# Truncated 5-simplexes

(Redirected from Truncated 5-simplex)
 Orthogonal projections in A5 Coxeter plane 5-simplex          Truncated 5-simplex          Bitruncated 5-simplex         In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.

There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex.

## Truncated 5-simplex

 Truncated 5-simplex Type Uniform 5-polytope Schläfli symbol t{3,3,3,3} Coxeter-Dynkin diagram                4-faces 12 6 {3,3,3} 6 t{3,3,3} Cells 45 30 {3,3} 15 t{3,3} Faces 80 60 {3}20 {6} Edges 75 Vertices 30 Vertex figure ( )v{3,3} Coxeter group A5 [3,3,3,3], order 720 Properties convex

The truncated 5-simplex has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells).

### Alternate names

• Truncated hexateron (Acronym: tix) (Jonathan Bowers)

### Coordinates

The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) or of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively.

## Bitruncated 5-simplex

 bitruncated 5-simplex Type Uniform 5-polytope Schläfli symbol 2t{3,3,3,3} Coxeter-Dynkin diagram              4-faces 12 6 2t{3,3,3} 6 t{3,3,3} Cells 60 45 {3,3} 15 t{3,3} Faces 140 80 {3} 60 {6} Edges 150 Vertices 60 Vertex figure { }v{3} Coxeter group A5 [3,3,3,3], order 720 Properties convex

### Alternate names

• Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)

### Coordinates

The vertices of the bitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,2) or of (0,0,1,2,2,2). These represent positive orthant facets of the bitruncated 6-orthoplex, and the tritruncated 6-cube respectively.

## Related uniform 5-polytopes

The truncated 5-simplex is one of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)