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, a 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 t_0,3{p,q,...}. This operation only exists for 4-polytopes {p,q,r} or higher.

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

For a regular {p,q,r} polychoron, 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.

For regular polychora/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 polychoron/honeycombs forms:

1. t_0,3{3,3,3}: Runcinated 5-cell
2. t_0,3{3,3,4}: Runcinated 16-cell (Same as runcinated 8-cell)
3. t_0,3{3,4,3}: Runcinated 24-cell
4. t_0,3{3,3,5}: Runcinated 120-cell (Same as runcinated 600-cell)
5. t_0,3{4,3,4}: Runcinated cubic honeycomb (Same as cubic honeycomb)

[edit] See also

• uniform polyhedron
• uniform polychoron
• Rectification (geometry)
• Truncation (geometry)
• Cantellation (geometry)

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