Runcinated 5-simplex


5-simplex
Runcinated 5-simplex
Runcitruncated 5-simplex

Birectified 5-simplex
Runcicantellated 5-simplex
Runcicantitruncated 5-simplex
Orthogonal projections in A5 Coxeter plane

In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.

There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations.

Contents

Runcinated 5-simplex

Runcinated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces 47 6 t03{3,3}
20 {3}x{3}
15 {}xt1{3,3}
6 t1{3,3}
Cells 255 45 {3,3}
180 {}x{3}
30 t1{3,3}
Faces 420 240 {3}
180 {4}
Edges 270
Vertices 60
Vertex figure
Coxeter group A5 [3,3,3,3], order 720
Properties convex

Alternate names

Coordinates

The vertices of the runcinated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,1,1,1,2) or of (0,1,1,1,2,2), seen as facets of a runciated 5-orthoplex, or a biruncinated 5-cube respectively.

Images

orthographic projections
Ak
Coxeter plane		 A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane		 A3 A2
Graph
Dihedral symmetry [4] [3]

Runcitruncated 5-simplex

Runcitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces 47 6 t0,1,3{3,3,3}
20 {3}x{6}
15 {}xt1{3,3}
6 t0,2{3,3,3}
Cells 255
Faces 570
Edges 540
Vertices 180
Vertex figure
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate names

Coordinates

The coordinates can be made in 6-space, as 180 permutations of:

(0,0,1,1,2,3)

This construction exists as one of 64 orthant facets of the runcitruncated 6-orthoplex.

Images

orthographic projections
Ak
Coxeter plane		 A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane		 A3 A2
Graph
Dihedral symmetry [4] [3]

Runcicantellated 5-simplex

Runcicantellated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,2,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces 47
Cells 255
Faces 570
Edges 540
Vertices 180
Vertex figure
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate names

Coordinates

The coordinates can be made in 6-space, as 180 permutations of:

(0,0,1,2,2,3)

This construction exists as one of 64 orthant facets of the runcicantellated 6-orthoplex.

Images

orthographic projections
Ak
Coxeter plane		 A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane		 A3 A2
Graph
Dihedral symmetry [4] [3]

Runcicantitruncated 5-simplex

Runcicantitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces 47 6 t0,1,2,3{3,3,3}
20 {3}x{6}
15 [{}xt0,1{3,3}
6 t0,1,2{3,3,3}
Cells 315 45 t0,1,2{3,3}
120 {}x{3}
120 {}x{6}
30 t{3,3}
Faces 810 120 {3}
450 {4}
240 {6}
Edges 900
Vertices 360
Vertex figure
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate names

Coordinates

The coordinates can be made in 6-space, as 360 permutations of:

(0,0,1,2,3,4)

This construction exists as one of 64 orthant facets of the runcicantitruncated 6-orthoplex.

Images

orthographic projections
Ak
Coxeter plane		 A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane		 A3 A2
Graph
Dihedral symmetry [4] [3]

Related uniform 5-polytopes

These polytopes are in a set of 19 uniform polytera 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)


t0
t1
t2
t0,1
t0,2
t1,2
t0,3

t1,3
t0,4
t0,1,2
t0,1,3
t0,2,3
t1,2,3
t0,1,4

t0,2,4
t0,1,2,3
t0,1,2,4
t0,1,3,4
t0,1,2,3,4

Notes

  1. ^ Klitizing, (x3o3o3x3o - spidtix)
  2. ^ Klitizing, (x3x3o3x3o - pattix)
  3. ^ Klitizing, (x3o3x3x3o - pirx)
  4. ^ Klitizing, (x3x3x3x3o - gippix)

References

External links