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{{Short description|Isogonal polytope with regular facets}}
{| class=wikitable align=right width=450
|+ Gosset's figures
!colspan=3|3D honeycombs
|-
|[[File:HC P1-P3.png|150px]]<BR>[[Tetrahedral-octahedral honeycomb|Simple tetroctahedric check]]
|[[File:Gyrated alternated cubic honeycomb.png|150px]]<BR>[[Gyrated alternated cubic honeycomb|Complex tetroctahedric check]]
|-
!colspan=3|4D polytopes
|-
|[[File:Schlegel half-solid rectified 5-cell.png|150px]]<BR>[[Rectified 5-cell|Tetroctahedric]]
|[[File:Rectified 600-cell schlegel halfsolid.png|150px]]<BR>[[Rectified 600-cell|Octicosahedric]]
|[[File:Ortho solid 969-uniform polychoron 343-snub.png|150px]]<BR>[[Snub 24-cell|Tetricosahedric]]
|}
In [[geometry]], by [[Thorold Gosset]]'s definition a '''semiregular polytope''' is usually taken to be a [[polytope]] that is [[isogonal figure|vertex-transitive]] and has all its [[facet (geometry)|facets]] being [[regular polytope]]s. [[E.L. Elte]] compiled a [[Emanuel_Lodewijk_Elte#Elte's semiregular polytopes of the first kind|longer list in 1912]] as ''The Semiregular Polytopes of the Hyperspaces'' which included a wider definition.
== Gosset's list ==
In [[three-dimensional space]] and below, the terms ''semiregular polytope'' and ''[[uniform polytope]]'' have identical meanings, because all uniform [[polygon]]s must be [[regular polygon|regular]]. However, since not all [[uniform polyhedra]] are [[regular polyhedra|regular]], the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions.
The three convex semiregular [[
;Gosset's 4-polytopes (with his names in parentheses):
:[[Rectified 5-cell]] (Tetroctahedric), {{CDD|node|3|node_1|3|node|3|node}}
:[[Rectified 600-cell]] (Octicosahedric), {{CDD|node|3|node_1|3|node|5|node}}
:[[Snub 24-cell]] (Tetricosahedric), {{CDD|node_h|3|node_h|4|node|3|node}}, {{CDD|node_h|3|node_h|3|node_h|4|node}} or {{CDD|node_h|3|node_h|split1|nodes_hh}}
;[[Semiregular E-polytope]]s in higher dimensions:
:[[5-demicube]] (5-ic semi-regular), a [[5-polytope]], {{CDD|node_h1|4|node|3|node|3|node|3|node}} ↔ {{CDD|nodes_10ru|split2|node|3|node|3|node}}
:[[2 21 polytope|2<sub>21</sub> polytope]] (6-ic semi-regular), a [[6-polytope]], {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}} or {{CDD|nodes_10r|3ab|nodes|split2|node|3|node}}
:[[3 21 polytope|3<sub>21</sub> polytope]] (7-ic semi-regular), a [[7-polytope]], {{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
:[[4 21 polytope|4<sub>21</sub> polytope]] (8-ic semi-regular), an [[8-polytope]], {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
==Euclidean honeycombs==
[[File:HC P1-P3.png|185px|thumb|The [[tetrahedral-octahedral honeycomb]] in Euclidean 3-space has alternating tetrahedral and octahedral cells.]]
Semiregular polytopes can be extended to semiregular [[honeycomb (geometry)|honeycombs]]. The semiregular Euclidean honeycombs are the [[tetrahedral-octahedral honeycomb]] (3D), [[gyrated alternated cubic honeycomb]] (3D) and the [[5 21 honeycomb|5<sub>21</sub> honeycomb]] (8D).
Gosset [[Convex uniform honeycomb|honeycombs]]:
#[[Tetrahedral-octahedral honeycomb]] or [[alternated cubic honeycomb]] (Simple tetroctahedric check), {{CDD|node_h1|4|node|3|node|4|node}} ↔ {{CDD|nodes_10ru|split2|node|4|node}} (Also [[quasiregular polytope]])
Semiregular E-honeycomb:
*[[5 21 honeycomb|5<sub>21</sub> honeycomb]] (9-ic check) (8D Euclidean honeycomb), {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
{{harvtxt|Gosset|1900}} additionally allowed Euclidean honeycombs as facets of higher-dimensional Euclidean honeycombs, giving the following additional figures:
#Hypercubic honeycomb prism, named by Gosset as the (''n'' – 1)-ic semi-check (analogous to a single rank or file of a chessboard)
#[[Convex uniform honeycomb#Frieze forms|Alternated hexagonal slab honeycomb]] (tetroctahedric semi-check), {{CDD|node_h|2x|node_h|6|node|3|node}}
==Hyperbolic honeycombs==
[[File:H3 4333-0100 center ultrawide.png|thumb|The [[hyperbolic tetrahedral-octahedral honeycomb]] has tetrahedral and two types of octahedral cells.]]
There are also hyperbolic uniform honeycombs composed of only regular cells {{harv|Coxeter|Whitrow|1950}}, including:
*[[Hyperbolic uniform honeycomb]]s, 3D honeycombs:
*# [[Alternated order-5 cubic honeycomb]], {{CDD|node_h1|4|node|3|node|5|node}} ↔ {{CDD|nodes_10ru|split2|node|5|node}} (Also [[quasiregular polytope]])
*# [[Hyperbolic tetrahedral-octahedral honeycomb|Tetrahedral-octahedral honeycomb]], {{CDD|label4|branch|3ab|branch_10l}}
*# [[Tetrahedron-icosahedron honeycomb]], {{CDD|label5|branch|3ab|branch_10l}}
*[[Paracompact uniform honeycomb]]s, 3D honeycombs, which include
*# [[Rectified order-6 tetrahedral honeycomb]], {{CDD
*# [[Rectified square tiling honeycomb]], {{CDD|node|4|node_1|4|node|3|node}}
*# [[
*# [[Alternated order-6 cubic honeycomb]], {{CDD|node_h1|4|node|3|node|6|node}} ↔ {{CDD|nodes_10ru|split2|node|6|node}} (Also quasiregular)
*# [[Alternated hexagonal tiling honeycomb]], {{CDD||node_h1|6|node|3|node|3|node}} ↔ {{CDD|branch_10ru|split2|node|3|node}}
*# [[Alternated
*# [[Alternated order-5 hexagonal tiling honeycomb]], {{CDD||node_h1|6|node|3|node|5|node}} ↔ {{CDD|branch_10ru|split2|node|5|node}}
*# [[Alternated order-6 hexagonal tiling honeycomb]], {{CDD||node_h1|6|node|3|node|6|node}} ↔ {{CDD|branch_10ru|split2|node|6|node}}
*# [[Alternated square tiling honeycomb]], {{CDD|node_h1|4|node|4|node|3|node}} ↔ {{CDD|nodes_10ru|split2-44|node|3|node}} (Also quasiregular)
*# [[Cubic-square tiling honeycomb]], {{CDD|label4|branch_10r|4a4b|branch}}
*# [[Order-4 square tiling honeycomb]], {{CDD|label4|branch_10r|4a4b|branch|label4}} = {{CDD|node_1|4|node|4|node|4|node}}
*# [[Tetrahedral-triangular tiling honeycomb]], {{CDD|label6|branch|3ab|branch_10l}}
*9D hyperbolic paracompact honeycomb:
*# [[6 21 honeycomb|6<sub>21</sub> honeycomb]] (10-ic check), {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
==See also==
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== References ==
*{{cite journal
| last1 = Blind | first1 = G.
| last2 = Blind | first2 = R. | author2-link = Roswitha Blind
| doi = 10.1007/BF02566640
| issue = 1
| journal = [[Commentarii Mathematici Helvetici]]
| mr = 1090169
| pages = 150–154
| title = The semiregular polytopes
| volume = 66
| year = 1991
| s2cid = 119695696
}}
* {{cite book | first = H. S. M. | last = Coxeter | authorlink = Harold Scott MacDonald Coxeter | year = 1973 | title = [[Regular Polytopes (book)|Regular Polytopes]] | edition = 3rd | publisher = Dover Publications | location = New York | isbn = 0-486-61480-8}}
* {{cite journal
| last1 = Coxeter | first1 = H. S. M. | author1-link = Harold Scott MacDonald Coxeter
| last2 = Whitrow | first2 = G. J.
| doi = 10.1098/rspa.1950.0070
| journal = [[Proceedings of the Royal Society]]
| mr = 0041576
| pages = 417–437
| title = World-structure and non-Euclidean honeycombs
| volume = 201
| year = 1950
| issue = 1066 | bibcode = 1950RSPSA.201..417C | s2cid = 120322123 }}
* {{Cite book | last = Elte | first = E. L. | authorlink = Emanuel Lodewijk Elte | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912 | isbn = 1-4181-7968-X}}
* {{cite journal | last=Gosset | first=Thorold | authorlink = Thorold Gosset | title = On the regular and semi-regular figures in space of ''n'' dimensions | journal = [[Messenger of Mathematics]] | volume = 29 | pages = 43–48 | year = 1900}}
* {{cite journal
| last = Makarov | first = P. V.
| department = Voprosy Diskret. Geom.
| journal = Mat. Issled. Akad. Nauk. Mold.
| mr = 958024
| pages = 139–150, 177
| title = On the derivation of four-dimensional semi-regular polytopes
| volume = 103
| year = 1988
}}
[[Category:Polytopes]]
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