Runcination

{{no footnotes|date=March 2015}}

File:Runcinated cubic honeycomb.png

In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) 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, and truncation.

It is represented by an extended Schläfli symbol t_0,3{p,q,...}. This operation only exists for 4-polytopes {p,q,r} or higher.

This operation is dual-symmetric for regular uniform 4-polytopes and 3-space convex uniform honeycombs.

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. The gaps between the separated edges become r-gonal prisms. The gaps between the separated vertices become {r,q} cells. The vertex figure for a regular 4-polytope {p,q,r} is an q-gonal antiprism (called an antipodium if p and r are different).

For regular 4-polytopes/honeycombs, this operation is also called expansion by 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 !Schläfli symbol Coxeter diagram !Name !Vertex figure !Image
colspan=4|Uniform 4-polytopes
align=center|t0,3{3,3,3} {{CDD|node_1|3|node|3|node|3|node_1}}	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 (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	80px	80px
align=center|t0,3{3,3,5} {{CDD|node_1|3|node|3|node|5|node_1}}	Runcinated 120-cell (Same as runcinated 600-cell)	80px |80px
colspan=4|Euclidean convex uniform honeycombs
align=center|t0,3{4,3,4} {{CDD|node_1|4|node|3|node|4|node_1}}	Runcinated cubic honeycomb (Same as cubic honeycomb)	80px |80px
colspan=4|Hyperbolic uniform honeycombs
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