Augmentation is the dual operation of truncation which replaces the faces of a polyhedron with pyramids of height h (where h may be positive, zero, or negative) having the face as the base (Cromwell 1997, p. 124 and 195-197). The operation is sometimes also called accretion, akisation (since it transforms a regular polygon to an n-akis polyhedron, i.e., quadruples the number of faces), capping, or cumulation.

B. Grünbaum used the terms elevatum and invaginatum for positive-height (outward-pointing) and negative-height (inward-pointing), respectively, pyramids used in augmentation.

The term "augmented" is also sometimes used in the more general context of affixing one polyhedral cap over the face of a base solid. An example is the Johnson solid called the augmented truncated cube, for which the affixed shape is a square cupola--not a pyramid.

Augmentation is implemented under the misnomer Stellate[poly, ratio] in the Wolfram Language package PolyhedronOperations` and is implemented in the Wolfram Language as AugmentedPolyhedron[poly].

Mineralogists give the following special names to augmented forms of regular solids (Berry and Mason 1959, pp. 124 and 127).

solidname of augmentationmineralogical name
cubetetrakis hexahedrontetrahexahedron
octahedronsmall triakis octahedrontrisoctahedron
tetrahedrontriakis tetrahedrontristetrahedron

Augmentation with h=0 gives a triangulated version of the original solid. Augmentation series from negative to positive augmentation heights are illustrated below for the Platonic solids.


The figure and table below give special solids formed by augmentation of given heights on Platonic solids with unit edge lengths.

Origami augmented tetrahedronOrigami augmented dodecahedron
Origami augmented dodecahedronOrigami augmented icosahedron

The top images above show an origami augmented tetrahedron and augmented dodecahedron. They are built using triangle edge modules and constructed in a manner similar to other solids described by Gurkewitz and Arnstein (1995, p. 53). The bottom left figure shows an inwardly augmented dodecahedron (Fusè 1990, pp. 126-129), while the right figure shows an augmented icosahedron constructed by E. W. Weisstein (Kasahara and Takahama 1987, p. 45).

See also

Elevatum, Diminished Polyhedron, Escher's Solid, Invaginatum, Midpoint Augmentation, Pyramid, Small Triakis Octahedron, Star Polyhedron, Stellation, Tetrakis Hexahedron, Triakis Tetrahedron, Truncation

Explore with Wolfram|Alpha


Berry, L. G. and Mason, B. Mineralogy: Concepts, Descriptions, Determinations. San Francisco, CA: W. H. Freeman, 1959.Cromwell, P. R. Polyhedra. New York: Cambridge University Press, 1997.Fusè, T. Unit Origami: Multidimensional Transformations. Tokyo: Japan Pub., 1990.Graziotti, U. Polyhedra, the Realm of Geometric Beauty. San Francisco, CA: 1962.Gurkewitz, R. and Arnstein, B. 3-D Geometric Origami: Modular Polyhedra. New York: Dover, 1995.Kasahara, K. and Takahama, T. Origami for the Connoisseur. Tokyo: Japan Publications, 1987.

Cite this as:

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

Subject classifications