# Runcinated tesseract

 Orthogonal projections in BC4 Coxeter plane TesseractTemplate:CDD Runcinated tesseract(Runcinated 16-cell)Template:CDD 16-cellTemplate:CDD Runcitruncated tesseract(Runcicantellated 16-cell)Template:CDD Runcitruncated 16-cell(Runcicantellated tesseract)Template:CDD Omnitruncated tesseract(Omnitruncated 16-cell)Template:CDD

In four-dimensional geometry, a runcinated tesseract (or runcinated 16-cell) is a convex uniform polychoron, being a runcination (a 3rd order truncation) of the regular tesseract.

There are 4 degrees of runcinations of the tesseract including with permutations truncations and cantellations.

## Runcinated tesseract

 Runcinated tesseract Schlegel diagram with 16 tetrahedra Type Uniform polychoron Schläfli symbol t0,3{4,3,3} Coxeter-Dynkin diagrams Template:CDD Cells 80 16 3.3.3 32 3.4.4 32 4.4.4 Faces 208 64 {3}144 {4} Edges 192 Vertices 64 Vertex figure Equilateral-triangular antipodium Symmetry group BC4, [3,3,4], order 384 Properties convex Uniform index 14 15 16

The runcinated tesseract has 16 tetrahedra, 32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron.

### Construction

The runcinated tesseract may be constructed by expanding the cells of a tesseract radially, and filling in the gaps with tetrahedra (vertex figures), cubes (face prisms), and triangular prisms (edge figures). The same process applied to a 16-cell also yields the same figure.

### Cartesian coordinates

The Cartesian coordinates of the vertices of the runcinated tesseract with edge length 2 are all permutations of:

${\displaystyle \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}})\right)}$

### Images

 Wireframe Wireframe with 16 tetrahedra. Wireframe with 32 triangular prisms.

### Structure

Eight of the cubical cells are connected to the other 24 cubical cells via all 6 square faces. The other 24 cubical cells are connected to the former 8 cells via only two opposite square faces; the remaining 4 faces are connected to the triangular prisms. The triangular prisms are connected to the tetrahedra via their triangular faces.

### Projections

The cube-first orthographic projection of the runcinated tesseract into 3-dimensional space has a (small) rhombicuboctahedral envelope. The images of its cells are laid out within this envelope as follows:

• The nearest and farthest cube from the 4d viewpoint projects to a cubical volume in the center of the envelope.
• Six cuboidal volumes connect this central cube to the 6 axial square faces of the rhombicuboctahedron. These are the images of 12 of the cubical cells (each pair of cubes share an image).
• The 18 square faces of the envelope are the images of the other cubical cells.
• The 12 wedge-shaped volumes connecting the edges of the central cube to the non-axial square faces of the envelope are the images of 24 of the triangular prisms (a pair of cells per image).
• The 8 triangular faces of the envelope are the images of the remaining 8 triangular prisms.
• Finally, the 8 tetrahedral volumes connecting the vertices of the central cube to the triangular faces of the envelope are the images of the 16 tetrahedra (again, a pair of cells per image).

This layout of cells in projection is analogous to the layout of the faces of the (small) rhombicuboctahedron under projection to 2 dimensions. The rhombicuboctahedron is also constructed from the cube or the octahedron in an analogous way to the runcinated tesseract. Hence, the runcinated tesseract may be thought of as the 4-dimensional analogue of the rhombicuboctahedron.

## Runcitruncated tesseract

 Runcitruncated tesseract Schlegel diagramcentered on a truncated cube,with cuboctahedral cells shown Type Uniform polychoron Schläfli symbol t0,1,3{4,3,3} Coxeter-Dynkin diagrams Template:CDD Cells 80 8 3.4.4 16 3.4.3.4 24 4.4.8 32 3.4.4 Faces 368 128 {3}192 {4}48 {8} Edges 480 Vertices 192 Vertex figure Rectangular pyramid Symmetry group BC4, [3,3,4], order 384 Properties convex Uniform index 18 19 20

The runcitruncated tesseract is bounded by 80 cells: 8 truncated cubes, 16 cuboctahedra, 24 octagonal prisms, and 32 triangular prisms.

### Construction

The runcitruncated tesseract may be constructed from the truncated tesseract by expanding the truncated cube cells outward radially, and inserting octagonal prisms between them. In the process, the tetrahedra expand into cuboctahedra, and triangular prisms fill in the remaining gaps.

The Cartesian coordinates of the vertices of the runcitruncated tesseract having an edge length of 2 is given by all permutations of:

${\displaystyle \left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)}$

### Projections

In the truncated cube first parallel projection of the runcitruncated tesseract into 3-dimensional space, the projection image is laid out as follows:

• The projection envelope is a non-uniform (small) rhombicuboctahedron, with 6 square faces and 12 rectangular faces.
• Two of the truncated cube cells project to a truncated cube in the center of the projection envelope.
• Six octagonal prisms connect this central truncated cube to the square faces of the envelope. These are the images of 12 of the octahedral prism cells, two cells to each image.
• The remaining 12 octahedral prisms are projected to the rectangular faces of the envelope.
• The 6 square faces of the envelope are the images of the remaining 6 truncated cube cells.
• Twelve right-angle triangular prisms connect the inner octagonal prisms. These are the images of 24 of the triangular prism cells. The remaining 8 triangular prisms project onto the triangular faces of the envelope.
• The 8 remaining volumes lying between the triangular faces of the envelope and the inner truncated cube are the images of the 16 cuboctahedral cells, a pair of cells to each image.

### Images

Stereographic projection with its 128 blue triangular faces and its 192 green quad faces.

## Runcitruncated 16-cell

 Runcitruncated 16-cell Schlegel diagramscentered on rhombicuboctahedron and truncated tetrahedron Type Uniform polychoron Schläfli symbol t0,1,3{3,3,4} Coxeter-Dynkin diagram Template:CDD Cells 80 8 3.4.4.4 16 3.6.6 24 4.4.4 32 4.4.6 Faces 368 64 {3}240 {4}64 {6} Edges 480 Vertices 192 Vertex figure Trapezoidal pyramid Symmetry group BC4, [3,3,4], order 384 Properties convex Uniform index 19 20 21

The runcitruncated 16-cell (or runcicantellated tesseract) is bounded by 80 cells: 8 rhombicuboctahedra, 16 truncated tetrahedra, 24 cubes, and 32 hexagonal prisms.

### Construction

The runcitruncated 16-cell may be constructed by contracting the small rhombicuboctahedral cells of the cantellated tesseract radially, and filling in the spaces between them with cubes. In the process, the octahedral cells expand into truncated tetrahedra (half of their triangular faces are expanded into hexagons by pulling apart the edges), and the triangular prisms expand into hexagonal prisms (each with its three original square faces joined, as before, to small rhombicuboctahedra, and its three new square faces joined to cubes).

The vertices of a runcitruncated 16-cell having an edge length of 2 is given by all permutations of the following Cartesian coordinates:

${\displaystyle \left(\pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)}$

### Structure

The small rhombicuboctahedral cells are joined via their 6 axial square faces to the cubical cells, and joined via their 12 non-axial square faces to the hexagonal prisms. The cubical cells are joined to the rhombicuboctahedra via 2 opposite faces, and joined to the hexagonal prisms via the remaining 4 faces. The hexagonal prisms are connected to the truncated tetrahedra via their hexagonal faces, and to the rhombicuboctahedra via 3 of their square faces each, and to the cubes via the other 3 square faces. The truncated tetrahedra are joined to the rhombicuboctahedra via their triangular faces, and the hexagonal prisms via their hexagonal faces.

### Projections

The following is the layout of the cells of the runcitruncated 16-cell under the parallel projection, small rhombicuboctahedron first, into 3-dimensional space:

• The projection envelope is a truncated icuboctahedron.
• Six of the small rhombicuboctahedra project onto the 6 octagonal faces of this envelope, and the other two project to a small rhombicuboctahedron lying at the center of this envelope.
• The 6 cuboidal volumes connecting the axial square faces of the central small rhombicuboctahedron to the center of the octagons correspond with the image of 12 of the cubical cells (each pair of the twelve share the same image).
• The remaining 12 cubical cells project onto the 12 square faces of the great rhombicuboctahedral envelope.
• The 8 volumes connecting the hexagons of the envelope to the triangular faces of the central rhombicuboctahedron are the images of the 16 truncated tetrahedra.
• The remaining 12 spaces connecting the non-axial square faces of the central small rhombicuboctahedron to the square faces of the envelope are the images of 24 of the hexagonal prisms.
• Finally, the last 8 hexagonal prisms project onto the hexagonal faces of the envelope.

This layout of cells is similar to the layout of the faces of the great rhombicuboctahedron under the projection into 2-dimensional space. Hence, the runcitruncated 16-cell may be thought of as one of the 4-dimensional analogues of the great rhombicuboctahedron. The other analogue is the omnitruncated tesseract.

## Omnitruncated tesseract

 Omnitruncated tesseract Schlegel diagram,centered on truncated cuboctahedron,truncated octahedral cells shown Type Uniform polychoron Schläfli symbol t0,1,2,3{3,3,4} Coxeter-Dynkin diagrams Template:CDD Cells 80 8 4.6.8 16 4.6.6 24 4.4.8 32 4.4.6 Faces 464 288 {4}128 {6}48 {8} Edges 768 Vertices 384 Vertex figure Chiral scalene tetrahedron Symmetry group BC4, [3,3,4], order 384 Properties convex Uniform index 20 21 22

The omnitruncated tesseract (or omnitruncated 16-cell) is bounded by 80 cells: 8 truncated cuboctahedra, 16 truncated octahedra, 24 octagonal prisms, and 32 hexagonal prisms.

### Construction

The omnitruncated tesseract can be constructed from the cantitruncated tesseract by radially displacing the truncated cuboctahedral cells so that octagonal prisms can be inserted between their octagonal faces. As a result, the triangular prisms expand into hexagonal prisms, and the truncated tetrahedra expand into truncated octahedra.

The Cartesian coordinates of the vertices of an omnitruncated tesseract having an edge length of 2 are given by all permutations of coordinates and sign of:

${\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}$

### Structure

The truncated cuboctahedra cells are joined to the octagonal prisms via their octagonal faces, the truncated octahedra via their hexagonal faces, and the hexagonal prisms via their square faces. The octagonal prisms are joined to the hexagonal prisms and the truncated octahedra via their square faces, and the hexagonal prisms are joined to the truncated octahedra via their hexagonal faces.

### Projections

In the truncated cuboctahedron first parallel projection of the omnitruncated tesseract into 3 dimensions, the images of its cells are laid out as follows:

• The projection envelope is in the shape of a non-uniform truncated cuboctahedron.
• Two of the truncated cuboctahedra project to the center of the projection envelope.
• The remaining 6 truncated cuboctahedra project to the (non-regular) octagonal faces of the envelope. These are connected to the central truncated cuboctahedron via 6 octagonal prisms, which are the images of the octagonal prism cells, a pair to each image.
• The 8 hexagonal faces of the envelope are the images of 8 of the hexagonal prisms.
• The remaining hexagonal prisms are projected to 12 non-regular hexagonal prism images, lying where a cube's edges would be. Each image corresponds to two cells.
• Finally, the 8 volumes between the hexagonal faces of the projection envelope and the hexagonal faces of the central truncated cuboctahedron are the images of the 16 truncated octahedra, two cells to each image.

This layout of cells in projection is similar to that of the runcitruncated 16-cell, which is analogous to the layout of faces in the octagon-first projection of the truncated cuboctahedron into 2 dimensions. Thus, the omnitruncated tesseract may be thought of as another analogue of the truncated cuboctahedron in 4 dimensions.

### Images

Perspective projections

Perspective projection centered on one of the truncated cuboctahedral cells, highlighted in yellow. Six of the surrounding octahedral prisms rendered in blue, and the remaining cells in green. Cells obscured from 4D viewpoint culled for clarity's sake.

Perspective projection centered on one of the truncated octahedral cells, highlighted in yellow. Four of the surrounding hexagonal prisms are shown in blue, with 4 more truncated octahedra on the other side of these prisms also shown in yellow. Cells obscured from 4D viewpoint culled for clarity's sake. Some of the other hexagonal and octagonal prisms may be discerned from this view as well.
Stereographic projections

Centered on truncated cuboctahedron

Centered on truncated octahedron

### Full snub tesseract

Vertex figure for the full snub tesseract

The full snub tesseract or full snub 16-cell, defined as an alternation of the omnitruncated tesseract, can not be made uniform, but it can be given Coxeter diagram Template:CDD, and symmetry [4,3,3]+, and constructed from 16 icosahedrons, 32 octahedra, 24 square antiprisms, 8 snub cubes and 192 tetrahedrons filling the gaps at the deleted vertices. It has 272 cells, 944 faces, 864 edges, and 192 vertices.[1]

## References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• H.S.M. Coxeter:
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Template:PolyCell
• http://www.polytope.de/nr17.html
• Template:KlitzingPolytopes x3o3o4x - sidpith, x3o3x4x - proh, x3x3o4x - prit