In four-dimensional
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a cantellated tesseract is a convex
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. Th ...
, being a
cantellation
In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...
(a 2nd order truncation) of the regular
tesseract
In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of e ...
.
There are four degrees of cantellations of the tesseract including with permutations truncations. Two are also derived from the 24-cell family.
Cantellated tesseract
The cantellated tesseract, bicantellated 16-cell, or small rhombated tesseract is a convex
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. Th ...
or 4-dimensional
polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
bounded by 56
cells: 8
small rhombicuboctahedra, 16
octahedra
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...
, and 32
triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A ...
s.
Construction
In the process of
cantellation
In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...
, a polytope's 2-faces are effectively shrunk. The
rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at ea ...
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.
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.
Cartesian coordinates
The
Cartesian coordinate
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
s of the vertices of a cantellated tesseract with edge length 2 is given by all permutations of:
:
Structure
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.
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.
Images
Projections
The following is the layout of the cantellated tesseract's cells under the parallel projection into 3-dimensional space, small rhombicuboctahedron first:
* The projection envelope is a
truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.
If the truncated cube has unit edge length, its dual triakis octahedron has edg ...
.
* The nearest and farthest small rhombicuboctahedral cells from the 4D viewpoint project to the volume of the same shape inscribed in the projection envelope.
* 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.
* 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.
* The remaining 8 triangular prisms project onto the triangular faces of the envelope.
* 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.
This layout of cells in projection is analogous to the layout of faces in the projection of the
truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.
If the truncated cube has unit edge length, its dual triakis octahedron has edg ...
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
In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract.
There are three truncations, including a bitruncation, and a tritruncation, which creates the ''truncated 16-cell''.
Truncated tesserac ...
.)
Another uniform 4-polytope with a similar layout of cells is the
runcitruncated 16-cell.
Cantitruncated tesseract
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the cantitruncated tesseract or great rhombated tesseract is a
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. Th ...
(or uniform 4-dimensional
polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
) that is bounded by 56
cells: 8
truncated cuboctahedra, 16
truncated tetrahedra
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron ...
, and 32
triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A ...
s.
Construction
The cantitruncated tesseract is constructed by the cantitruncation of the
tesseract
In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of e ...
.
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.
Alternatively, a ''uniform'' cantitruncated tesseract may be constructed by placing 8 uniform
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
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
s of its vertices as all permutations of:
:
Structure
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.
The truncated tetrahedra correspond with the tesseract's vertices, and the triangular prisms correspond with the tesseract's edges.
Images
Projections
In the truncated cuboctahedron first parallel projection into 3 dimensions, the cells of the cantitruncated tesseract are laid out as follows:
* The projection envelope is a non-uniform
truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.
If the truncated cube has unit edge length, its dual triakis octahedron has edg ...
, with longer edges between octagons and shorter edges in the 8 triangles.
* The irregular octagonal faces of the envelope correspond with the images of 6 of the 8 truncated cuboctahedral cells.
* 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.
* 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.
* The remaining 8 triangular prisms project onto the triangular faces of the projection envelope.
* The remaining 8 spaces, corresponding to a cube's corners, are the images of the 16 truncated tetrahedra, a pair to each space.
This layout of cells in projection is similar to that of the cantellated tesseract.
Alternative names
*Cantitruncated tesseract (
Norman W. Johnson)
*Cantitruncated 4-cube
*Cantitruncated 8-cell
*Cantitruncated octachoron
*Great (George Olshevsky)
*Grit (Jonathan Bowers: for great rhombated tesseract)
*012-ambo tesseract (
John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
)
Related uniform polytopes
It is second in a series of cantitruncated hypercubes:
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
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
'', (3rd edition, 1973), Dover edition, , 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, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'',
ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'',
ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45*
John H. Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 26. pp. 409: Hemicubes: 1
n1)
*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
*
* o3x3o4x - srit, o3x3x4x - grit
Paper model of cantitruncated tesseractcreated using nets generated by
Stella4D software
{{Polytopes
4-polytopes
Truncated tilings