User:Tomruen/Stellations of 16-cell

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In 4-dimensional geometry, a stellation of a 16-cell is a 4-polytope constructed as volumes between hyperplane of the facets of a regular 16-cell.

Vertices[edit]

There are 4 sets of vertices as intersections of 4 hyperplanes:

Set Typical member Vertices Distance from center Convex polytope Graph
1 (2,0,0,0) 8 2 {3,3,4}
2 (1,1,1,1) 16 2 {4,3,3}
3 (2,2,2,0) 32 2√3 r{4,3,3}
4 (4,2,2,2) 64 2√7 variation of t0,3{4,3,3}

1-regions[edit]

These 4 sets define 7 1-regions (line segments):

Set 1 1 1 2 1 3 1 4 2 3 3 3 3 4
Length 2√2 2 2√2 4 2 2√2 2√2

2-regions[edit]

The 4 sets define 8 types of 2-regions (triangles|

Set 1 1 1 1 1 2 1 1 3 1 2 3 1 3 3 1 3 4 2 3 4 3 3 4
Type Equilateral Isosceles right Equilateral Isosceles right Equilateral Isosceles right Isosceles right Equilateral

3-regions[edit]

The 4 sets define 7 types of 3-regions (polyhedra):

Type Description Types of faces Number
1 1 1 1 Regular tetrahedron 1 1 1 4
1 1 1 2 Tetrahedron 1 1 1 1
1 1 2 3
1 1 2 3 Tetrahedron 1 1 2 1
1 1 3 1
1 2 3 2
1 1 3 3 Regular tetrahedron 1 1 3 2
1 3 3 2
1 2 3 3 Tetrahedron 1 2 3 2
1 3 3 1
2 3 3 1
2 3 3 3 4 Triangular dipyramid 2 3 3 3
3 3 4 3
1 3 3 4 Tetrahedron 1 3 3 1
1 3 4 2
3 3 4 1

layers[edit]

There are 5 layers (polychora):

Layer Component Structure Number of components
a 16-cell 1 1 1 1 1
b Simplex 1 1 1 1
1 1 1 2		 16
c Simplex 1 1 2 3
 32
d Simplex 1 1 2 3
1 1 3 3
1 2 3 3		 96
e 1 2 3 3
2 3 3 3 4
1 3 3 4		 64

Stellations[edit]

There are 6 stellated forms:

Name Composition Description
A a 8 vertices of 16-cell
B a∪b Compound of 2 tesseracts
Partially regular [2γ4]β4
C a∪b∪c 24 vertices of 24-cell
D a∪b∪c∪d
e e
De a∪b∪c∪d∪e

References[edit]

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