Augmentation is the dual operation of truncation which replaces the faces of a polyhedron with pyramids
of height
(where
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
-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).
solid | name of augmentation | mineralogical name |
cube | tetrakis hexahedron | tetrahexahedron |
octahedron | small triakis octahedron | trisoctahedron |
tetrahedron | triakis tetrahedron | tristetrahedron |
Augmentation with
gives a triangulated version of the original solid. Augmentation series from negative
to positive augmentation heights are illustrated below for the Platonic
solids.
![CumulatedPlatonicSeries](images/eps-svg/CumulatedPlatonicSeries_1000.png)
The figure and table below give special solids formed by augmentation of given heights on Platonic solids with unit edge lengths.
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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).