Runcinated tesseracts

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4-cube t0.svg
Tesseract
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4-cube t03.svg
Runcinated tesseract
(Runcinated 16-cell)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
4-cube t3.svg
16-cell
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
4-cube t013.svg
Runcitruncated tesseract
(Runcicantellated 16-cell)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
4-cube t023.svg
Runcitruncated 16-cell
(Runcicantellated tesseract)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
4-cube t0123.svg
Omnitruncated tesseract
(Omnitruncated 16-cell)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Orthogonal projections in BC4 Coxeter plane

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 variations of runcinations of the tesseract including with permutations truncations and cantellations.


Runcinated tesseract[edit]

Runcinated tesseract
Schlegel half-solid runcinated 8-cell.png
Schlegel diagram with 16 tetrahedra
Type Uniform polychoron
Schläfli symbol t0,3{4,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells 80 16 3.3.3 Tetrahedron.png
32 3.4.4 Triangular prism.png
32 4.4.4 Hexahedron.png
Faces 208 64 {3}
144 {4}
Edges 192
Vertices 64
Vertex figure Runcinated 8-cell verf.png
Equilateral-triangular antipodium
Symmetry group BC4, [3,3,4], order 384
Properties convex
Uniform index 14 15 16

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

Construction[edit]

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[edit]

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

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

Images[edit]

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph 4-cube t03.svg 4-cube t03 B3.svg 4-cube t03 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph 4-cube t03 F4.svg 4-cube t03 A3.svg
Dihedral symmetry [12/3] [4]
Schlegel diagrams
Runci tessaract1.png
Wireframe
Runci tessaract2.png
Wireframe with 16 tetrahedra.
Runci tessaract3.png
Wireframe with 32 triangular prisms.

Structure[edit]

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[edit]

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[edit]

Runcitruncated tesseract
Schlegel half-solid runcitruncated 8-cell.png
Schlegel diagram
centered on a truncated cube,
with cuboctahedral cells shown
Type Uniform polychoron
Schläfli symbol t0,1,3{4,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells 80 8 3.4.4 Truncated hexahedron.png
16 3.4.3.4 Cuboctahedron.png
24 4.4.8 Octagonal prism.png
32 3.4.4 Triangular prism.png
Faces 368 128 {3}
192 {4}
48 {8}
Edges 480
Vertices 192
Vertex figure Runcitruncated 8-cell verf.png
Rectangular pyramid
Symmetry group BC4, [3,3,4], order 384
Properties convex
Uniform index 18 19 20

The runcitruncated tesseract, runcicantellated 16-cell, or prismatorhombated hexadecachoron is bounded by 80 cells: 8 truncated cubes, 16 cuboctahedra, 24 octagonal prisms, and 32 triangular prisms.

Construction[edit]

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:

\left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right)

Projections[edit]

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[edit]

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph 4-cube t013.svg 4-cube t013 B3.svg 4-cube t013 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph 4-cube t013 F4.svg 4-cube t013 A3.svg
Dihedral symmetry [12/3] [4]

Runci trunc tessaract.png
Stereographic projection with its 128 blue triangular faces and its 192 green quad faces.

Runcitruncated 16-cell[edit]

Runcitruncated 16-cell
Runcitruncated 16-cell.pngSchlegel half-solid runcitruncated 16-cell.png
Schlegel diagrams
centered on rhombicuboctahedron and truncated tetrahedron
Type Uniform polychoron
Schläfli symbol t0,1,3{3,3,4}
Coxeter-Dynkin diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells 80 8 3.4.4.4 Small rhombicuboctahedron.png
16 3.6.6 Truncated tetrahedron.png
24 4.4.4 Hexahedron.png
32 4.4.6 Hexagonal prism.png
Faces 368 64 {3}
240 {4}
64 {6}
Edges 480
Vertices 192
Vertex figure Runcitruncated 16-cell verf.png
Trapezoidal pyramid
Symmetry group BC4, [3,3,4], order 384
Properties convex
Uniform index 19 20 21

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

Construction[edit]

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:

\left(\pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right)

Images[edit]

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph 4-cube t023.svg 4-cube t023 B3.svg 4-cube t023 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph 4-cube t023 F4.svg 4-cube t023 A3.svg
Dihedral symmetry [12/3] [4]

Structure[edit]

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[edit]

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 cuboctahedron.
  • 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[edit]

Omnitruncated tesseract
Schlegel half-solid omnitruncated 8-cell.png
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 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells 80 8 4.6.8 Great rhombicuboctahedron.png
16 4.6.6 Truncated octahedron.png
24 4.4.8 Octagonal prism.png
32 4.4.6 Hexagonal prism.png
Faces 464 288 {4}
128 {6}
48 {8}
Edges 768
Vertices 384
Vertex figure Omnitruncated 8-cell verf.png
Chiral scalene tetrahedron
Symmetry group BC4, [3,3,4], order 384
Properties convex
Uniform index 20 21 22

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

Construction[edit]

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:

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

Structure[edit]

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[edit]

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[edit]

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph 4-cube t0123.svg 4-cube t0123 B3.svg 4-cube t0123 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph 4-cube t0123 F4.svg 4-cube t0123 A3.svg
Dihedral symmetry [12/3] [4]
Perspective projections
Omnitruncated tesseract-perspective-great rhombicuboctahedron-first-01.png
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.
Omnitruncated tesseract-perspective-truncated octahedron-first.png
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
Omnitruncated tesseract stereographic (tCO).png
Centered on truncated cuboctahedron
Omnitruncated tesseract stereographic (tO).png
Centered on truncated octahedron
Net
Great disprismatotesseractihexadecachoron net.png
Omnitruncated tesseract
Dual gidpith net.png
Dual to omnitruncated tesseract

Full snub tesseract[edit]

Vertex figure for the full snub tesseract

The full snub tesseract or omnisnub tesseract, defined as an alternation of the omnitruncated tesseract, can not be made uniform, but it can be given Coxeter diagram CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png, 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]

Related uniform polytopes[edit]

Name tesseract rectified
tesseract
truncated
tesseract
cantellated
tesseract
runcinated
tesseract
bitruncated
tesseract
cantitruncated
tesseract
runcitruncated
tesseract
omnitruncated
tesseract
Coxeter
diagram
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schläfli
symbol
{4,3,3} t1{4,3,3}
r{4,3,3}
t0,1{4,3,3}
t{4,3,3}
t0,2{4,3,3}
rr{4,3,3}
t0,3{4,3,3} t1,2{4,3,3}
2t{4,3,3}
t0,1,2{4,3,3}
tr{4,3,3}
t0,1,3{4,3,3} t0,1,2,3{4,3,3}
Schlegel
diagram
Schlegel wireframe 8-cell.png Schlegel half-solid rectified 8-cell.png Schlegel half-solid truncated tesseract.png Schlegel half-solid cantellated 8-cell.png Schlegel half-solid runcinated 8-cell.png Schlegel half-solid bitruncated 8-cell.png Schlegel half-solid cantitruncated 8-cell.png Schlegel half-solid runcitruncated 8-cell.png Schlegel half-solid omnitruncated 8-cell.png
B4 4-cube t0.svg 4-cube t1.svg 4-cube t01.svg 4-cube t02.svg 4-cube t03.svg 4-cube t12.svg 4-cube t012.svg 4-cube t013.svg 4-cube t0123.svg
 
Name 16-cell rectified
16-cell
truncated
16-cell
cantellated
16-cell
runcinated
16-cell
bitruncated
16-cell
cantitruncated
16-cell
runcitruncated
16-cell
omnitruncated
16-cell
Coxeter
diagram
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schläfli
symbol
{3,3,4} t1{3,3,4}
r{3,3,4}
t0,1{3,3,4}
t{3,3,4}
t0,2{3,3,4}
rr{3,3,4}
t0,3{3,3,4} t1,2{3,3,4}
2t{3,3,4}
t0,1,2{3,3,4}
tr{3,3,4}
t0,1,3{3,3,4} t0,1,2,3{3,3,4}
Schlegel
diagram
Schlegel wireframe 16-cell.png Schlegel half-solid rectified 16-cell.png Schlegel half-solid truncated 16-cell.png Schlegel half-solid cantellated 16-cell.png Schlegel half-solid runcinated 16-cell.png Schlegel half-solid bitruncated 16-cell.png Schlegel half-solid cantitruncated 16-cell.png Schlegel half-solid runcitruncated 16-cell.png Schlegel half-solid omnitruncated 16-cell.png
B4 4-cube t3.svg 4-cube t2.svg 4-cube t23.svg 4-cube t13.svg 4-cube t03.svg 4-cube t12.svg 4-cube t123.svg 4-cube t023.svg 4-cube t0123.svg

Notes[edit]

References[edit]