# Runcination
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> Source: https://en.wikipedia.org/wiki/Runcination
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{{no footnotes|date=March 2015}}
thumb|A ''runcinated cubic honeycomb'' (partial) - The original cells (purple cubes) are reduced in size. Faces become new blue cubic cells. Edges become new red cubic cells. Vertices become new cubic cells (hidden).
In [geometry](/source/geometry), '''runcination''' is an operation that cuts a [regular polytope](/source/regular_polytope) (or [honeycomb](/source/Honeycomb_(geometry))) simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers.{{fact|date=December 2012|reason=Has this term ever been used in a reliable source?}}
It is a higher-order truncation operation, following [cantellation](/source/Cantellation_(geometry)) and [truncation](/source/Truncation_(geometry)).
It is represented by an extended [Schläfli symbol](/source/Schl%C3%A4fli_symbol) t0,3{p,q,...}. This operation only exists for [4-polytope](/source/4-polytope)s {p,q,r} or higher.
This operation is dual-symmetric for regular [uniform 4-polytope](/source/uniform_4-polytope)s and [3-space](/source/3-space) [convex uniform honeycomb](/source/convex_uniform_honeycomb)s.
For a regular {p,q,r} 4-polytope, the original {p,q} cells remain, but become separated. The gaps at the separated faces become ['''p'''-gonal prisms](/source/Prism_(geometry)). The gaps between the separated edges become '''r'''-gonal prisms. The gaps between the separated vertices become {r,q} cells. The [vertex figure](/source/vertex_figure) for a regular 4-polytope {p,q,r} is an ''q''-gonal [antiprism](/source/antiprism) (called an ''antipodium'' if ''p'' and ''r'' are different).
For regular 4-polytopes/honeycombs, this operation is also called '''[expansion](/source/Expansion_(geometry))''' by [Alicia Boole Stott](/source/Alicia_Boole_Stott), as imagined by moving the cells of the regular form away from the center, and filling in new faces in the gaps for each opened vertex and edge.
'''Runcinated 4-polytopes/honeycombs forms:'''
{| class=wikitable
[Coxeter diagram](/source/Coxeter_diagram)
!Name
!Image
|-
!colspan=4|[Uniform 4-polytope](/source/Uniform_4-polytope)s
|-
|align=center|t0,3{3,3,3}
{{CDD|node_1|3|node|3|node|3|node_1}}
||[Runcinated 5-cell](/source/Runcinated_5-cell)||80px
||80px
|-
|align=center|t0,3{3,3,4}
{{CDD|node_1|3|node|3|node|4|node_1}}
||[Runcinated 16-cell](/source/Runcinated_16-cell)
(Same as '''runcinated 8-cell''')||80px
||80px80px
|-
|align=center|t0,3{3,4,3}
{{CDD|node_1|3|node|4|node|3|node_1}}
||[Runcinated 24-cell](/source/Runcinated_24-cell)||80px
||80px
|-
|align=center|t0,3{3,3,5}
{{CDD|node_1|3|node|3|node|5|node_1}}
||[Runcinated 120-cell](/source/Runcinated_120-cell)
(Same as '''runcinated 600-cell''')||80px
|80px
|-
!colspan=4|Euclidean [convex uniform honeycomb](/source/convex_uniform_honeycomb)s
|-
|align=center|t0,3{4,3,4}
{{CDD|node_1|4|node|3|node|4|node_1}}
||'''Runcinated cubic honeycomb'''
(Same as [cubic honeycomb](/source/cubic_honeycomb))||80px
|80px
|-
!colspan=4|Hyperbolic [uniform honeycombs](/source/Convex_uniform_honeycombs_in_hyperbolic_space)
|-
|align=center|t0,3{4,3,5}
{{CDD|node_1|4|node|3|node|5|node_1}}
||'''Runcinated order-5 cubic honeycomb'''||80px||
|-
|align=center|t0,3{3,5,3}
{{CDD|node_1|3|node|5|node|3|node_1}}
||'''Runcinated icosahedral honeycomb'''||80px||
|-
|align=center|t0,3{5,3,5}
{{CDD|node_1|5|node|3|node|5|node_1}}
||'''Runcinated order-5 dodecahedral honeycomb'''||80px||
|}
== See also ==
* [Uniform polyhedron](/source/Uniform_polyhedron)
* [Uniform 4-polytope](/source/Uniform_4-polytope)
* [Rectification (geometry)](/source/Rectification_(geometry))
* [Truncation (geometry)](/source/Truncation_(geometry))
* [Cantellation (geometry)](/source/Cantellation_(geometry))
== References ==
* [Coxeter, H.S.M.](/source/Coxeter) ''[Regular Polytopes](/source/Regular_Polytopes_(book))'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} (pp. 145–154 Chapter 8: Truncation, p 210 Expansion)
* [Norman Johnson](/source/Norman_Johnson_(mathematician)) ''Uniform Polytopes'', Manuscript (1991)
** [N.W. Johnson](/source/Norman_Johnson_(mathematician)): ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
* [John H. Conway](/source/John_Horton_Conway), Heidi Burgiel, [Chaim Goodman-Strauss](/source/Chaim_Goodman-Strauss), ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26)
== External links ==
* {{mathworld | urlname = Expansion | title = Expansion}}
Category:Polytopes
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Adapted from the Wikipedia article [Runcination](https://en.wikipedia.org/wiki/Runcination) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Runcination?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.