg|70px|Tetrakis hexahedron]]
|  24 isosceles triangles
|  36
|  14
| 143.130°
|  O<sub>h</sub>
|- align="center"
| [[deltoidal icositetrahedron]]
| [[Image:deltoidalicositetrahedron.jpg|70px|Deltoidal icositetrahedron]]
|  24 [[Kite (geometry)|kites]]
|  48
|  26
| 138.118°
|  O<sub>h</sub>
|- align="center"
| [[disdyakis dodecahedron]]
| [[Image:disdyakisdodecahedron.jpg|70px|Disdyakis dodecahedron]]
|  48 [[Scalene triangle|scalene triangles]]
|  72
|  26
| 155.082°
|  O<sub>h</sub>
|- align="center"
| [[pentagonal icositetrahedron]]
| [[Image:pentagonalicositetrahedronccw.jpg|70px|Pentagonal icositetrahedron (Ccw)]]
|  24 [[Pentagon|pentagons]]
|  60
|  38
| 136.309°
|  O
|- align="center"
| [[rhombic triacontahedron]]
| [[Image:rhombictriacontahedron.svg|70px|Rhombic triacontahedron]]
|  30 [[Rhombus|rhombi]]
|  60
|  32
| 144°
|  I<sub>h</sub>
|- align="center"
| [[triakis icosahedron]]
| [[Image:triakisicosahedron.jpg|70px|Triakis icosahedron]]
|  60 isosceles triangles
|  90
|  32
| 160.613°
|  I<sub>h</sub>
|- align="center"
| [[pentakis dodecahedron]]
| [[Image:pentakisdodecahedron.jpg|70px|Pentakis dodecahedron]]
|  60 isosceles triangles
|  90
|  32
| 156.719°
|  I<sub>h</sub>
|- align="center"
| [[deltoidal hexecontahedron]]
| [[Image:deltoidalhexecontahedron.jpg|70px|Deltoidal hexecontahedron]]
|  60 kites
|  120
|  62
| 154.121°
|  I<sub>h</sub>
|- align="center"
| [[disdyakis triacontahedron]]
| [[Image:disdyakistriacontahedron.jpg|70px|Disdyakis triacontahedron]]
|  120 scalene triangles
|  180
|  62
| 164.888°
|  I<sub>h</sub>
|- align="center"
| [[pentagonal hexecontahedron]]
| [[Image:pentagonalhexecontahedronccw.jpg|70px|Pentagonal hexecontahedron (Ccw)]]
|  60 pentagons
|  150
|  92
| 153.179°
|  I
|}

== References ==
{{sfn whitelist|CITEREFWilliams1979}}
=== Footnotes ===
{{reflist|20em}}

=== Works cited ===
{{refbegin|30em}}
* {{citation
 | last1 = Brigaglia | first1 = Aldo
 | last2 = Palladino | first2 = Nicla
 | last3 = Vaccaro | first3 = Maria Alessandra
 | editor1-last = Emmer | editor1-first = Michele
 | editor2-last = Abate | editor2-first = Marco
 | contribution = Historical notes on star geometry in mathematics, art and nature
 | doi = 10.1007/978-3-319-93949-0_17
 | pages = 197–211
 | publisher = Springer International Publishing
 | title = Imagine Math 6: Between Culture and Mathematics
 | year = 2018| isbn = 978-3-319-93948-3
 | hdl = 10447/325250
 | hdl-access = free
 }}.
* {{citation
 | last1 = Çolak | first1 = Zeynep
 | last2 = Gelişgen | first2 = Özcan
 | year = 2015
 | title = New Metrics for Deltoidal Hexacontahedron and Pentakis Dodecahedron
 | journal = Sakarya University Journal of Science
 | volume = 19
 | issue = 3
 | pages = 353&ndash;360
 | doi = 10.16984/saufenbilder.03497
 | doi-broken-date = 1 July 2025
 | url = https://dergipark.org.tr/en/pub/saufenbilder/issue/20705/221184
}}
* {{citation
 | last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy
 | last2 = Rollett | first2 = A. P.
 | edition = 2nd
 | location = Oxford
 | mr = 0124167
 | publisher = Clarendon Press
 | title = Mathematical Models
 | title-link = Mathematical Models (Cundy and Rollett)
 | year = 1961}}.
* {{citation
 | last = Diudea | first = M. V.
 | year = 2018
 | title = Multi-shell Polyhedral Clusters
 | series = Carbon Materials: Chemistry and Physics
 | volume = 10
 | publisher = [[Springer Science+Business Media|Springer]]
 | isbn = 978-3-319-64123-2
 | doi = 10.1007/978-3-319-64123-2
 | url = https://books.google.com/books?id=p_06DwAAQBAJ
 }}.
* {{citation
 | last = Fredriksson | first = Albin
 | title = Optimizing for the Rupert property
 | journal = [[The American Mathematical Monthly]]
 | pages = 255–261
 | volume = 131
 | issue = 3
 | year = 2024
 | doi = 10.1080/00029890.2023.2285200
 | arxiv = 2210.00601
 }}.
* {{citation
 | last1 = Gailiunas | first1 = P.
 | last2 = Sharp | first2 = J.
 | doi = 10.1080/00207390500064049
 | issue = 6
 | journal = International Journal of Mathematical Education in Science and Technology
 | pages = 617–642
 | title = Duality of polyhedra
 | volume = 36
 | year = 2005| bibcode = 2005IJMES..36..617G
 | s2cid = 120818796
 }}.
* {{citation
 | last1 = Heil | first1 = E.
 | last2 = Martini | first2 = H.
 | editor-last1 = Gruber | editor-first = P. M.
 | editor-last2 = Wills | editor-first2 = J. M.
 | url = https://books.google.com/books?id=M2viBQAAQBAJ
 | publisher = North Holland
 | title = Handbook of Convex Geometry
 | contribution = Special convex bodies
 | year = 1993
| isbn = 978-0-08-093439-6
 }}
* {{citation
 | last = Wenninger | first = Magnus | author-link = Magnus Wenninger
 | title = Dual Models
 | publisher = [[Cambridge University Press]]
 | isbn = 978-0-521-54325-5
 | mr = 730208
 | year = 1983
}}
* {{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
{{refend}}

==External links==
{{commons category|Catalan solids}}
* {{MathWorld | urlname=CatalanSolid | title=Catalan Solids}}
* {{mathworld | urlname = Isohedron | title = Isohedron}}
* [http://dmccooey.com/polyhedra/Catalan.html Catalan Solids] &ndash; at Visual Polyhedra
* [http://www.georgehart.com/virtual-polyhedra/archimedean-duals-info.html Archimedean duals] &ndash; at Virtual Reality Polyhedra
* [https://web.archive.org/web/20051201035511/http://ibiblio.org/e-notes/3Dapp/Catalan.htm Interactive Catalan Solid] in Java
* [https://orbi.uliege.be/handle/2268/194785 Download link for Catalan's original 1865 publication] &ndash; with beautiful figures, PDF format
{{Catalan solids}}
{{Polyhedron navigator}}

{{DEFAULTSORT:Catalan Solid}}
[[Category:Catalan solids|*]]
[[Category:Polyhedra]]