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. The term originates from Kepler's names for the Archimedean solids.

In general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation.

A special kind of truncation, usually implied, is a uniform truncation, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths. There are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra.

A regular polygon can also be represented by its Coxeter-Dynkin diagram, CDel node 1.pngCDel n.pngCDel node.png, and its uniform truncation CDel node 1.pngCDel n.pngCDel node 1.png, and its complete truncation CDel node.pngCDel n.pngCDel node 1.png. The graph CDel node.pngCDel n.pngCDel node.png represents Coxeter group I2(n), with each node representing a mirror, and the edge representing the angle π/n between the mirrors, and a circle is given around one or both mirrors to show which ones are active.

When "truncation" applies to platonic solids or regular tilings, usually "uniform truncation" is implied, which means truncating until the original faces become regular polygons with twice as many sides as the original form.

Another type of truncation, cantellation, cuts edges and vertices, removing the original edges, replacing them with rectangles, removing the original vertices, and replacing them with the faces of the dual of the original regular polyhedra or tiling.

Higher dimensional polytopes have higher truncations. Runcination cuts faces, edges, and vertices. In 5 dimensions, sterication cuts cells, faces, and edges.

Edge-truncation is a beveling, or chamfer for polyhedra, similar to cantellation, but retaining the original vertices, and replacing edges by hexagons. In 4-polytopes, edge-truncation replaces edges with elongated bipyramid cells.

A uniform alternation of a truncated cuboctahedron gives a nonuniform snub cube.

Alternation or partial truncation removes only some of the original vertices.

Diminishment is a more general term used in reference to Johnson solids for the removal of one or more vertices, edges, or faces of a polytope, without disturbing the other vertices. For example, the tridiminished icosahedron starts with a regular icosahedron with 3 vertices removed.

Other partial truncations are symmetry-based; for example, the tetrahedrally diminished dodecahedron.

Types of truncations shown on an edge isolated from a larger polygon or polyhedron with red and blue vertices. The edge reverses direction after complete truncation.

The linear truncation process can be generalized by allowing parametric truncations that are negative, or that go beyond the midpoint of the edges, causing self-intersecting star polyhedra, and can parametrically relate to some of the regular star polygons and uniform star polyhedra.