Runcination 📖 Wikipedia

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 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:

Schläfli symbol Coxeter diagram	Name	Vertex figure	Image
Uniform 4-polytopes
t0,3{3,3,3}	Runcinated 5-cell
t0,3{3,3,4}	Runcinated 16-cell (Same as runcinated 8-cell)
t0,3{3,4,3}	Runcinated 24-cell
t0,3{3,3,5}	Runcinated 120-cell (Same as runcinated 600-cell)
Euclidean convex uniform honeycombs
t0,3{4,3,4}	Runcinated cubic honeycomb (Same as cubic honeycomb)
Hyperbolic uniform honeycombs
t0,3{4,3,5}	Runcinated order-5 cubic honeycomb
t0,3{3,5,3}	Runcinated icosahedral honeycomb
t0,3{5,3,5}	Runcinated order-5 dodecahedral honeycomb

• Uniform polyhedron
• Uniform 4-polytope
• Rectification (geometry)
• Truncation (geometry)
• Cantellation (geometry)

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation, p 210 Expansion)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)

• Weisstein, Eric W. "Expansion". MathWorld.