Truncation is the removal of portions of solids falling outside a set of symmetrically placed planes.

The operation implemented as Truncate[polyhedron, r] in the Wolfram Language package PolyhedronOperations` displaces points along the edges of a polyhedron by a ratio r<=1/2, where r is the fraction of the edge length at which to truncate, and then fills in the resulting holes with new polygons. While this is not true truncation, the operation is equivalent to truncation for regular solids and r<=1/2. The operation is implemented in the Wolfram Language as TruncatedPolyhedron[poly].

The dual operation of truncation is augmentation, which is the operation of replacing facial polygons with pyramids.


As illustrated above, the five Platonic solids belong to one of the following three truncation series (which, in the first two cases, carry the solid to its dual polyhedron).

Furthermore, seven of the 13 Archimedean solids can be constructed from Platonic solids by truncation, which the four remaining Archimedeans requiring either expansion or snubification.

See also

Augmentation, Dürer's Solid, Expansion, Pyramid, Snubification, Stellation, Truncated Cube, Truncated Dodecahedron, Truncated Icosahedron, Truncated Octahedron, Truncated Tetrahedron, Vertex Figure

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Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 138, 1987.Cromwell, P. R. Polyhedra. New York: Cambridge University Press, pp. 80-83, 124, and 194, 1997.Pugh, A. Polyhedra: A Visual Approach. Berkeley: University of California Press, pp. 76-80, 1976.

Referenced on Wolfram|Alpha


Cite this as:

Weisstein, Eric W. "Truncation." From MathWorld--A Wolfram Web Resource.

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