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| |+ Four cantellations
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| |- align=center valign=top
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| |[[File:4-cube t0.svg|150px]]<BR>[[tesseract]]<BR>{{CDD|node_1|4|node|3|node|3|node}}
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| |[[File:4-cube t02.svg|150px]]<BR>Cantellated tesseract<BR>{{CDD|node_1|4|node|3|node_1|3|node}}
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| |[[File:4-cube t13.svg|150px]]<BR>Cantellated 16-cell<BR>([[Rectified 24-cell]])<BR>{{CDD|node|4|node_1|3|node|3|node_1}}
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| |- align=center valign=top
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| |[[File:4-cube t3.svg|150px]]<BR>[[16-cell]]<BR>{{CDD|node|4|node|3|node|3|node_1}}
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| |[[File:4-cube t012.svg|150px]]<BR>Cantitruncated tesseract<BR>{{CDD|node_1|4|node_1|3|node_1|3|node}}
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| |[[File:4-cube t123.svg|150px]]<BR>Cantitruncated 16-cell<BR>([[Truncated 24-cell]])<BR>{{CDD|node|4|node_1|3|node_1|3|node_1}}
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| |-
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| !colspan=3|[[Orthogonal projection]]s in A<sub>4</sub> [[Coxeter plane]]
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| |}
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| In four-dimensional [[geometry]], a '''cantellated tesseract''' is a convex [[uniform polychoron]], being a [[cantellation]] (a 2nd order truncation) of the regular [[tesseract]].
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| There are four degrees of cantellations of the tesseract including with permutations truncations. Two are also derived from the 24-cell family.
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| {{TOC left}}
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| {{-}}
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| ==Cantellated tesseract==
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| {| class="wikitable" align="right" style="margin-left:10px" width="280"
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| |-
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| !bgcolor=#e7dcc3 colspan=3|Cantellated tesseract
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| |-
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| |bgcolor=#ffffff align=center colspan=3|[[Image:Schlegel half-solid cantellated 8-cell.png|280px]]<BR>[[Schlegel diagram]]<BR>Centered on rhombicuboctahedron<BR>octahedral cells shown
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| |-
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| |bgcolor=#e7dcc3|Type
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| |colspan=2|[[Uniform polychoron]]
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| |-
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]
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| |colspan=2|rr{4,3,3}
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
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| |colspan=2|{{CDD|node_1|4|node|3|node_1|3|node}}
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| |-
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| |bgcolor=#e7dcc3|Cells
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| |56
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| |8 [[Small rhombicuboctahedron|''3.4.4.4'']] [[Image:Small rhombicuboctahedron.png|20px]]<BR>16 [[Octahedron|''3.3.3.3'']] [[Image:Octahedron.png|20px]]<br>32 [[Triangular prism|''3.4.4'']] [[Image:Triangular prism.png|20px]]
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| |-
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| |bgcolor=#e7dcc3|Faces
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| |248
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| |128 [[triangle|{3}]]<BR>120 [[square (geometry)|{4}]]
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| |-
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| |bgcolor=#e7dcc3|Edges
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| |colspan=2|288
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| |-
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| |bgcolor=#e7dcc3|Vertices
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| |colspan=2|96
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| |-
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| |bgcolor=#e7dcc3|[[Vertex figure]]
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| |colspan=2|[[Image:Cantellated 8-cell verf.png|80px]]<BR>Square wedge
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
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| |colspan=2|BC<sub>4</sub>, [3,3,4], order 384
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| |-
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| |bgcolor=#e7dcc3|Properties
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| |colspan=2|[[Convex polytope|convex]]
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| |-
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| |bgcolor=#e7dcc3|Uniform index
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| |colspan=2|''[[Truncated tesseract|13]]'' 14 ''[[Runcinated tesseract|15]]''
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| |}
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| The '''cantellated tesseract''' (or '''bicantellated 16-cell''') is a convex [[uniform polychoron]] or 4-dimensional [[polytope]] bounded by 56 [[cell (mathematics)|cells]]: 8 [[small rhombicuboctahedron|small rhombicuboctahedra]], 16 [[octahedron|octahedra]], and 32 [[triangular prism]]s.
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| ===Construction===
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| In the process of [[Cantellation (geometry)|cantellation]], a polytope's 2-faces are effectively shrunk. The [[rhombicuboctahedron]] can be called a cantellated cube, since if its six faces are shrunk in their respective planes, each vertex will separate into the three vertices of the rhombicuboctahedron's triangles, and each edge will separate into two of the opposite edges of the rhombicuboctahedrons twelve non-axial squares.
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| When the same process is applied to the tesseract, each of the eight cubes becomes a rhombicuboctahedron in the described way. In addition however, since each cube's edge was previously shared with two other cubes, the separating edges form the three parallel edges of a triangular prism—32 triangular prisms, since there were 32 edges. Further, since each vertex was previously shared with three other cubes, the vertex would split into 12 rather than three new vertices. However, since some of the shrunken faces continues to be shared, certain pairs of these 12 potential vertices are identical to each other, and therefore only 6 new vertices are created from each original vertex (hence the cantellated tesseract's 96 vertices compared to the tesseract's 16). These six new vertices form the vertices of an octahedron—16 octahedra, since the tesseract had 16 vertices.
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| ===Cartesian coordinates===
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| The [[Cartesian coordinate]]s of the vertices of a cantellated tesseract with edge length 2 is given by all permutations of:
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| :<math>\left(\pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right)</math>
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| ===Structure===
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| The 8 small rhombicuboctahedral cells are joined to each other via their axial square faces. Their non-axial square faces, which correspond with the edges of a cube, are connected to the triangular prisms. The triangular faces of the small rhombicuboctahedra and the triangular prisms are connected to the 16 octahedra.
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| Its structure can be imagined by means of the tesseract itself: the rhombicuboctahedra are analogous to the tesseract's cells, the triangular prisms are analogous to the tesseract's edges, and the octahedra are analogous to the tesseract's vertices.
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| === Images ===
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| {{4-cube Coxeter plane graphs|t02|150}}
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| {| class="wikitable"
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| |[[Image:Cantellated tesseract1.png|200px]]<BR>Wireframe
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| |[[Image:Cantellated tesseract2.png|200px]]<BR>16 [[octahedron|octahedra]] shown.
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| |[[Image:Cantellated tesseract3.png|200px]]<BR>32 [[triangular prism]]s shown.
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| |}
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| ===Projections===
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| The following is the layout of the cantellated tesseract's cells under the parallel projection into 3-dimensional space, small rhombicuboctahedron first:
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| * The projection envelope is a [[truncated cube]].
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| * The nearest and farthest small rhombicuboctahedral cells from the 4D viewpoint project to the volume of the same shape inscribed in the projection envelope.
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| * The axial squares of this central small rhombicuboctahedron touches the centers of the 6 octagons of the envelope. The octagons are the image of the other 6 small rhombicuboctahedral cells.
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| * The 12 wedge-shaped volumes connecting the non-axial square faces of the central small rhombicuboctahedron to the neighbouring octagons are the images of 24 of the triangular prisms.
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| * The remaining 8 triangular prisms project onto the triangular faces of the envelope.
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| * Between the triangular faces of the envelope and the triangular faces of the central small rhombicuboctahedron are 8 octahedral volumes, which are the images of the 16 octahedral cells.
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| This layout of cells in projection is analogous to the layout of faces in the projection of the [[truncated cube]] into 2 dimensions. Hence, the cantellated tesseract may be thought of as an analogue of the truncated cube in 4 dimensions. (It is not the only possible analogue; another close candidate is the [[truncated tesseract]].)
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| Another uniform polychoron with a similar layout of cells is the [[runcitruncated 16-cell]].
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| ==Cantitruncated tesseract==
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| {| class="wikitable" align="right" style="margin-left:10px" width="280"
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| |-
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| |bgcolor=#e7dcc3 align=center colspan=3|'''Cantitruncated tesseract'''
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| |colspan=3 align=center|[[Image:Cantitruncated tesseract stella4d.png|280px]]<BR>[[Schlegel diagram]] centered on [[truncated cuboctahedron]] cell with [[octagon]]al faces hidden.
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| |-
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| |bgcolor=#e7dcc3|Type
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| |colspan=2|[[Uniform polychoron]]
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| |-
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]
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| |colspan=2|tr{4,3,3}
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s
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| |colspan=2|{{CDD|node_1|4|node_1|3|node_1|3|node}}
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| |-
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| |bgcolor=#e7dcc3|Cells
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| |56
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| |8 [[truncated cuboctahedron|''4.6.8'']] [[Image:Great rhombicuboctahedron.png|20px]]<BR>16 [[truncated tetrahedron|''3.6.6'']] [[Image:Truncated tetrahedron.png|20px]]<BR>32 [[triangular prism|''3.4.4'']] [[Image:Triangular prism.png|20px]]
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| |-
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| |bgcolor=#e7dcc3|Faces
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| |248
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| |64 [[triangle|{3}]], 96 [[square (geometry)|{4}]]<BR>64 [[hexagon|{6}]], 24 [[octagon|{8}]]
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| |-
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| |bgcolor=#e7dcc3|Edges
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| |colspan=2|384
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| |-
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| |bgcolor=#e7dcc3|Vertices
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| |colspan=2|192
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| |-
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| |bgcolor=#e7dcc3|[[Vertex figure]]
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| |colspan=2|[[Image:Cantitruncated 8-cell verf.png|80px]]<BR>[[Sphenoid (geometry)|Sphenoid]]
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
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| |colspan=2|BC<sub>4</sub>, [3,3,4], order 384
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| |-
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| |bgcolor=#e7dcc3|Properties
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| |colspan=2|[[Convex polytope|convex]]
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| |-
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| |bgcolor=#e7dcc3|Uniform index
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| |colspan=2|''[[Truncated 16-cell|17]]'' 18 ''[[Runcitruncated tesseract|19]]''
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| |}
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| In [[geometry]], the '''cantitruncated tesseract''' is a [[uniform polychoron]] (or uniform 4-dimensional [[polytope]]) that is bounded by 56 [[cell (mathematics)|cells]]: 8 [[Truncated cuboctahedron|truncated cuboctahedra]], 16 [[truncated tetrahedron|truncated tetrahedra]], and 32 [[triangular prism]]s.
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| ===Construction===
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| The cantitruncated tesseract is constructed by the cantitruncation of the [[tesseract]].
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| Cantitruncation is often thought of as rectification followed by truncation. However, the result of this construction would be a polytope which, while its structure would be very similar to that given by cantitruncation, not all of its faces would be uniform.
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| Alternatively, a ''uniform'' cantitruncated tesseract may be constructed by placing 8 uniform [[Truncated cuboctahedron|truncated cuboctahedra]] in the hyperplanes of a tesseract's cells, displaced along the coordinate axes such that their octagonal faces coincide. For an edge length of 2, this construction gives the [[Cartesian coordinate]]s of its vertices as all permutations of:
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| :<math>\left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2}),\ \pm(1+2\sqrt{2})\right)</math>
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| ===Structure===
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| The 8 truncated cuboctahedra are joined to each other via their octagonal faces, in an arrangement corresponding to the 8 cubical cells of the tesseract. They are joined to the 16 truncated tetrahedra via their hexagonal faces, and their square faces are joined to the square faces of the 32 triangular prisms. The triangular faces of the triangular prisms are joined to the truncated tetrahedra.
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| The truncated tetrahedra correspond with the tesseract's vertices, and the triangular prisms correspond with the tesseract's edges.
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| === Images===
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| {{4-cube Coxeter plane graphs|t012|150}}
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| {| class=wikitable width=320
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| |- align=center valign=top
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| |[[Image:Cantitruncated tesseract.png|320px]]<BR>A [[stereographic projection]] of the cantellated tesseract, as a tiling on a [[3-sphere]], with its 64 blue triangles, 96 green squares and 64 red hexagonal faces (the octagonal faces are not drawn).
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| |}
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| ===Projections===
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| In the truncated cuboctahedron first parallel projection into 3 dimensions, the cells of the cantitruncated tesseract are laid out as follows:
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| * The projection envelope is a non-uniform [[truncated cube]], with longer edges between octagons and shorter edges in the 8 triangles.
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| * The irregular octagonal faces of the envelope correspond with the images of 6 of the 8 truncated cuboctahedral cells.
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| * The other two truncated cuboctahedral cells project to a truncated cuboctahedron inscribed in the projection envelope. The octagonal faces touch the irregular octagons of the envelope.
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| * In the spaces corresponding to a cube's edges lie 12 volumes in the shape of irregular triangular prisms. These are the images, one per pair, of 24 of the triangular prism cells.
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| * The remaining 8 triangular prisms project onto the triangular faces of the projection envelope.
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| * The remaining 8 spaces, corresponding to a cube's corners, are the images of the 16 truncated tetrahedra, a pair to each space.
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| This layout of cells in projection is similar to that of the cantellated tesseract.
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| === Alternative names ===
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| *Cantitruncated tesseract ([[Norman Johnson (mathematician)|Norman W. Johnson]])
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| *Cantitruncated 4-cube
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| *Cantitruncated 8-cell
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| *Cantitruncated octachoron
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| *Great prismatotesseractihexadecachoron ([[George Olshevsky]])
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| *Grit (Jonathan Bowers: for great rhombated tesseract)
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| *012-ambo tesseract ([[John Horton Conway|John Conway]])
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| == Related uniform polytopes ==
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| {{Tesseract family}}
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| == References==
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| * [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
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| * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
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| ** Coxeter, ''[[Regular Polytopes (book)|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)
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| ** 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)
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| ** '''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]
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| *** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
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| *** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
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| *** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
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| * [[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. 409: Hemicubes: 1<sub>n1</sub>)
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| * [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
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| ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
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| * {{PolyCell | urlname = section2.html| title = 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 14, 18}}
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| * {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} o3x3o4x - srit, o3x3x4x - grit
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| * [http://www.software3d.com/Grit.php Paper model of cantitruncated tesseract] created using nets generated by [[Stella (software)|Stella4D]] software
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| {{Polytopes}}
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| [[Category:Four-dimensional geometry]]
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| [[Category:Polychora]]
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