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{{short description|13 polyhedra; duals of the Archimedean solids}}
{{short description|13 polyhedra; duals of the Archimedean solids}}
[[File:Catalan-18.jpg|thumb|Set of Catalan solids]]
{{multiple image
[[File:DormanLuke.svg|thumb|upright|The [[rhombic dodecahedron]]'s construction, the dual polyhedron of a [[cuboctahedron]], by [[Dorman Luke construction]]]]
| align = right | total_width=500
| image1 = Polyhedron truncated 4a dual max.png |width1=3800|height1=3655
| image2 = Polyhedron snub 6-8 left dual max.png |width2=3833|height2=3895
| image3 = Polyhedron great rhombi 12-20 dual max.png |width3=3928|height3=3773
| footer = [[Triakis tetrahedron]], [[pentagonal icositetrahedron]] and [[disdyakis triacontahedron]].
}}


The Catalan solids are the [[dual polyhedron]] of [[Archimedean solids]], a set of thirteen polyhedrons with highly symmetric forms [[semiregular polyhedron]]s in which two or more polygonal of their faces are met at a vertex.{{sfnp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}} 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.{{sfnp|Wenninger|1983|p=1|loc=Basic notions about stellation and duality}} One way to construct the Catalan solids is by using the method of [[Dorman Luke construction]].<ref>{{multiref
{{multiple image
|{{harvp|Cundy|Rollett|1961|p=117}}
| align = right | total_width=500
|{{harvp|Wenninger|1983|p=30}}
| image1 = Dual compound truncated 4a max.png |width1=3784|height1=3648
}}</ref>
| image2 = Dual compound snub 6-8 left max.png |width2=3908|height2=3928
| image3 = Dual compound great rhombi 12-20 max.png |width3=3924|height3=3792
| footer = The solids above (dark) shown together with their duals (light). The visible parts of the Catalan solids are regular [[Pyramid (geometry)|pyramids]].
}}


These solids are [[face-transitive]] or ''isohedron'' 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. They have constant [[dihedral angle]]s, meaning the angle between any two of their faces are the same.{{sfnp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}} Additionally, both Catalan solids [[rhombic dodecahedron]] and [[rhombic triacontahedron]] are [[edge-transitive]], meaning there is a transitive between the edges preserving their symmetrical.{{cn}} These solids were also already discovered by [[Johannes Kepler]] during the study of [[zonohedron]]s, until [[Eugene Catalan]] first completed the list of the thirteen solids in 1865.<ref>{{multiref
[[File:Polyhedron 6-8 dual vertexconfig.png|thumb|A [[rhombic dodecahedron]] with its [[face configuration]].]]
|{{harvp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}}
|{{harvp|Martini|Heil|1993|p=[https://books.google.com/books?id=M2viBQAAQBAJ&pg=PA352 352]}}
}}</ref> The [[pentagonal icositetrahedron]] and the [[pentagonal hexecontahedron]] are [[Chirality (mathematics)|chiral]] because they are dual to the [[snub cube]] and [[snub dodecahedron]] respectively, which are chiral. That being said, these two solids are not identical when being mirrored.


Eleven of the thirteen Catalan solids have the [[Rupert property]] (a copy of the same or larger shape solid can be passed through a hole in the solid).{{sfnp|Fredriksson|2024}}
In [[mathematics]], a '''Catalan solid''', or '''Archimedean dual''', is a polyhedron that is [[dual polyhedron|dual]] to an [[Archimedean solid]]. There are 13 Catalan solids. They are named after the [[Belgium|Belgian]] mathematician [[Eugène Catalan]], who first described them in 1865.

The Catalan solids are all [[Convex polytope|convex]]. They are [[face-transitive]] but not [[vertex-transitive]]. This is because the dual Archimedean solids are vertex-transitive and not face-transitive. Note that unlike [[Platonic solid]]s and [[Archimedean solid]]s, the faces of Catalan solids are ''not'' [[regular polygon]]s. However, the [[vertex figure]]s of Catalan solids are regular, and they have constant [[dihedral angle]]s. Being face-transitive, Catalan solids are [[isohedra]].

Additionally, two of the Catalan solids are [[edge-transitive]]: the [[rhombic dodecahedron]] and the [[rhombic triacontahedron]]. These are the [[Quasiregular polyhedron#Quasiregular duals|duals]] of the two [[Quasiregular polyhedron|quasi-regular]] Archimedean solids.

Just as [[Prism (geometry)|prisms]] and [[antiprisms]] are generally not considered Archimedean solids, [[bipyramid]]s and [[trapezohedron|trapezohedra]] are generally not considered Catalan solids, despite being face-transitive.

Two of the Catalan solids are [[chirality (mathematics)|chiral]]: the [[pentagonal icositetrahedron]] and the [[pentagonal hexecontahedron]], dual to the chiral [[snub cube]] and [[snub dodecahedron]]. These each come in two [[enantiomorphs]]. Not counting the enantiomorphs, bipyramids, and trapezohedra, there are a total of 13 Catalan solids.

Eleven of the 13 Catalan solids have the [[Rupert property]]: a copy of the solid, of the same or larger shape, can be passed through a hole in the solid.
<ref name=fred>{{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
}}</ref>


== List of Catalan solids and their duals ==
{| class="wikitable sortable"
{| class="wikitable sortable"
|+ The thirteen Catalan solids
! [[Conway polyhedron notation|Conway name]]
! Name
! [[Archimedean_solid|Archimedean dual]]
! Image
! [[Face configuration|Face<br>polygon]]
! Orthogonal<br>wireframes
! Pictures
! Face angles (°)
! Dihedral angle (°)
!Midradius<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Archimedean Solid |url=https://mathworld.wolfram.com/ |access-date=2022-07-02 |website=mathworld.wolfram.com |language=en}}</ref>
! Faces
! Faces
! Edges
! Edges
! Vertices
! Vert
! Dihedral angle{{sfnp|Williams|1979}}
! [[Symmetry group|Sym.]]
! [[List of spherical symmetry groups#Polyhedral symmetry|Point group]]
|- align=center
|- align="center"
| [[triakis tetrahedron]]<br>"kT"
| [[truncated tetrahedron]]
| [[triakis tetrahedron]]
| [[Image:triakistetrahedron.jpg|70px|Triakis tetrahedron]]
| [[Isosceles triangle|Isosceles]]<br>[[File:DU02 facets.png|40px]]<br>V3.6.6
| 12 [[Isosceles triangle|isosceles triangles]]
| [[File:Dual tetrahedron t01 ae.png|60px]][[File:Dual tetrahedron t01 A2.png|60px]][[File:Dual tetrahedron t01.png|60px]]
| [[Image:Polyhedron truncated 4a dual max.png|70px|Triakis tetrahedron]][[Image:triakistetrahedron.jpg|70px|Triakis tetrahedron]]
| 112.885<br>33.557<br>33.557
| 129.521
|1.0607
| 12
| 18
| 18
| 8
| 8
| 129.521°
| T<sub>d</sub>
| T<sub>d</sub>
|- align=center
|- align="center"
| [[rhombic dodecahedron]]<br>"jC"
| [[rhombic dodecahedron]]
| [[Image:rhombicdodecahedron.jpg|70px|Rhombic dodecahedron]]
| [[cuboctahedron]]
| 12 [[rhombus]]
| [[Rhombus]]<br>[[File:DU07 facets.png|40px]]<br>V3.4.3.4
| [[File:Dual cube t1 v.png|60px]] [[File:Dual cube t1.png|60px]][[File:Dual cube t1 B2.png|60px]]
| [[Image:Polyhedron 6-8 dual max.png|70px|Rhombic dodecahedron]][[Image:rhombicdodecahedron.jpg|70px|Rhombic dodecahedron]]
| 70.529<br>109.471<br>70.529<br>109.471
| 120
|0.8660
| 12
| 24
| 24
| 14
| 14
| 120°
| O<sub>h</sub>
| O<sub>h</sub>
|- align=center
|- align="center"
| [[triakis octahedron]]<br>"kO"
| [[triakis octahedron]]
| [[Image:triakisoctahedron.jpg|70px|Triakis octahedron]]
| [[truncated cube]]
| 24 isosceles triangles
| [[Isosceles triangle|Isosceles]]<br>[[File:DU09 facets.png|40px]]<br>V3.8.8
| [[File:Dual truncated cube t01 e88.png|60px]][[File:Dual truncated cube t01.png|60px]][[File:Dual truncated cube t01 B2.png|60px]]
| [[Image:Polyhedron truncated 6 dual max.png|70px|Triakis octahedron]][[Image:triakisoctahedron.jpg|70px|Triakis octahedron]]
| 117.201<br>31.400<br>31.400
| 147.350
|1.7071
| 24
| 36
| 36
| 14
| 14
| 147.350°
| O<sub>h</sub>
| O<sub>h</sub>
|- align=center
|- align="center"
| [[tetrakis hexahedron]]<br>"kC"
| [[tetrakis hexahedron]]
| [[Image:tetrakishexahedron.jpg|70px|Tetrakis hexahedron]]
| [[truncated octahedron]]
| 24 isosceles triangles
| [[Isosceles triangle|Isosceles]]<br>[[File:DU08 facets.png|40px]]<br>V4.6.6
| [[File:Dual cube t12 e66.png|60px]][[File:Dual cube t12.png|60px]][[File:Dual cube t12 B2.png|60px]]
| [[Image:Polyhedron truncated 8 dual max.png|70px|Tetrakis hexahedron]][[Image:tetrakishexahedron.jpg|70px|Tetrakis hexahedron]]
| 83.621<br>48.190<br>48.190
| 143.130
|1.5000
| 24
| 36
| 36
| 14
| 14
| 143.130°
| O<sub>h</sub>
| O<sub>h</sub>
|- align=center
|- align="center"
| [[deltoidal icositetrahedron]]<br>"oC"
| [[deltoidal icositetrahedron]]
| [[Image:deltoidalicositetrahedron.jpg|70px|Deltoidal icositetrahedron]]
| [[rhombicuboctahedron]]
| [[Kite (geometry)|Kite]]<br>[[File:DU10 facets.png|40px]]<br>V3.4.4.4
| 24 [[Kite (geometry)|kites]]
| [[File:Dual cube t02 f4b.png|60px]][[File:Dual cube t02.png|60px]][[File:Dual cube t02 B2.png|60px]]
| [[Image:Polyhedron small rhombi 6-8 dual max.png|70px|Deltoidal icositetrahedron]][[Image:deltoidalicositetrahedron.jpg|70px|Deltoidal icositetrahedron]]
| 81.579<br>81.579<br>81.579<br>115.263
| 138.118
|1.3066
| 24
| 48
| 48
| 26
| 26
| 138.118°
| O<sub>h</sub>
| O<sub>h</sub>
|- align=center
|- align="center"
| [[disdyakis dodecahedron]]<br>"mC"
| [[disdyakis dodecahedron]]
| [[Image:disdyakisdodecahedron.jpg|70px|Disdyakis dodecahedron]]
| [[truncated cuboctahedron]]
| [[Scalene triangle|Scalene]]<br>[[File:DU11 facets.png|40px]]<br>V4.6.8
| 48 [[Scalene triangle|scalene triangles]]
| [[File:Dual cube t012 f4.png|60px]][[File:Dual cube t012.png|60px]][[File:Dual cube t012 B2.png|60px]]
| [[Image:Polyhedron great rhombi 6-8 dual max.png|70px|Disdyakis dodecahedron]][[Image:disdyakisdodecahedron.jpg|70px|Disdyakis dodecahedron]]
| 87.202<br>55.025<br>37.773
| 155.082
|2.2630
| 48
| 72
| 72
| 26
| 26
| 155.082°
| O<sub>h</sub>
| O<sub>h</sub>
|- align=center
|- align="center"
| [[pentagonal icositetrahedron]]<br>"gC"
| [[pentagonal icositetrahedron]]
| [[Image:pentagonalicositetrahedronccw.jpg|70px|Pentagonal icositetrahedron (Ccw)]]
| [[snub cube]]
| [[Pentagon]]<br>[[File:DU12 facets.png|40px]]<br>V3.3.3.3.4
| 24 [[Pentagon|pentagons]]
| [[File:Dual snub cube e1.png|60px]][[File:Dual snub cube A2.png|60px]][[File:Dual snub cube B2.png|60px]]
| [[Image:Polyhedron snub 6-8 right dual max.png|70px|Pentagonal icositetrahedron]][[Image:pentagonalicositetrahedronccw.jpg|70px|Pentagonal icositetrahedron (Ccw)]]
| 114.812<br>114.812<br>114.812<br>114.812<br>80.752
| 136.309
|1.2472
| 24
| 60
| 60
| 38
| 38
| 136.309°
| O
| O
|- align=center
|- align="center"
| [[rhombic triacontahedron]]<br>"jD"
| [[rhombic triacontahedron]]
| [[Image:rhombictriacontahedron.svg|70px|Rhombic triacontahedron]]
| [[icosidodecahedron]]
| 30 rhombus
| [[Rhombus]]<br>[[File:DU24 facets.png|40px]]<br>V3.5.3.5
| [[File:Dual dodecahedron t1 e.png|60px]][[File:Dual dodecahedron t1 A2.png|60px]][[File:Dual dodecahedron t1 H3.png|60px]]
| [[Image:Polyhedron 12-20 dual max.png|70px|Rhombic triacontahedron]][[Image:rhombictriacontahedron.svg|70px|Rhombic triacontahedron]]
| 63.435<br>116.565<br>63.435<br>116.565
| 144
|1.5388
| 30
| 60
| 60
| 32
| 32
| 144°
| I<sub>h</sub>
| I<sub>h</sub>
|- align=center
|- align="center"
| [[triakis icosahedron]]<br>"kI"
| [[triakis icosahedron]]
| [[Image:triakisicosahedron.jpg|70px|Triakis icosahedron]]
| [[truncated dodecahedron]]
| 60 isosceles triangles
| [[Isosceles triangle|Isosceles]]<br>[[File:DU26 facets.png|40px]]<br>V3.10.10
| [[File:Dual dodecahedron t12 exx.png|60px]][[File:Dual dodecahedron t12 A2.png|60px]][[File:Dual dodecahedron t12 H3.png|60px]]
| [[Image:Polyhedron truncated 12 dual max.png|70px|Triakis icosahedron]][[Image:triakisicosahedron.jpg|70px|Triakis icosahedron]]
| 119.039<br>30.480<br>30.480
| 160.613
|2.9271
| 60
| 90
| 90
| 32
| 32
| 160.613°
| I<sub>h</sub>
| I<sub>h</sub>
|- align=center
|- align="center"
| [[pentakis dodecahedron]]<br>"kD"
| [[pentakis dodecahedron]]
| [[Image:pentakisdodecahedron.jpg|70px|Pentakis dodecahedron]]
| [[truncated icosahedron]]
| 60 isosceles triangles
| [[Isosceles triangle|Isosceles]]<br>[[File:DU25 facets.png|40px]]<br>V5.6.6
| [[File:Dual dodecahedron t01 e66.png|60px]][[File:Dual dodecahedron t01 A2.png|60px]][[File:Dual dodecahedron t01 H3.png|60px]]
| [[Image:Polyhedron truncated 20 dual max.png|70px|Pentakis dodecahedron]][[Image:pentakisdodecahedron.jpg|70px|Pentakis dodecahedron]]
| 68.619<br>55.691<br>55.691
| 156.719
|2.4271
| 60
| 90
| 90
| 32
| 32
| 156.719°
| I<sub>h</sub>
| I<sub>h</sub>
|- align=center
|- align="center"
| [[deltoidal hexecontahedron]]<br>"oD"
| [[deltoidal hexecontahedron]]
| [[Image:deltoidalhexecontahedron.jpg|70px|Deltoidal hexecontahedron]]
| [[rhombicosidodecahedron]]
| 60 kites
| [[Kite (geometry)|Kite]]<br>[[File:DU27 facets.png|40px]]<br>V3.4.5.4
| [[File:Dual dodecahedron t02 f4.png|60px]][[File:Dual dodecahedron t02 A2.png|60px]][[File:Dual dodecahedron t02 H3.png|60px]]
| [[File:Polyhedron small rhombi 12-20 dual max.png|70px|Deltoidal hexecontahedron]][[Image:deltoidalhexecontahedron.jpg|70px|Deltoidal hexecontahedron]]
| 86.974<br>67.783<br>86.974<br>118.269
| 154.121
|2.1763
| 60
| 120
| 120
| 62
| 62
| 154.121°
| I<sub>h</sub>
| I<sub>h</sub>
|- align=center
|- align="center"
| [[disdyakis triacontahedron]]<br>"mD"
| [[disdyakis triacontahedron]]
| [[Image:disdyakistriacontahedron.jpg|70px|Disdyakis triacontahedron]]
| [[truncated icosidodecahedron]]
| 120 scalene triangles
| [[Scalene triangle|Scalene]]<br>[[File:DU28 facets.png|40px]]<br>V4.6.10
| [[File:Dual dodecahedron t012 f4.png|60px]][[File:Dual dodecahedron t012 A2.png|60px]][[File:Dual dodecahedron t012 H3.png|60px]]
| [[Image:Polyhedron great rhombi 12-20 dual max.png|70px|Disdyakis triacontahedron]][[Image:disdyakistriacontahedron.jpg|70px|Disdyakis triacontahedron]]
| 88.992<br>58.238<br>32.770
| 164.888
|3.7694
| 120
| 180
| 180
| 62
| 62
| 164.888°
| I<sub>h</sub>
| I<sub>h</sub>
|- align=center
|- align="center"
| [[pentagonal hexecontahedron]]<br>"gD"
| [[pentagonal hexecontahedron]]
| [[Image:pentagonalhexecontahedronccw.jpg|70px|Pentagonal hexecontahedron (Ccw)]]
| [[snub dodecahedron]]
| 60 pentagons
| [[Pentagon]]<br>[[File:DU29 facets.png|40px]]<br>V3.3.3.3.5
| [[File:Dual snub dodecahedron e1.png|60px]][[File:Dual snub dodecahedron A2.png|60px]][[File:Dual snub dodecahedron H2.png|60px]]
| [[Image:Polyhedron snub 12-20 right dual max.png|70px|Pentagonal hexecontahedron]][[Image:pentagonalhexecontahedronccw.jpg|70px|Pentagonal hexecontahedron (Ccw)]]
| 118.137<br>118.137<br>118.137<br>118.137<br>67.454
| 153.179
|2.0971
| 60
| 150
| 150
| 92
| 92
| 153.179°
| I
| I
|}
|}


== References ==
{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
=== Footnotes ===
! bgcolor="#ddd" | ordered by size
{{reflist|20em}}
|-
|
The Catalan solids' [[Archimedean solid|Archimedean]] duals, all shown with the same edge length. Sorted by [[midradius]] in descending order:


=== Works cited ===
[[File:Catalan Solids to Scale.svg]]

All faces of Catalan solids, same scale as above:

[[File:Catalan Facets.svg]]
|}

== Symmetry ==
The Catalan solids, along with their dual [[Archimedean solids]], can be grouped in those with tetrahedral, octahedral and icosahedral symmetry.
For both octahedral and icosahedral symmetry there are six forms. The only Catalan solid with genuine tetrahedral symmetry is the [[triakis tetrahedron]] (dual of the [[truncated tetrahedron]]). The [[rhombic dodecahedron]] and [[tetrakis hexahedron]] have octahedral symmetry, but they can be colored to have only tetrahedral symmetry. [[Rectification (geometry)|Rectification]] and snub also exist with tetrahedral symmetry, but they are [[Platonic solid|Platonic]] instead of Archimedean, so their duals are Platonic instead of Catalan. (They are shown with brown background in the table below.)

{| class="wikitable"
|+ [[Tetrahedral symmetry]]
!Archimedean<br><small>(Platonic)</small>
|style="background-color: #decdbc;"|[[Image:Polyhedron 4-4.png|100px]]
|style="border-right:1px solid #f9f9f9;"|[[Image:Polyhedron truncated 4a max.png|100px]]
|[[Image:Polyhedron truncated 4b max.png|100px]]
|[[Image:Polyhedron small rhombi 4-4 max.png|100px]]
|[[Image:Polyhedron great rhombi 4-4 max.png|100px]]
|style="background-color: #decdbc;"|[[Image:Polyhedron snub 4-4 left max.png|100px]]
|-
!Catalan<br><small>(Platonic)</small>
|style="background-color: #decdbc;"|[[Image:Polyhedron 4-4 dual blue.png|99px]]
|style="border-right:1px solid #f9f9f9;"|[[Image:Polyhedron truncated 4a dual max.png|100px]]
|[[Image:Polyhedron truncated 4b dual max.png|100px]]
|[[Image:Polyhedron small rhombi 4-4 dual max.png|100px]]
|[[Image:Polyhedron great rhombi 4-4 dual max.png|100px]]
|style="background-color: #decdbc;"|[[Image:Polyhedron snub 4-4 left dual max.png|99px]]
|}

{| class="wikitable"
|+ [[Octahedral symmetry]]
!Archimedean
|[[Image:Polyhedron 6-8 max.png|100px]]
|[[Image:Polyhedron truncated 6 max.png|100px]]
|[[Image:Polyhedron truncated 8 max.png|100px]]
|[[Image:Polyhedron small rhombi 6-8 max.png|100px]]
|[[Image:Polyhedron great rhombi 6-8 max.png|100px]]
|[[Image:Polyhedron snub 6-8 left max.png|100px]]
|-
!Catalan
|[[Image:Polyhedron 6-8 dual max.png|100px]]
|[[Image:Polyhedron truncated 6 dual max.png|100px]]
|[[Image:Polyhedron truncated 8 dual max.png|100px]]
|[[Image:Polyhedron small rhombi 6-8 dual max.png|100px]]
|[[Image:Polyhedron great rhombi 6-8 dual max.png|100px]]
|[[Image:Polyhedron snub 6-8 left dual max.png|100px]]
|}

{| class="wikitable"
|+ [[Icosahedral symmetry]]
!Archimedean
|[[Image:Polyhedron 12-20 max.png|100px]]
|[[Image:Polyhedron truncated 12 max.png|100px]]
|[[Image:Polyhedron truncated 20 max.png|100px]]
|[[Image:Polyhedron small rhombi 12-20 max.png|100px]]
|[[Image:Polyhedron great rhombi 12-20 max.png|100px]]
|[[Image:Polyhedron snub 12-20 left max.png|100px]]
|-
!Catalan
|[[Image:Polyhedron 12-20 dual max.png|100px]]
|[[Image:Polyhedron truncated 12 dual max.png|100px]]
|[[Image:Polyhedron truncated 20 dual max.png|100px]]
|[[Image:Polyhedron small rhombi 12-20 dual max.png|100px]]
|[[Image:Polyhedron great rhombi 12-20 dual max.png|100px]]
|[[Image:Polyhedron snub 12-20 left dual max.png|100px]]
|}

==Geometry==
{{unsourced section|date=October 2024}}
All [[dihedral angles]] of a Catalan solid are equal. Denoting their value by <math>\theta</math> , and denoting the face angle at the vertices where <math>p</math> faces meet by <math>\alpha_p</math>, we have
:<math>\sin(\theta/2)=\cos(\pi/p)/\cos(\alpha_p/2)</math>.
This can be used to compute <math>\theta</math> and <math>\alpha_p</math>, <math>\alpha_q</math>, ... , from <math>p</math>, <math>q</math> ... only.

===Triangular faces===
Of the 13 Catalan solids, 7 have triangular faces. These are of the form Vp.q.r, where p, q and r take their values among 3, 4, 5, 6, 8 and 10. The angles <math>\alpha_p</math>, <math>\alpha_q</math> and <math>\alpha_r</math> can be computed in the following way. Put <math>a = 4\cos^2(\pi/p)</math>, <math>b = 4\cos^2(\pi/q)</math>, <math>c = 4\cos^2(\pi/r)</math> and put
:<math>S = -a^2-b^2-c^2+2 a b + 2 b c + 2 c a</math>.
Then
:<math>\cos(\alpha_p) = \frac{S}{2 b c} - 1 </math>,
:<math>\sin(\alpha_p/2) = \frac{-a+b+c}{2\sqrt{b c}}</math>.
For <math>\alpha_q</math> and <math>\alpha_r</math> the expressions are similar of course. The [[dihedral angle]] <math>\theta</math> can be computed from
:<math>\cos(\theta)=1- 2 a b c/S</math>.
Applying this, for example, to the [[disdyakis triacontahedron]] (<math>p=4</math>, <math>q=6</math> and <math>r=10</math>, hence <math>a = 2</math>, <math>b = 3</math> and <math>c = \phi + 2</math>, where <math>\phi</math> is the [[golden ratio]]) gives <math>\cos(\alpha_4)=\frac{2-\phi}{6(2+\phi)}= \frac{7-4\phi}{30}</math> and <math>\cos(\theta) = \frac{-10-7\phi}{14+5\phi}=\frac{-48\phi-155}{241}</math>.

===Quadrilateral faces===
Of the 13 Catalan solids, 4 have quadrilateral faces. These are of the form Vp.q.p.r, where p, q and r take their values among 3, 4, and 5. The angle <math>\alpha_p</math>can be computed by the following formula:
:<math>\cos(\alpha_p)= \frac{2\cos^2(\pi/p)-\cos^2(\pi/q)-\cos^2(\pi/r)}{2\cos^2(\pi/p)+2\cos(\pi/q)\cos(\pi/r)}</math>.
From this, <math>\alpha_q</math>, <math>\alpha_r</math> and the dihedral angle can be easily computed. Alternatively, put <math>a = 4\cos^2(\pi/p)</math>, <math>b = 4\cos^2(\pi/q)</math>, <math>c = 4\cos^2(\pi/p)+4\cos(\pi/q)\cos(\pi/r)</math>. Then <math>\alpha_p</math> and <math>\alpha_q</math> can be found by applying the formulas for the triangular case. The angle <math>\alpha_r</math> can be computed similarly of course.
The faces are [[Kite (geometry)|kites]], or, if <math>q=r</math>, [[rhombi]].
Applying this, for example, to the [[deltoidal icositetrahedron]] (<math>p=4</math>, <math>q=3</math> and <math>r=4</math>), we get <math>\cos(\alpha_4)=\frac{1}{2}-\frac{1}{4}\sqrt{2}</math>.

===Pentagonal faces===
Of the 13 Catalan solids, 2 have pentagonal faces. These are of the form Vp.p.p.p.q, where p=3, and q=4 or 5. The angle <math>\alpha_p</math>can be computed by solving a degree three equation:
:<math>8\cos^2(\pi/p)\cos^3(\alpha_p)-8\cos^2(\pi/p)\cos^2(\alpha_p)+\cos^2(\pi/q)=0</math>.

===Metric properties===
For a Catalan solid <math>\bf C</math> let <math>\bf A</math> be the dual with respect to the [[midsphere]] of <math>\bf C</math>. Then <math>\bf A</math> is an Archimedean solid with the same midsphere. Denote the length of the edges of <math>\bf A</math> by <math>l</math>. Let <math>r</math> be the [[inradius]] of the faces of <math>\bf C</math>, <math>r_m</math> the midradius of <math>\bf C</math> and <math>\bf A</math>, <math>r_i</math> the inradius of <math>\bf C</math>, and <math>r_c</math> the circumradius of <math>\bf A</math>. Then these quantities can be expressed in <math>l</math> and the dihedral angle <math>\theta</math> as follows:
:<math>r^2=\frac{l^2}{8}(1-\cos\theta)</math>,

:<math>r_m^2=\frac{l^2}{4}\frac{1-\cos\theta}{1+\cos\theta}</math>,

:<math>r_i^2=\frac{l^2}{8}\frac{(1-\cos\theta)^2}{1+\cos\theta}</math>,

:<math>r_c^2=\frac{l^2}{2}\frac{1}{1+\cos\theta}</math>.

These quantities are related by <math>r_m^2=r_i^2+r^2</math>, <math>r_c^2=r_m^2+l^2/4</math> and <math>r_i r_c=r_m^2</math>.

As an example, let <math>\bf A</math> be a cuboctahedron with edge length <math>l=1</math>. Then <math>\bf C</math> is a rhombic dodecahedron. Applying the formula for quadrilateral faces with <math>p=4</math> and <math>q=r=3</math> gives <math>\cos \theta=-1/2</math>, hence <math>r_i=3/4</math>, <math>r_m=\frac{1}{2}\sqrt{3}</math>, <math>r_c=1</math>, <math>r=\frac{1}{4}\sqrt{3}</math>.

All vertices of <math>\bf C</math> of type <math>p</math> lie on a sphere with radius <math>r_{c,p}</math> given by
:<math>r_{c,p}^2=r_i^2+\frac{2r^2}{1-\cos\alpha_p}</math>,
and similarly for <math>q,r,\ldots</math>.

Dually, there is a sphere which touches all faces of <math>\bf A</math> which are regular <math>p</math>-gons (and similarly for <math>q,r,\ldots</math>) in their center. The radius <math>r_{i,p}</math> of this sphere is given by
:<math>r_{i,p}^2=r_m^2-\frac{l^2}{4}\cot^2(\pi/p)</math>.

These two radii are related by <math>r_{i,p}r_{c,p}=r_m^2</math>. Continuing the above example: <math>\cos\alpha_3=-1/3</math> and <math>\cos\alpha_4=1/3</math>, which gives <math>r_{c,3}=\frac{3}{8}\sqrt{6}</math>, <math>r_{c,4}=\frac{3}{4}\sqrt{2}</math>, <math>r_{i,3}=\frac{1}{3}\sqrt{6}</math> and <math>r_{i,4}=\frac{1}{2}\sqrt{2}</math>.

If <math>P</math> is a vertex of <math>\bf C</math> of type <math>p</math>, <math>e</math> an edge of <math>\bf C</math> starting at <math>P</math>, and <math>P^\prime</math> the point where the edge <math>e</math> touches the midsphere of <math>\bf C</math>, denote the distance <math>P P^\prime</math> by <math>l_p</math>. Then the edges of <math>\bf C</math> joining vertices of type <math>p</math> and type <math>q</math> have length <math>l_{p, q} = l_p + l_q</math>. These quantities can be computed by
:<math>l_p=\frac{l}{2}\frac{\cos(\pi/p)}{\sin(\alpha_p/2)}</math>,
and similarly for <math>q, r, \ldots</math>. Continuing the above example: <math>\sin(\alpha_3/2)=\frac{1}{3}\sqrt{6}</math>, <math>\sin(\alpha_4/2)=\frac{1}{3}\sqrt{3}</math>, <math>l_3=\frac{1}{8}\sqrt{6}</math>, <math>l_4=\frac{1}{4}\sqrt{6}</math>, so the edges of the rhombic dodecahedron have length <math>l_{3,4}=\frac{3}{8}\sqrt{6}</math>.

The dihedral angles <math>\alpha_{p, q}</math>between <math>p</math>-gonal and <math>q</math>-gonal faces of <math>\bf A</math> satisfy
:<math>\cos \alpha_{p,q} = \frac{l^2}{4}\frac{\cot(\pi/p)\cot(\pi/q)}{r_m^2}-\frac{r_{i, p}r_{i, q}}{r_m^2} = \frac{l_p l_q-r_m^2}{r_{c,p}r_{c,q}}</math>.

Finishing the rhombic dodecahedron example, the dihedral angle <math>\alpha_{3,4}</math> of the cuboctahedron is given by <math>\cos \alpha_{3,4}=-\frac{1}{3}\sqrt{3}</math>.

===Construction===

The face of any Catalan polyhedron may be obtained from the [[vertex figure]] of the dual [[Archimedean solid]] using the [[Uniform dual polyhedron#Dorman Luke construction|Dorman Luke construction]].<ref>{{harvtxt|Cundy|Rollett|1961}}, p.&nbsp; 117; {{harvtxt|Wenninger|1983}}, p.&nbsp;30.</ref>

===Application to other solids===

All of the formulae of this section apply to the [[Platonic solids]], and [[bipyramids]] and [[trapezohedra]] with equal dihedral angles as well, because they can be derived from the constant dihedral angle property only. For the [[pentagonal trapezohedron]], for example, with faces V3.3.5.3, we get <math>\cos(\alpha_3)=\frac{1}{4}-\frac{1}{4}\sqrt{5}</math>, or <math>\alpha_3=108^{\circ}</math>. This is not surprising: it is possible to cut off both apexes in such a way as to obtain a [[regular dodecahedron]].

== See also ==
* [[List of uniform tilings]] Shows dual uniform polygonal tilings similar to the Catalan solids
* [[Conway polyhedron notation]] A notational construction process
* [[Archimedean solid]]
* [[Johnson solid]]

==Notes==
{{reflist}}

== References ==
* [[Eugène Catalan]] ''Mémoire sur la Théorie des Polyèdres.'' J. l'École Polytechnique (Paris) 41, 1-71, 1865.
* {{citation
* {{citation
| last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy
| last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy
Line 389: Line 145:
| title-link = Mathematical Models (Cundy and Rollett)
| title-link = Mathematical Models (Cundy and Rollett)
| year = 1961}}.
| year = 1961}}.
* {{citation
| last = Diudea | first = M. V.
| year = 2018
| title = Multi-shell Polyhedral Clusters
| 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
* {{citation
| last1 = Gailiunas | first1 = P.
| last1 = Gailiunas | first1 = P.
Line 400: Line 176:
| year = 2005| s2cid = 120818796
| year = 2005| s2cid = 120818796
}}.
}}.
* {{citation
* [[Alan Holden]] ''Shapes, Space, and Symmetry''. New York: Dover, 1991.
| last1 = Heil | first1 = E.
* {{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=[[Cambridge University Press]] | isbn=978-0-521-54325-5 | mr=730208 | year=1983}} (The thirteen semiregular convex polyhedra and their duals)
| last2 = Martil | 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
}}
* {{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)
* {{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
* {{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7 }} Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms


==External links==
==External links==

Revision as of 13:30, 23 October 2024

Set of Catalan solids
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]

These solids are face-transitive or isohedron 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. They have constant dihedral angles, meaning the angle between any two of their faces are the same.[1] Additionally, both Catalan solids rhombic dodecahedron and rhombic triacontahedron are edge-transitive, meaning there is a transitive between the edges preserving their symmetrical.[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 being said, these two solids are not identical when being mirrored.

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

The thirteen Catalan solids
Name Image Faces Edges Vertices Dihedral angle[6] Point group
triakis tetrahedron Triakis tetrahedron 12 isosceles triangles 18 8 129.521° Td
rhombic dodecahedron Rhombic dodecahedron 12 rhombus 24 14 120° Oh
triakis octahedron Triakis octahedron 24 isosceles triangles 36 14 147.350° Oh
tetrakis hexahedron Tetrakis hexahedron 24 isosceles triangles 36 14 143.130° Oh
deltoidal icositetrahedron Deltoidal icositetrahedron 24 kites 48 26 138.118° Oh
disdyakis dodecahedron Disdyakis dodecahedron 48 scalene triangles 72 26 155.082° Oh
pentagonal icositetrahedron Pentagonal icositetrahedron (Ccw) 24 pentagons 60 38 136.309° O
rhombic triacontahedron Rhombic triacontahedron 30 rhombus 60 32 144° Ih
triakis icosahedron Triakis icosahedron 60 isosceles triangles 90 32 160.613° Ih
pentakis dodecahedron Pentakis dodecahedron 60 isosceles triangles 90 32 156.719° Ih
deltoidal hexecontahedron Deltoidal hexecontahedron 60 kites 120 62 154.121° Ih
disdyakis triacontahedron Disdyakis triacontahedron 120 scalene triangles 180 62 164.888° Ih
pentagonal hexecontahedron Pentagonal hexecontahedron (Ccw) 60 pentagons 150 92 153.179° I

References

Footnotes

  1. ^ a b Diudea (2018), p. 39.
  2. ^ Wenninger (1983), p. 1, Basic notions about stellation and duality.
  3. ^
  4. ^
  5. ^ Fredriksson (2024).
  6. ^ 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.; Martil, 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)
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