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# Cantellation

Cantellation, also known as (polyhedron) expansion (Stott 1910, not to be confused with general geometric expansion) is the process of radially displacing the edges or faces of a polyhedron while keeping their orientations and sizes constant then filling in the gaps with new faces (Ball and Coxeter 1987, pp. 139-140). This procedure was devised by Stott (1910), and can be used to construct all 11 amphichiral (out of 13 total) Archimedean solids. The opposite operation of polyhedron expansion (i.e., inward expansion) can ne called polyhedron contraction. Expansion is a special case of snubification in which no twist occurs.

The term "cantellation" is sometimes reserved for the -dimensional version of the operation corresponding to polyhedron expansion.

The following table summarizes some expansions of some unit edge length Platonic and Archimedean solids, where is the displacement and is the golden ratio.

Affine Transformation, Central Dilation, Convex Hull, Dilation, Geometric Contraction, Homothetic, Polyhedron, Snubification, Transformation, Truncation

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## References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 1987.Coxeter, H. S. M. and Greitzer, S. L. "Dilation." §4.7 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 94-95, 1967.Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 13, 1999.Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910.

Cantellation

## Cite this as:

Weisstein, Eric W. "Cantellation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cantellation.html