Runcination

File:Runcinated cubic honeycomb.png

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 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} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	Runcinated 5-cell	File:Runcinated 5-cell verf.svg	File:Schlegel half-solid runcinated 5-cell.png
t0,3{3,3,4} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png	Runcinated 16-cell (Same as runcinated 8-cell)	File:Runcinated 8-cell verf.png	File:Schlegel half-solid runcinated 16-cell.pngFile:Schlegel half-solid runcinated 8-cell.png
t0,3{3,4,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	Runcinated 24-cell	File:Runcinated 24-cell verf.png	File:Runcinated 24-cell Schlegel halfsolid.png
t0,3{3,3,5} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png	Runcinated 120-cell (Same as runcinated 600-cell)	File:Runcinated 120-cell verf.png	File:Runcinated 120-cell.png
Euclidean convex uniform honeycombs
t0,3{4,3,4} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png	Runcinated cubic honeycomb (Same as cubic honeycomb)	File:Runcinated cubic honeycomb verf.png	File:Runcinated cubic honeycomb.png
Hyperbolic uniform honeycombs
t0,3{4,3,5} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png	Runcinated order-5 cubic honeycomb	File:Runcinated order-5 cubic honeycomb verf.png
t0,3{3,5,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	Runcinated icosahedral honeycomb	File:Runcinated icosahedral honeycomb verf.png
t0,3{5,3,5} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png	Runcinated order-5 dodecahedral honeycomb	File:Runcinated order-5 dodecahedral honeycomb verf.png

See also

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

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, (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, (Chapter 26)