WikiMili The Free Encyclopedia

Rectified tesseract | ||
---|---|---|

Schlegel diagram Centered on cuboctahedron tetrahedral cells shown | ||

Type | Uniform 4-polytope | |

Schläfli symbol | r{4,3,3} = 2r{3,3 ^{1,1}}h _{3}{4,3,3} | |

Coxeter-Dynkin diagrams | ||

Cells | 24 | 8 (3.4.3.4) 16 ( 3.3.3) |

Faces | 88 | 64 {3} 24 {4} |

Edges | 96 | |

Vertices | 32 | |

Vertex figure | (Elongated equilateral-triangular prism) | |

Symmetry group | B_{4} [3,3,4], order 384D _{4} [3^{1,1,1}], order 192 | |

Properties | convex, edge-transitive | |

Uniform index | 10 11 12 |

In geometry, the **rectified tesseract**, **rectified 8-cell** is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It has half the vertices of a runcinated tesseract, with its **runcic tesseract**.

**Geometry** is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

In geometry, a **uniform 4-polytope** is a 4-polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In elementary geometry, a **polytope** is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions *n* as an *n*-dimensional polytope or ** n-polytope**. Flat sides mean that the sides of a (

- Construction
- Images
- Projections
- Alternative names
- Related uniform polytopes
- Runcic cubic polytopes
- Tesseract polytopes
- References

It has two uniform constructions, as a *rectified 8-cell* r{4,3,3} and a cantellated demitesseract, rr{3,3^{1,1}}, the second alternating with two types of tetrahedral cells.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC_{8}.

**Emanuel Lodewijk Elte** was a Dutch mathematician. He is noted for discovering and classifying semiregular polytopes in dimensions four and higher.

The rectified tesseract may be constructed from the tesseract by truncating its vertices at the midpoints of its edges.

In geometry, the **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 eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

In Euclidean geometry, **rectification** or **complete-truncation** is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

The Cartesian coordinates of the vertices of the rectified tesseract with edge length 2 is given by all permutations of:

Coxeter plane | B_{4} | B_{3} / D_{4} / A_{2} | B_{2} / D_{3} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [8] | [6] | [4] |

Coxeter plane | F_{4} | A_{3} | |

Graph | |||

Dihedral symmetry | [12/3] | [4] |

Wireframe | 16 tetrahedral cells |

In the cuboctahedron-first parallel projection of the rectified tesseract into 3-dimensional space, the image has the following layout:

- The projection envelope is a cube.
- A cuboctahedron is inscribed in this cube, with its vertices lying at the midpoint of the cube's edges. The cuboctahedron is the image of two of the cuboctahedral cells.
- The remaining 6 cuboctahedral cells are projected to the square faces of the cube.
- The 8 tetrahedral volumes lying at the triangular faces of the central cuboctahedron are the images of the 16 tetrahedral cells, two cells to each image.

In geometry, a **cube** is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

- Rit (Jonathan Bowers: for rectified tesseract)
- Ambotesseract (Neil Sloane & John Horton Conway)
- Rectified tesseract/Runcic tesseract (Norman W. Johnson)
- Runcic 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope
- Rectified 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope

Runcic n-cubes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

n | 4 | 5 | 6 | 7 | 8 | ||||||

[1^{+},4,3^{n-2}]= [3,3 ^{n-3,1}] | [1^{+},4,3^{2}]= [3,3 ^{1,1}] | [1^{+},4,3^{3}]= [3,3 ^{2,1}] | [1^{+},4,3^{4}]= [3,3 ^{3,1}] | [1^{+},4,3^{5}]= [3,3 ^{4,1}] | [1^{+},4,3^{6}]= [3,3 ^{5,1}] | ||||||

Runcic figure | |||||||||||

Coxeter | = | = | = | = | = | ||||||

Schläfli | h_{3}{4,3^{2}} | h_{3}{4,3^{3}} | h_{3}{4,3^{4}} | h_{3}{4,3^{5}} | h_{3}{4,3^{6}} |

B4 symmetry polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Name | tesseract | rectified tesseract | truncated tesseract | cantellated tesseract | runcinated tesseract | bitruncated tesseract | cantitruncated tesseract | runcitruncated tesseract | omnitruncated tesseract | ||

Coxeter diagram | = | = | |||||||||

Schläfli symbol | {4,3,3} | t_{1}{4,3,3}r{4,3,3} | t_{0,1}{4,3,3}t{4,3,3} | t_{0,2}{4,3,3}rr{4,3,3} | t_{0,3}{4,3,3} | t_{1,2}{4,3,3}2t{4,3,3} | t_{0,1,2}{4,3,3}tr{4,3,3} | t_{0,1,3}{4,3,3} | t_{0,1,2,3}{4,3,3} | ||

Schlegel diagram | |||||||||||

B_{4} | |||||||||||

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 | = | = | = | = | = | = | |||||

Schläfli symbol | {3,3,4} | t_{1}{3,3,4}r{3,3,4} | t_{0,1}{3,3,4}t{3,3,4} | t_{0,2}{3,3,4}rr{3,3,4} | t_{0,3}{3,3,4} | t_{1,2}{3,3,4}2t{3,3,4} | t_{0,1,2}{3,3,4}tr{3,3,4} | t_{0,1,3}{3,3,4} | t_{0,1,2,3}{3,3,4} | ||

Schlegel diagram | |||||||||||

B_{4} |

In geometry, a **cuboctahedron** is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is the only radially equilateral convex polyhedron.

In geometry, a **4-polytope** is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells.

In geometry, the **5-cell** is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a **C _{5}**,

In four-dimensional geometry, a **16-cell** is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called **C _{16}**,

In four-dimensional geometry, a **runcinated 5-cell** is a convex uniform 4-polytope, being a runcination of the regular 5-cell.

In four-dimensional geometry, a **runcinated tesseract** is a convex uniform 4-polytope, being a runcination of the regular tesseract.

In four-dimensional geometry, a **cantellated tesseract** is a convex uniform 4-polytope, being a cantellation of the regular tesseract.

The **cubic honeycomb** or **cubic cellulation** is the only proper regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a **cubille**.

The **order-5 cubic honeycomb** is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

In geometry, a **truncated tesseract** is a uniform 4-polytope formed as the truncation of the regular tesseract.

In geometry, the **rectified 24-cell** or **rectified icositetrachoron** is a uniform 4-dimensional polytope, which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by rectification of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra.

In four-dimensional geometry, a **cantellated 24-cell** is a convex uniform 4-polytope, being a cantellation of the regular 24-cell.

In five-dimensional geometry, a **5-cube** is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

In geometry, an **8-cube** is an eight-dimensional hypercube (8-cube). It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.

In geometry, a **9-cube** is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.

In geometry, a **10-cube** is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

In seven-dimensional geometry, a **rectified 7-orthoplex** is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.

In six-dimensional geometry, a **runcic 5-cube** or is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

- H.S.M. Coxeter:
- H.S.M. Coxeter,
*Regular Polytopes*, 3rd Edition, Dover New York, 1973 **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, ISBN 978-0-471-01003-6- (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]

- (Paper 22) H.S.M. Coxeter,

- H.S.M. Coxeter,
- Norman Johnson
*Uniform Polytopes*, Manuscript (1991)- N.W. Johnson:
*The Theory of Uniform Polytopes and Honeycombs*, Ph.D. (1966)

- N.W. Johnson:
- 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 11 , George Olshevsky.
- Klitzing, Richard. "4D uniform polytopes (polychora) o4x3o3o - rit".

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.