Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.

Truncation in regular polyhedra and tilings

When the term applies to truncating platonic solids or regular tilings, usually "uniform truncation" is implied, which means to truncate until the original faces become regular polygons with double the sides.

This sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube and a rectified cube. The final polyhedron is a cuboctahedron.

The middle image is the uniform truncated cube. It is represented by an extended Schläfli symbol t_0,1{p,q,...}.

Other truncations

In quasiregular polyhedra, a "truncation" is a more qualitative term where some other adjustments are made to adjust truncated faces to become regular. These are sometimes called rhombitruncations.

For example, the truncated cuboctahedron is not really a truncation since the cut vertices of the cuboctahedron would form rectangular faces rather than squares, so a wider operation is needed to "adjust" the polyhedron to fit desired squares.

In the quasiregular duals, an "alternate truncation" operation only truncates alternate vertices. (This operation can also apply to any zonohedron which have even-sided faces.)

Uniform polyhedron and tiling examples

This table shows the truncation progression between the regular forms, with the rectified forms (full truncation) in the center. Comparable faces are colored red and yellow to show the continuum in the sequences.

Truncation in polychora and honeycomb tessellation

A regular polychoron or tessellation {p,q,r}, truncated becomes a uniform polychoron or tessellation with 2 cells: truncated {p,q}, and {q,r} cells are created on the truncated section.

See: uniform polychoron and convex uniform honeycomb.

See also

* uniform polyhedron
* uniform polychoron
* Bitruncation (geometry)
* Rectification (geometry)
* Alternation (geometry)
* Conway polyhedron notation

References

* Coxeter, H.S.M. "Regular Polytopes", (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp.145-154 Chapter 8: Truncation)

External links

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