Runcination

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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, 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.[citation needed]

It is a higher order truncation operation, following cantellation, and truncation.

It is represented by an extended Schläfli symbol t0,3{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 taking the cells of the regular form moving them away from the center and filling in new faces in the gaps for each opened vertex and edge.

Runcinated 4-polytopes/honeycombs forms:

Schläfli symbol
Coxeter diagram
Name Vertex figure Image
Uniform 4-polytopes
t0,3{3,3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcinated 5-cell Runcinated 5-cell verf.png Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Runcinated 16-cell
(Same as runcinated 8-cell)
Runcinated 8-cell verf.png Schlegel half-solid runcinated 16-cell.pngSchlegel half-solid runcinated 8-cell.png
t0,3{3,4,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcinated 24-cell Runcinated 24-cell verf.png Runcinated 24-cell Schlegel halfsolid.png
t0,3{3,3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
Runcinated 120-cell
(Same as runcinated 600-cell)
Runcinated 120-cell verf.png Runcinated 120-cell.png
Euclidean convex uniform honeycombs
t0,3{4,3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Runcinated cubic honeycomb
(Same as cubic honeycomb)
Runcinated cubic honeycomb verf.png Runcinated cubic honeycomb.png
Hyperbolic uniform honeycombs
t0,3{4,3,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
Runcinated order-5 cubic honeycomb Runcinated order-5 cubic honeycomb verf.png
t0,3{3,5,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcinated icosahedral honeycomb Runcinated icosahedral honeycomb verf.png
t0,3{5,3,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
Runcinated order-5 dodecahedral honeycomb Runcinated order-5 dodecahedral honeycomb verf.png

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