Catalan solid: Difference between revisions
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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]] | [[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 '''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 dual, 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 | 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 dual, 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|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]}} | |{{harvp|Heil|Martini|1993|p=[https://books.google.com/books?id=M2viBQAAQBAJ&pg=PA352 352]}} | ||
}}</ref> | }}</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, starting from the set of Platonic solids, all faces of which are attached by pyramids. These examples are the [[Kleetope]] 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. | 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. | ||
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=== Works cited === | === 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 = 353–360 | |||
| doi = 10.16984/saufenbilder.03497 | |||
| doi-broken-date = 25 February 2025 | |||
| url = https://dergipark.org.tr/en/pub/saufenbilder/issue/20705/221184 | |||
}} | |||
* {{citation | * {{citation | ||
| last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy | | last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy | ||
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}} | }} | ||
* {{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9) | * {{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9) | ||
{{refend}} | |||
==External links== | ==External links== | ||
Revision as of 03:57, 26 June 2025
The Catalan solids are the dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices.Template:Sfnp The faces of the Catalan solids correspond by duality to the vertices of Archimedean solids, and vice versa.Template:Sfnp
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 dual, the Archimedean solids, are vertex-transitive but not face-transitive. Each Catalan solid has constant dihedral angles, meaning the angle between any two adjacent faces is the same.Template:Sfnp Additionally, two Catalan solids, the rhombic dodecahedron and rhombic triacontahedron, are edge-transitive, meaning their edges are symmetric to each other.Script error: No such module "Unsubst". Some Catalan solids were discovered by Johannes Kepler during his study of zonohedra, and Eugene Catalan completed the list of the thirteen solids in 1865.[1]
In general, each face of a dual uniform polyhedron (including the Catalan solid) can be constructed by using the Dorman Luke construction.[2] Some of the Catalan solids can be constructed, starting from the set of Platonic solids, all faces of which are attached by pyramids. These examples are the Kleetope of Platonic solids: triakis tetrahedron, tetrakis hexahedron, triakis octahedron, triakis icosahedron, and pentakis dodecahedron.[3]
Two Catalan solids, the pentagonal icositetrahedron and the pentagonal hexecontahedron, are 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.Template:Sfnp
References
Footnotes
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Works cited
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- Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
External links
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- Catalan Solids – at Visual Polyhedra
- Archimedean duals – at Virtual Reality Polyhedra
- Interactive Catalan Solid in Java
- Download link for Catalan's original 1865 publication – with beautiful figures, PDF format