Runcination

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title: "Runcination" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polytopes"] topic_path: "general/polytopes" source: "https://en.wikipedia.org/wiki/Runcination" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::figure[src="https://upload.wikimedia.org/wikipedia/commons/7/78/Runcinated_cubic_honeycomb.png" caption="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.

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 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: ::data[format=table] | Schläfli symbol Coxeter diagram | Name | Vertex figure | Image | Uniform 4-polytopes | Euclidean convex uniform honeycombs | Hyperbolic uniform honeycombs | |---|---|---|---|---|---|---| | t0,3{3,3,3} | |Runcinated 5-cell | [[File:Runcinated 5-cell verf.svg|80px]] | |[[File:Schlegel half-solid runcinated 5-cell.png|80px]] | | | | | t0,3{3,3,4} | |Runcinated 16-cell (Same as runcinated 8-cell) | [[File:Runcinated 8-cell verf.png|80px]] | |[[File:Schlegel half-solid runcinated 16-cell.png|80px]][[File:Schlegel half-solid runcinated 8-cell.png|80px]] | | | | | t0,3{3,4,3} | |Runcinated 24-cell | [[File:Runcinated 24-cell verf.svg|80px]] | |[[File:Runcinated 24-cell Schlegel halfsolid.png|80px]] | | | | | t0,3{3,3,5} | |Runcinated 120-cell (Same as runcinated 600-cell) | [[File:Runcinated 120-cell verf.png|80px]] | [[File:Runcinated 120-cell.png|80px]] | | | | | t0,3{4,3,4} | |Runcinated cubic honeycomb (Same as cubic honeycomb) | [[File:Runcinated cubic honeycomb verf.png|80px]] | [[File:Runcinated cubic honeycomb.png|80px]] | | | | | t0,3{4,3,5} | |Runcinated order-5 cubic honeycomb | [[File:Runcinated order-5 cubic honeycomb verf.png|80px]] | | | | | | t0,3{3,5,3} | |Runcinated icosahedral honeycomb | [[File:Runcinated icosahedral honeycomb verf.png|80px]] | | | | | | t0,3{5,3,5} | |Runcinated order-5 dodecahedral honeycomb | [[File:Runcinated order-5 dodecahedral honeycomb verf.png|80px]] | | | | | ::

References

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::

polytopes