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 c ...

, a truncated 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 ...

formed as the

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathb ...

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 three truncations, including a

bitruncation In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves. Bitruncated regular pol ...

, and a tritruncation, which creates the ''truncated 16-cell''.

Truncated tesseract

The truncated tesseract is bounded by 24

cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...

: 8 truncated cubes, and 16

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

Alternate names

* Truncated tesseract ( Norman W. Johnson) * Truncated tesseract (Acronym tat) (George Olshevsky, and Jonathan Bowers)

Construction

The truncated tesseract may be constructed by truncating the vertices of the

1/(\sqrt+2)

of the edge length. A regular tetrahedron is formed at each truncated vertex. 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 truncated tesseract having edge length 2 is given by all permutations of: :

\left(\pm1,\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+\sqrt)\right)

Projections

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

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

. * Two of the truncated cube cells project onto a truncated cube inscribed in the cubical envelope. * The other 6 truncated cubes project onto the square faces of the envelope. * The 8 tetrahedral volumes between the envelope and the triangular faces of the central truncated cube are the images of the 16 tetrahedra, a pair of cells to each image.

Images

Related polytopes

The '' truncated

'', is third in a sequence of truncated

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...

Bitruncated tesseract

The bitruncated tesseract, bitruncated 16-cell, or tesseractihexadecachoron is constructed by a

operation applied to the

. It can also be called a runcicantic tesseract with half the vertices of a runcicantellated tesseract with a construction.

Alternate names

* Bitruncated tesseract/Runcicantic tesseract ( Norman W. Johnson) * Tesseractihexadecachoron (Acronym tah) (George Olshevsky, and Jonathan Bowers)

Construction

A tesseract is bitruncated by truncating its

beyond their midpoints, turning the eight

s into eight truncated octahedra. These still share their square faces, but the hexagonal faces form truncated tetrahedra which share their triangular faces with each other. The

s of the vertices of a bitruncated tesseract having edge length 2 is given by all permutations of: :

\left(0,\ \pm\sqrt,\ \pm2\sqrt,\ \pm2\sqrt\right)

Structure

The truncated octahedra are connected to each other via their square faces, and to the truncated tetrahedra via their hexagonal faces. The truncated tetrahedra are connected to each other via their triangular faces.

Projections

Stereographic projections

The truncated-octahedron-first projection of the bitruncated tesseract into 3D space has a truncated cubical envelope. Two of the truncated octahedral cells project onto a truncated octahedron inscribed in this envelope, with the square faces touching the centers of the octahedral faces. The 6 octahedral faces are the images of the remaining 6 truncated octahedral cells. The remaining gap between the inscribed truncated octahedron and the envelope are filled by 8 flattened truncated tetrahedra, each of which is the image of a pair of truncated tetrahedral cells.

Related polytopes

The ''

bitruncated In geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves. Bitruncated regular polyt ...

'' is second in a sequence of bitruncated

Truncated 16-cell

The truncated 16-cell, truncated hexadecachoron, cantic tesseract which is bounded by 24

: 8 regular octahedra, and 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 ...

. It has half the vertices of a

cantellated tesseract In four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular tesseract. There are four degrees of cantellations of the tesseract including with permutations t ...

with construction . It is related to, but not to be confused with, the

24-cell In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, o ...

, which is a

regular 4-polytope In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star reg ...

bounded by 24 regular octahedra.

Alternate names

* Truncated 16-cell/Cantic tesseract ( Norman W. Johnson) * Truncated hexadecachoron (Acronym thex) (George Olshevsky, and Jonathan Bowers)Klitzing, (x3x3o4o - thex)

Construction

The truncated 16-cell may be constructed from the

16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the ...

by truncating its vertices at 1/3 of the edge length. This results in the 16 truncated tetrahedral cells, and introduces the 8 octahedra (vertex figures). (Truncating a 16-cell at 1/2 of the edge length results in the

, which has a greater degree of symmetry because the truncated cells become identical with the vertex figures.) The

s of the vertices of a truncated 16-cell having edge length √2 are given by all permutations, and sign combinations of : (0,0,1,2) An alternate construction begins with a

demitesseract In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mi ...

with vertex coordinates (±3,±3,±3,±3), having an even number of each sign, and truncates it to obtain the permutations of : (1,1,3,3), with an even number of each sign.

Structure

The truncated tetrahedra are joined to each other via their hexagonal faces. The octahedra are joined to the truncated tetrahedra via their triangular faces.

Projections

Centered on octahedron

The octahedron-first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure: * The projection envelope is a

truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 ...

. * The 6 square faces of the envelope are the images of 6 of the octahedral cells. * An octahedron lies at the center of the envelope, joined to the center of the 6 square faces by 6 edges. This is the image of the other 2 octahedral cells. * The remaining space between the envelope and the central octahedron is filled by 8 truncated tetrahedra (distorted by projection). These are the images of the 16 truncated tetrahedral cells, a pair of cells to each image. This layout of cells in projection is analogous to the layout of faces in the projection of the

into 2-dimensional space. Hence, the truncated 16-cell may be thought of as the 4-dimensional analogue of the truncated octahedron.

Centered on truncated tetrahedron

The truncated tetrahedron first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure: * The projection envelope is a truncated cube. * The nearest truncated tetrahedron to the 4D viewpoint projects to the center of the envelope, with its triangular faces joined to 4 octahedral volumes that connect it to 4 of the triangular faces of the envelope. * The remaining space in the envelope is filled by 4 other truncated tetrahedra. * These volumes are the images of the cells lying on the near side of the truncated 16-cell; the other cells project onto the same layout except in the dual configuration. * The six octagonal faces of the projection envelope are the images of the remaining 6 truncated tetrahedral cells.

Images

Related polytopes

A truncated 16-cell, as a cantic 4-cube, is related to the dimensional family of cantic n-cubes:

Related uniform polytopes

Related uniform polytopes in demitesseract symmetry

Related uniform polytopes in tesseract symmetry

Notes

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 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 o ...

, 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) * * o3o3o4o - tat, o3x3x4o - tah, x3x3o4o - thex

External links

Paper model of truncated tesseract
created using nets generated by Stella4D software {{Polytopes 4-polytopes