Catalan solid: Difference between revisions
imported>Dedhert.Jr →Works cited: ref begin end |
imported>Fgnievinski removed Category:Polyhedra using HotCat already in eponym cat |
||
| Line 6: | Line 6: | ||
== The solids == | == 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 | 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|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]}} | ||
| Line 15: | Line 15: | ||
|{{harvp|Cundy|Rollett|1961|p=117}} | |{{harvp|Cundy|Rollett|1961|p=117}} | ||
|{{harvp|Wenninger|1983|p=30}} | |{{harvp|Wenninger|1983|p=30}} | ||
}}</ref> Some of the Catalan solids can be constructed | }}</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|Brigaglia|Palladino|Vaccaro|2018}} | ||
|{{harvp|Çolak|Gelişgen|2015}} | |{{harvp|Çolak|Gelişgen|2015}} | ||
| Line 171: | Line 171: | ||
| pages = 353–360 | | pages = 353–360 | ||
| doi = 10.16984/saufenbilder.03497 | | doi = 10.16984/saufenbilder.03497 | ||
| doi-broken-date = | | doi-broken-date = 1 July 2025 | ||
| url = https://dergipark.org.tr/en/pub/saufenbilder/issue/20705/221184 | | url = https://dergipark.org.tr/en/pub/saufenbilder/issue/20705/221184 | ||
}} | }} | ||
| Line 215: | Line 215: | ||
| title = Duality of polyhedra | | title = Duality of polyhedra | ||
| volume = 36 | | volume = 36 | ||
| year = 2005| s2cid = 120818796 | | year = 2005| bibcode = 2005IJMES..36..617G | ||
| s2cid = 120818796 | |||
}}. | }}. | ||
* {{citation | * {{citation | ||
| Line 253: | Line 254: | ||
{{DEFAULTSORT:Catalan Solid}} | {{DEFAULTSORT:Catalan Solid}} | ||
[[Category:Catalan solids|*]] | [[Category:Catalan solids|*]] | ||
Latest revision as of 05:12, 22 November 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 duals, 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 by adding pyramids to the faces of Platonic solids. These examples are Kleetopes 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
<templatestyles src="Reflist/styles.css" />
Script error: No such module "Check for unknown parameters".
Works cited
<templatestyles src="Refbegin/styles.css" />
- Script error: No such module "citation/CS1"..
- Script error: No such module "citation/CS1".
- Script error: No such module "citation/CS1"..
- Script error: No such module "citation/CS1"..
- Script error: No such module "citation/CS1"..
- Script error: No such module "citation/CS1"..
- Script error: No such module "citation/CS1".
- Script error: No such module "citation/CS1".
- Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
External links
- Script error: No such module "Template wrapper".
- Script error: No such module "Template wrapper".
- 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