| Runcicantellated 24-cell honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform 4-honeycomb |
| Schläfli symbols | t0,2,3{3,4,3,3} s2,3{3,4,3,3} |
| Coxeter diagrams |      |
| 4-face type | t0,1,3{3,3,4}  |
| Cell type | |
| Face type | |
| Vertex figure | |
| Coxeter groups | , [3,4,3,3] |
| Properties | Vertex transitive |
In four-dimensional Euclidean geometry, the runcicantellated 24-cell honeycomb is a uniform space-filling honeycomb.
The [3,4,3,3], , Coxeter group generates 31 permutations of uniform tessellations, 28 are unique in this family and ten are shared in the [4,3,3,4] and [4,3,31,1] families. The alternation (13) is also repeated in other families.
| F4 honeycombs | |||
|---|---|---|---|
| Extended symmetry | Extended diagram | Order | Honeycombs |
| [3,3,4,3] | | ×1 | |
| [3,4,3,3] | | ×1 |
|
| [(3,3)[3,3,4,3*]] =[(3,3)[31,1,1,1]] =[3,4,3,3] |       =  =  | ×4 | |
Regular and uniform honeycombs in 4-space:
In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes. It is not semiregular by Gosset's definition of regular facets, but all of its cells (ridges) are regular, either tetrahedra or icosahedra.
In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells.
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.
In four-dimensional Euclidean geometry, the rectified tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space. It is constructed by a rectification of a tesseractic honeycomb which creates new vertices on the middle of all the original edges, rectifying the cells into rectified tesseracts, and adding new 16-cell facets at the original vertices. Its vertex figure is an octahedral prism, {3,4}×{}.
In four-dimensional Euclidean geometry, the steriruncitruncated tesseractic honeycomb is a uniform space-filling honeycomb.
In four-dimensional Euclidean geometry, the stericantellated tesseractic honeycomb is a uniform space-filling honeycomb.
In four-dimensional Euclidean geometry, the omnitruncated tesseractic honeycomb is a uniform space-filling honeycomb. It has omnitruncated tesseract, truncated cuboctahedral prism, and 8-8 duoprism facets in an irregular 5-cell vertex figure.
In four-dimensional Euclidean geometry, the cantellated tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space. It is constructed by a cantellation of a tesseractic honeycomb creating cantellated tesseracts, and new 24-cell and octahedral prism facets at the original vertices.
In four-dimensional Euclidean geometry, the runcicantellated tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.
In four-dimensional Euclidean geometry, the stericantitruncated tesseractic honeycomb is a uniform space-filling honeycomb. It is composed of runcitruncated 16-cell, cantitruncated tesseract, rhombicuboctahedral prism, truncated cuboctahedral prism, and 4-8 duoprism facets, arranged around an irregular 5-cell vertex figure.
In four-dimensional Euclidean geometry, the steriruncic tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.
In four-dimensional Euclidean geometry, the stericantic tesseractic honeycomb is a uniform space-filling tessellation in Euclidean 4-space.
In four-dimensional Euclidean geometry, the cantellated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a cantellation of the regular 24-cell honeycomb, containing rectified tesseract, cantellated 24-cell, and tetrahedral prism cells.
In four-dimensional Euclidean geometry, the cantitruncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a cantitruncation of the regular 24-cell honeycomb, containing truncated tesseract, cantitruncated 24-cell, and tetrahedral prism cells.
In four-dimensional Euclidean geometry, the runcinated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a runcination of the regular 24-cell honeycomb, containing runcinated 24-cell, 24-cell, octahedral prism, and 3-3 duoprism cells.
In four-dimensional Euclidean geometry, the runcinated 16-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a runcination of the regular 16-cell honeycomb, containing Rectified 24-cell, runcinated tesseract, cuboctahedral prism, and 3-3 duoprism cells.
In four-dimensional Euclidean geometry, the stericated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a sterication of the regular 24-cell honeycomb, containing 24-cell, 16-cell, octahedral prism, tetrahedral prism, and 3-3 duoprism cells.
In four-dimensional Euclidean geometry, the bitruncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a bitruncation of the regular 24-cell honeycomb, constructed by truncated tesseract and bitruncated 24-cell cells.
In four-dimensional Euclidean geometry, the steritruncated 16-cell honeycomb is a uniform space-filling honeycomb, with runcinated 24-cell, truncated 16-cell, octahedral prism, 3-6 duoprism, and truncated tetrahedral prism cells.
In four-dimensional Euclidean geometry, the stericantitruncated 16-cell honeycomb is a uniform space-filling honeycomb.
| 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 |
