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{| class=wikitable align=right width=450
|- align=center valign=top
|[[File:4-cube t0.svg|150px]]<BR>[[Tesseract]]<BR>{{CDD|node_1|4|node|3|node|3|node}}
|[[File:4-cube t03.svg|150px]]<BR>Runcinated tesseract<BR>(Runcinated 16-cell)<BR>{{CDD|node_1|4|node|3|node|3|node_1}}
|[[File:4-cube t3.svg|150px]]<BR>[[16-cell]]<BR>{{CDD|node|4|node|3|node|3|node_1}}
|- align=center valign=top
|[[File:4-cube t013.svg|150px]]<BR>Runcitruncated tesseract<BR>(Runcicantellated 16-cell)<BR>{{CDD|node_1|4|node_1|3|node|3|node_1}}
|[[File:4-cube t023.svg|150px]]<BR>Runcitruncated 16-cell<BR>(Runcicantellated tesseract)<BR>{{CDD|node_1|4|node|3|node_1|3|node_1}}
|[[File:4-cube t0123.svg|150px]]<BR>Omnitruncated tesseract<BR>(Omnitruncated 16-cell)<BR>{{CDD|node_1|4|node_1|3|node_1|3|node_1}}
|-
!colspan=3|[[Orthogonal projection]]s in BC<sub>4</sub> [[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 degrees of runcinations of the tesseract including with permutations truncations and cantellations.

{{TOC left}}
{{-}}

==Runcinated tesseract==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Runcinated tesseract'''
|-
|bgcolor=#ffffff align=center colspan=3|[[File:Schlegel half-solid runcinated 8-cell.png|220px]]<BR>[[Schlegel diagram]] with 16 tetrahedra
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform polychoron]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|t<sub>0,3</sub>{4,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s
|colspan=2|{{CDD|node_1|4|node|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells
|80
|16 [[Tetrahedron|''3.3.3'']] [[File:Tetrahedron.png|20px]]<BR>32 [[Triangular prism|''3.4.4'']] [[File:Triangular prism.png|20px]]<BR>32 [[cube|''4.4.4'']] [[File:Hexahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|208
|64 [[triangle|{3}]]<BR>144 [[square (geometry)|{4}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|192
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|64
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Runcinated 8-cell verf.png|80px]]<BR>Equilateral-triangular antipodium
|-
|bgcolor=#e7dcc3|[[Symmetry group]]
|colspan=2|BC<sub>4</sub>, [3,3,4], order 384
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Cantellated tesseract|14]]'' 15 ''[[Bitruncated tesseract|16]]''
|}

The '''runcinated tesseract''' has 16 [[tetrahedron|tetrahedra]], 32 [[cube]]s, and 32 [[triangular prism]]s. 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:

:<math>\left(\pm 1,\ \pm 1,\ \pm 1,\ \pm(1+\sqrt{2})\right)</math>

===Images ===
{{4-cube Coxeter plane graphs|t03|150}}

{| class="wikitable"
|+ [[Schlegel diagram]]s
|[[File:Runci tessaract1.png|180px]]<BR>Wireframe
|[[File:Runci tessaract2.png|180px]]<BR>Wireframe with 16 [[tetrahedron|tetrahedra]].
|[[File:Runci tessaract3.png|180px]]<BR>Wireframe with 32 [[triangular prism]]s.
|}

===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) [[rhombicuboctahedron|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==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Runcitruncated tesseract'''
|-
|bgcolor=#ffffff align=center colspan=3|[[File:Schlegel half-solid runcitruncated 8-cell.png|250px]]<BR>[[Schlegel diagram]]<BR>centered on a truncated cube,<BR>with cuboctahedral cells shown
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform polychoron]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|t<sub>0,1,3</sub>{4,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s
|colspan=2|{{CDD|node_1|4|node_1|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells
|80
|8 [[Truncated cube|''3.4.4'']] [[File:Truncated hexahedron.png|20px]]<BR>16 [[Cuboctahedron|''3.4.3.4'']] [[File:Cuboctahedron.png|20px]]<br>24 [[Octagonal prism|''4.4.8'']] [[File:Octagonal prism.png|20px]]<br>32 [[Triangular prism|''3.4.4'']] [[File:Triangular prism.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|368
|128 [[triangle|{3}]]<BR>192 [[square (geometry)|{4}]]<br>48 [[octagon|{8}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|480
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|192
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Runcitruncated 8-cell verf.png|80px]]<BR>Rectangular pyramid
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
|colspan=2|BC<sub>4</sub>, [3,3,4], order 384
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Cantitruncated tesseract|18]]'' 19 ''[[Runcitruncated 16-cell|20]]''
|}
The '''runcitruncated tesseract''' is bounded by 80 cells: 8 [[truncated cube]]s, 16 [[cuboctahedron|cuboctahedra]], 24 [[octagonal prism]]s, and 32 [[triangular prism]]s.

===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 [[tetrahedron|tetrahedra]] expand into cuboctahedra, and triangular prisms fill in the remaining gaps.

The [[Cartesian coordinate]]s of the vertices of the runcitruncated tesseract having an edge length of&nbsp;2 is given by all permutations of:

:<math>\left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right)</math>

===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 ===
{{4-cube Coxeter plane graphs|t013|150}}

[[File:Runci trunc tessaract.png|300px]]<BR>[[Stereographic projection]] with its 128 blue triangular faces and its 192 green quad faces.

==Runcitruncated 16-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="320"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Runcitruncated 16-cell'''
|-
|bgcolor=#ffffff align=center colspan=3|[[File:Runcitruncated 16-cell.png|150px]][[File:Schlegel half-solid runcitruncated 16-cell.png|150px]]<BR>[[Schlegel diagram]]s<BR>centered on [[rhombicuboctahedron]] and [[truncated tetrahedron]]
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform polychoron]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|t<sub>0,1,3</sub>{3,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
|colspan=2|{{CDD|node_1|4|node|3|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells
|80
|8 [[rhombicuboctahedron|''3.4.4.4'']] [[File:Small rhombicuboctahedron.png|20px]]<BR>16 [[Truncated tetrahedron|''3.6.6'']] [[File:Truncated tetrahedron.png|20px]]<br>24 [[Cube|''4.4.4'']] [[File:Hexahedron.png|20px]]<br>32 [[Hexagonal prism|''4.4.6'']] [[File:Hexagonal prism.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|368
|64 [[triangle|{3}]]<BR>240 [[square (geometry)|{4}]]<br>64 [[hexagon|{6}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|480
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|192
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Runcitruncated 16-cell verf.png|80px]]<BR>Trapezoidal pyramid
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
|colspan=2|BC<sub>4</sub>, [3,3,4], order 384
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Runcitruncated tesseract|19]]'' 20 ''[[Omnitruncated tesseract|21]]''
|}

The '''runcitruncated 16-cell''' (or '''runcicantellated tesseract''') is bounded by 80 [[cell (mathematics)|cells]]: 8 [[rhombicuboctahedron|rhombicuboctahedra]], 16 [[truncated tetrahedron|truncated tetrahedra]], 24 [[cube]]s, and 32 [[hexagonal prism]]s.

===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 coordinate]]s:

:<math>\left(\pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right)</math>

=== Images ===
{{4-cube Coxeter plane graphs|t023|150}}

===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 [[cell (mathematics)|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==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Omnitruncated tesseract'''
|-
|bgcolor=#ffffff align=center colspan=3|[[File:Schlegel half-solid omnitruncated 8-cell.png|250px]]<BR>[[Schlegel diagram]],<BR>centered on truncated cuboctahedron,<BR>truncated octahedral cells shown
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform polychoron]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|t<sub>0,1,2,3</sub>{3,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s
|colspan=2|{{CDD|node_1|4|node_1|3|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells
|80
|8 [[Truncated cuboctahedron|''4.6.8'']] [[File:Great rhombicuboctahedron.png|20px]]<BR>16 [[Truncated octahedron|''4.6.6'']] [[File:Truncated octahedron.png|20px]]<br>24 [[Octagonal prism|''4.4.8'']] [[File:Octagonal prism.png|20px]]<br>32 [[Hexagonal prism|''4.4.6'']] [[File:Hexagonal prism.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|464
|288 [[square (geometry)|{4}]]<br>128 [[hexagon|{6}]]<br>48 [[octagon|{8}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|768
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|384
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Omnitruncated 8-cell verf.png|80px]]<BR>Chiral scalene tetrahedron
|-
|bgcolor=#e7dcc3|[[Symmetry group]]
|colspan=2|BC<sub>4</sub>, [3,3,4], order 384
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Runcitruncated 16-cell|20]]'' 21 ''[[24-cell|22]]''
|}

The '''omnitruncated tesseract''' (or '''omnitruncated 16-cell''') is bounded by 80 [[cell (mathematics)|cells]]: 8 [[Truncated cuboctahedron|truncated cuboctahedra]], 16 [[truncated octahedron|truncated octahedra]], 24 [[octagonal prism]]s, and 32 [[hexagonal prism]]s.

===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 coordinate]]s of the vertices of an omnitruncated tesseract having an edge length of&nbsp;2 are given by all permutations of coordinates and sign of:

:<math>\left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right)</math>

===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 ===
{{4-cube Coxeter plane graphs|t0123|150}}

{| class="wikitable" width=640
!colspan=2|Perspective projections
|-
|align=center valign=top|[[File:Omnitruncated tesseract-perspective-great rhombicuboctahedron-first-01.png]]<BR>Perspective projection centered on one of the [[Truncated cuboctahedron|truncated cuboctahedral]] cells, highlighted in yellow. Six of the surrounding [[octahedral prism]]s rendered in blue, and the remaining cells in green. Cells obscured from 4D viewpoint culled for clarity's sake.
|align=center valign=top|[[File:Omnitruncated tesseract-perspective-truncated octahedron-first.png|320px]]<BR>Perspective projection centered on one of the [[truncated octahedron|truncated octahedral]] cells, highlighted in yellow. Four of the surrounding [[hexagonal prism]]s 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.
|-
!colspan=2|Stereographic projections
|-
|align=center width=320|[[File:Omnitruncated tesseract stereographic (tCO).png|200px]]<BR>Centered on truncated cuboctahedron
|align=center width=320|[[File:Omnitruncated tesseract stereographic (tO).png|200px]]<BR>Centered on truncated octahedron
|-
|}

=== Full snub tesseract ===
[[File:Snub tesseract verf.png|80px|thumb|Vertex figure for the '''full snub tesseract''']]
The '''full snub tesseract''' or '''full snub 16-cell''', defined as an [[Alternation (geometry)|alternation]] of the omnitruncated tesseract, can not be made uniform, but it can be given Coxeter diagram {{CDD|node_h|4|node_h|3|node_h|3|node_h}}, and [[Coxeter notation|symmetry]] [4,3,3]<sup>+</sup>, and constructed from 16 [[icosahedron]]s, 32 [[octahedra]], 24 [[square antiprism]]s, 8 [[snub cube]]s and 192 [[tetrahedron]]s filling the gaps at the deleted vertices. It has 272 cells, 944 faces, 864 edges, and 192 vertices.<ref>http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm</ref>

== Related uniform polytopes ==
{{Tesseract family}}

== Notes==
{{reflist}}

== References ==
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: 
** Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.&nbsp;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.&nbsp;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 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (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 Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp.&nbsp;409: Hemicubes: 1<sub>n1</sub>)
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* {{PolyCell | urlname = section2.html| title = 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 15, 19, 20, and 21}}
* http://www.polytope.de/nr17.html
* {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} x3o3o4x - sidpith, x3o3x4x - proh, x3x3o4x - prit

{{Polytopes}}

[[Category:Four-dimensional geometry]]
[[Category:Polychora]]