Rectified 24-cell honeycomb | |
---|---|
(No image) | |
Type | Uniform 4-honeycomb |
Schläfli symbol | r{3,4,3,3} rr{3,3,4,3} r2r{4,3,3,4} r2r{4,3,31,1} |
Coxeter-Dynkin diagrams |
|
4-face type |
Tesseract
Rectified 24-cell |
Cell type |
Cube
Cuboctahedron |
Face type |
Square Triangle |
Vertex figure |
Tetrahedral prism |
Coxeter groups | , [3,4,3,3] , [4,3,3,4] , [4,3,31,1 , [31,1,1,1] |
Properties | Vertex transitive |
In four-dimensional Euclidean geometry, the rectified 24-cell honeycomb is a uniform space-filling honeycomb. It is constructed by a rectification of the regular 24-cell honeycomb, containing tesseract and rectified 24-cell cells.
There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored rectified 24-cell and tesseract facets. The tetrahedral prism vertex figure contains 4 rectified 24-cells capped by two opposite tesseracts.
Coxeter group |
Coxeter diagram |
Facets | Vertex figure | Vertex figure symmetry (order) |
---|---|---|---|---|
= [3,4,3,3] |
4: 1: |
, [3,3,2] (48) | ||
3: 1: 1: |
, [3,2] (12) | |||
= [4,3,3,4] |
2,2: 1: |
, [2,2] (8) | ||
= [31,1,3,4] |
1,1: 2: 1: |
, [2] (4) | ||
= [31,1,1,1 |
1,1,1,1: 1: |
, [] (2) |
Regular and uniform honeycombs in 4-space:
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E2 | Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)- honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |
Rectified 24-cell honeycomb | |
---|---|
(No image) | |
Type | Uniform 4-honeycomb |
Schläfli symbol | r{3,4,3,3} rr{3,3,4,3} r2r{4,3,3,4} r2r{4,3,31,1} |
Coxeter-Dynkin diagrams |
|
4-face type |
Tesseract
Rectified 24-cell |
Cell type |
Cube
Cuboctahedron |
Face type |
Square Triangle |
Vertex figure |
Tetrahedral prism |
Coxeter groups | , [3,4,3,3] , [4,3,3,4] , [4,3,31,1 , [31,1,1,1] |
Properties | Vertex transitive |
In four-dimensional Euclidean geometry, the rectified 24-cell honeycomb is a uniform space-filling honeycomb. It is constructed by a rectification of the regular 24-cell honeycomb, containing tesseract and rectified 24-cell cells.
There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored rectified 24-cell and tesseract facets. The tetrahedral prism vertex figure contains 4 rectified 24-cells capped by two opposite tesseracts.
Coxeter group |
Coxeter diagram |
Facets | Vertex figure | Vertex figure symmetry (order) |
---|---|---|---|---|
= [3,4,3,3] |
4: 1: |
, [3,3,2] (48) | ||
3: 1: 1: |
, [3,2] (12) | |||
= [4,3,3,4] |
2,2: 1: |
, [2,2] (8) | ||
= [31,1,3,4] |
1,1: 2: 1: |
, [2] (4) | ||
= [31,1,1,1 |
1,1,1,1: 1: |
, [] (2) |
Regular and uniform honeycombs in 4-space:
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E2 | Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)- honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |