Special cut

The special cuts are a set of 314,248,344 4-dimensional Blind polytopes formed by diminishing the hexacosichoron (600-cell), i.e. removing non-adjacent sets of vertices and taking the convex hull of the remaining vertices. Each diminishing removes 20 tetrahedra and replaces them with an icosahedron. The number of diminishings is called the size of the special cut. Roughly speaking, the special cuts are the 4-dimensional analogue of the three Johnson solids formed by diminishing the icosahedron (diminished icosahedron, metabidiminished icosahedron, and tridiminished icosahedron) plus the pentagonal antiprism. They have no obvious generalization to higher dimensions.

In 2008 Mathieu Dutour Sikirić and Wendy Myrvold enumerated the complete list.

Some special cuts are highly symmetrical. The semiregular snub disicositetrachoron (snub 24-cell) has a maximum size of 24, and a maximum possible symmetry group order of 576. A special cut is called maximal if no further diminishings can be done; the minimum size of a maximal special cut is 10.

312,809,673 (about 99.54%) of the special cuts are asymmetrical. Asymmetrical special cuts range from size 3 (where there is only one such special cut) to size 21 inclusive (count 22).