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eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube. There are unique 8 degrees of rectifications
Rectified_8-cubes
birectified 5-cube are located in the square face centers of the 5-cube. Rectified penteract (acronym: rin) (Jonathan Bowers) The rectified 5-cube may be constructed
Rectified_5-cubes
centers of the 7-cube. rectified hepteract (acronym: rasa) (Jonathan Bowers) Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin
Rectified_7-cubes
Geometrical Shape
birectified 6-cube are located in the square face centers of the 6-cube. Rectified hexeract (acronym: rax) (Jonathan Bowers) The rectified 6-cube may be constructed
Rectified_6-cubes
geometry, a rectified 9-cube is a convex uniform 9-polytope, being a rectification of the regular 9-cube. There are 9 rectifications of the 9-cube. The zeroth
Rectified_9-cubes
geometry, a rectified 10-cube is a convex uniform 10-polytope, being a rectification of the regular 10-cube. There are 10 rectifications of the 10-cube, with
Rectified_10-cubes
Rectified octacross Rectified diacosipentacontahexazetton (acronym: rek) (Jonathan Bowers) There are two Coxeter groups associated with the rectified
Rectified_8-orthoplexes
rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes,
Rectified_24-cell
Convex polyhedron with 38 faces
In geometry, the rectified truncated cube is a polyhedron, constructed as a rectified, truncated cube. It has 38 faces: 8 equilateral triangles, 24 isosceles
Rectified_truncated_cube
5-demicube 5-cube, Rectified 5-cube, 5-cube, Truncated 5-cube, Cantellated 5-cube, Runcinated 5-cube, Stericated 5-cube 5-orthoplex, Rectified 5-orthoplex
List_of_mathematical_shapes
Polytope contained by 7-polytope facets
convex regular 8-polytopes: {3,3,3,3,3,3,3} - 8-simplex {4,3,3,3,3,3,3} - 8-cube {3,3,3,3,3,3,4} - 8-orthoplex There are no nonconvex regular 8-polytopes.
Uniform_8-polytope
Natural number
regular compounds. The cuboctahedron, on the other hand, is a rectified cube or rectified octahedron, and one of only two convex quasiregular polyhedra
8
6-demicube 6-cube, Rectified 6-cube, Truncated 6-cube, Cantellated 6-cube, Runcinated 6-cube, Stericated 6-cube, Pentellated 6-cube 6-orthoplex, Rectified 6-orthoplex
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex. There are unique 3 degrees
Rectified_8-simplexes
Polyhedron with 8 triangles and 6 squares
A cuboctahedron, rectified cube, or rectified octahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices
Cuboctahedron
Polyhedron with eight triangular faces
vertices at which four sides meet, and twelve edges. Its dual polyhedron is a cube. It can be formed as the convex hull of the six axis-parallel unit vectors
Octahedron
Uniform 8-polytope
eight-dimensional geometry, a cantic 8-cube or truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube. Truncated demiocteract
Cantic_8-cube
Polytope
makes the dual of rectified 5-orthoplex. The intersection of the 5-cube and 5-orthoplex compound is the uniform birectified 5-cube: = ∩ . The compound
Compound of 5-cube and 5-orthoplex
Compound_of_5-cube_and_5-orthoplex
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
Rectified_tesseract
Type of tesseract
truncated 16-cell. The truncated tesseract is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra. Truncated tesseract (Norman W. Johnson) Truncated
Truncated_tesseract
Operation in Euclidean geometry
rectified form. New vertices are placed at the center of the edges of the original polygon. Each platonic solid and its dual have the same rectified polyhedron
Rectification_(geometry)
Only regular space-filling tessellation of the cube
honeycomb) 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
Cubic_honeycomb
Near-miss Johnson solid with 92 faces
polyhedra: Near-miss Johnson solid Rectified truncated tetrahedron Rectified truncated octahedron Rectified truncated cube Rectified truncated dodecahedron "PolyHédronisme"
Rectified truncated icosahedron
Rectified_truncated_icosahedron
as a first rectification of a 5-dimensional cross polytope. Rectified pentacross Rectified triacontaditeron (32-faceted 5-polytope) Acronym: rat (Jonathan
Rectified_5-orthoplexes
sequence of honeycombs with square tiling cells: The rectified square tiling honeycomb, t1{4,4,3}, has cube and square tiling facets, with a triangular prism
Square_tiling_honeycomb
Polytope in 8-dimensional geometry
orthoplexes. The rectified 421 can be seen as a rectification of the 421 polytope, creating new vertices on the center of edges of the 421. Rectified
4_21_polytope
semiregular polytope, labeling it as S1 5. Rectified hexateron (Acronym: rix) (Jonathan Bowers) The vertices of the rectified 5-simplex can be more simply positioned
Rectified_5-simplexes
the 6-orthoplex. The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb. or Rectified hexacross Rectified hexacontatetrapeton
Rectified_6-orthoplexes
disprismatotesseractihexadecachoron has 16 tetrahedra, 32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron.
Runcinated_tesseracts
Uniform 6-polytope
(dual of E6 lattice). Birectified 221 polytope Rectified pentacontatetrapeton (Acronym: ram) - rectified 54-facetted polypeton (Jonathan Bowers) Vertices
1_22_polytope
Regular tiling of hyperbolic 3-space
of a sequence of polychora and honeycombs with dodecahedral cells: The rectified order-4 dodecahedral honeycomb, , has alternating octahedron and icosidodecahedron
Order-4 dodecahedral honeycomb
Order-4_dodecahedral_honeycomb
constructed from this honeycomb. or rectified heptacross rectified hecatonicosaoctaexon (Acronym: rez) (Jonathan Bowers) - rectified 128-faceted polyexon There
Rectified_7-orthoplexes
Tiling of hyperbolic 3-space by uniform polyhedra
figures, made from 8 cubes and 12 p-gonal prisms at a vertex for any integer p. In the case p = 4 (a regular icosahedron), all cells are cubes and the result
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
Group of polytopes
alternation with half symmetry). There are two regular forms, the 8-orthoplex and 8-cube, with 16 and 256 vertices respectively. They can be visualized as
B8_polytope
Uniform 6-dimensional polytope
uniform polypeta are regular polypeta: the 6-simplex {3,3,3,3,3}, the 6-cube (hexeract) {4,3,3,3,3}, and the 6-orthoplex (hexacross) {3,3,3,3,4}. Regular
Uniform_6-polytope
Regular tiling of hyperbolic 3-space
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
Order-5_cubic_honeycomb
Uniform 8 dimensional polytope
Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube: (±1,±1,±1,±1,±1,±1,±1,±1) with an odd number
8-demicube
generated with 2 or more rings of the Coxeter group : , , , , . The rectified cubic-octahedral honeycomb is a compact uniform honeycomb, constructed
Cubic-octahedral_honeycomb
of tetrahedron and icosahedron cells. (The other two are the rectified 5-cell and rectified 600-cell.) Snub icositetrachoron Snub demitesseract Semi-snub
Snub_24-cell
Uniform Polytope
group orders. The rectified 231 is a rectification of the 231 polytope, creating new vertices on the center of edge of the 231. Rectified
2_31_polytope
Convex polyhedron with 38 faces
In geometry, the rectified truncated octahedron is a convex polyhedron, constructed as a rectified, truncated octahedron. It has 38 faces: 24 isosceles
Rectified truncated octahedron
Rectified_truncated_octahedron
Class of 4-dimensional polytopes
Space of n Dimensions. In four dimensions, this gives the rectified 5-cell, the rectified 600-cell, and the snub 24-cell. 1910: Alicia Boole Stott, in
Uniform_4-polytope
Convex uniform 4-polytope
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
Cantellated_tesseract
Group of irregular uniform polytopes
and all rectified (single-ring) Coxeter–Dynkin diagrams. The family of n-simplices contain Gosset–Elte figures of the form 0ij as all rectified forms of
Gosset–Elte_figures
Polytope constructed from alternation of a hypercube
the vertices of {4,3,...,3}. The vertex figures of demihypercubes are rectified n-simplexes. They are represented by Coxeter-Dynkin diagrams of three
Demihypercube
Convex uniform 7-polytope in seven-dimensional geometry
it as S1 7. Rectified octaexon (Acronym: roc) (Jonathan Bowers) The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations
Rectified_7-simplexes
is cell-transitive, consisting of 48 truncated cubes, and also edge-transitive, with 3 truncated cubes cells per edge and with one triangle and two octagons
Truncated_24-cells
Seven-dimensional geometric object
convex regular 7-polytopes: {3,3,3,3,3,3} - 7-simplex {4,3,3,3,3,3} - 7-cube {3,3,3,3,3,4} - 7-orthoplex There are no nonconvex regular 7-polytopes. The
Uniform_7-polytope
Convex polyhedron with 92 faces
In geometry, the rectified truncated dodecahedron is a convex polyhedron, constructed as a rectified, truncated dodecahedron. It has 92 faces: 20 equilateral
Rectified truncated dodecahedron
Rectified_truncated_dodecahedron
Geometric operation on a regular polytope
to regular tilings and honeycombs. Cantellating a polyhedron is also rectifying its rectification. Cantellation (for polyhedra and tilings) is also called
Cantellation_(geometry)
heptacomb Rectified pentacontahexa-hecatonicosihexa-exic heptacomb Acronym: lanquoh (Jonathan Bowers) 8-polytope 331 honeycomb Klitzing
1_33_honeycomb
Type of polyhedron
also be constructed as a rectified rhombicuboctahedron. Expanded rhombic dodecahedron Rectified rhombicuboctahedron Rectified small rhombicuboctahedron
Expanded_cuboctahedron
cube (), and octahedron (). The Hessian is analogous to the tetrahedron, like the cube is a double tetrahedron, and the octahedron as a rectified tetrahedron
Hessian_polyhedron
Convex polyhedron with 20 faces
In geometry, the rectified truncated tetrahedron is a polyhedron, constructed as a rectified, truncated tetrahedron. It has 20 faces: 4 equilateral triangles
Rectified truncated tetrahedron
Rectified_truncated_tetrahedron
Archimedean solid with 26 faces
constructed from a cube by drawing a smaller one in the middle of each face, parallel to the cube's edges. After removing the edges of a cube, the squares may
Rhombicuboctahedron
Type of geometrical object
with one or more rings. Twelve cases are shown below: ten single-ring (rectified) forms, and two truncations. Bowers-style acronym names are given in parentheses
Uniform_10-polytope
Uniform 6-polytope
in dimensional series 22k. The rectified 221 has 216 vertices, and 126 facets: 72 rectified 5-simplices, and 27 rectified 5-orthoplexes and 27 5-demicubes
2_21_polytope
In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells
Rectified_600-cell
eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex. There are four unique degrees
Truncated_8-simplexes
Uniform polytope
triangle, doubled into a prism: {3,3}×{3}×{}. Rectified pentacontahexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym: lanq) (Jonathan
1_32_polytope
heptellated 9-orthoplex. A second construction in 9-space, from the center of a rectified 9-orthoplex is given by coordinate permutations of: (1,-1,0,0,0,0,0,0
Heptellated_8-simplexes
Regular 5-polytope
called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead. Cartesian coordinates for the vertices of a
5-demicube
Uniform polychoron
(Thorold Gosset) Dispentachoron Rectified 5-cell (Norman W. Johnson) Rectified 4-simplex Fully truncated 4-simplex Rectified pentachoron (Acronym: rap) (Jonathan
Rectified_5-cell
Type of geometric object
Coxeter-Dynkin diagrams with one or more rings. Eleven cases are shown below: Nine rectified forms and 2 truncations. Bowers-style acronym names are given in parentheses
Uniform_9-polytope
Removal of alternate vertices
be snubbed if its truncation has only even-sided faces. All truncated rectified polyhedra can be snubbed, not just from regular polyhedra. The snub square
Alternation_(geometry)
Isogonal polyhedron with regular faces
which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate
Uniform_polyhedron
Type of polyhedron
all regular faces, and 4.6.8 vertex figure. The triangle and squares of the rhombicuboctahedron can be independently rectified or truncated, creating four
Truncated_rhombicuboctahedron
In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry. The three simplest forms are the 421, 241, and 142 polytopes, composed of
E8_polytope
Uniform 7-dimensional polytope
3-sphere tiling, a tetrahedral hosohedron.) Rectified hecatonicosihexa-pentacosiheptacontahexa-exon as a rectified 126-576 facetted polyexon (acronym: ranq)
3_21_polytope
Type of uniform space-filling tessellation
densest known sphere packing in 5 dimensions. The 40 vertices of the rectified 5-orthoplex vertex figure of the 5-demicubic honeycomb reflect the kissing
5-demicubic_honeycomb
First crewed space mission to orbit the Moon
to reignite in orbit. Without assurances that these problems had been rectified, NASA administrators could not justify risking a crewed mission until
Apollo_8
alternation of the regular 8-cubic honeycomb. It is composed of two different types of facets. The 8-cubes become alternated into 8-demicubes h{4,3,3,3,3,3
8-demicubic_honeycomb
the square faces. The 24-cell facets exist between these vertices as rectified 16-cells. If the coordinates of the tesseractic honeycomb are integers
24-cell_honeycomb
Four-dimensional shape
bipyramid exists as the cells of the dual of the uniform rectified 5-simplex, and rectified 5-cube or the dual of any uniform 5-polytope with a tetrahedral
Tetrahedral_bipyramid
Isogonal polytope with uniform facets
and so on. An extended Schläfli symbol can be used for representing rectified forms, with a single subscript: k-th rectification = tk{p1, p2, ..., pn−1}
Uniform_polytope
Aut(E6) [18/2] E6 [12] D5 [8] D4 / A2 [6] A5 [6] D3 / A3 [4] 1 221 Icosihepta-heptacontadipeton (jak) 2 Rectified 221 Rectified icosihepta-heptacontadipeton
E6_polytope
Tessellation of convex uniform polyhedron cells
group: [6,3[3]] or . 7 are half symmetry forms of [6,3,6]: ↔ . There are 8 forms, 1 unique, generated by ring permutations of the Coxeter group: . Two
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
Regular object in four dimensional geometry
polytopes constructed from the 24-cell. The 24-cell can also be derived as a rectified 16-cell: Octacube (sculpture) Uniform 4-polytope § The F4 family Coxeter
24-cell
5-dimensional geometric object
5-simplex honeycomb, . The 5-demicube honeycomb, , vertex figure is a rectified 5-orthoplex and facets are the 5-orthoplex and 5-demicube. Pyramidal 5-polytopes
5-polytope
In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube. A uniform 7-polytope
Cantic_7-cube
In 8-dimensional geometry, there are 135 uniform polytopes with A8 symmetry. There is one self-dual regular form, the 8-simplex with 9 vertices. Each can
A8_polytope
Concept in euclidean geometry
centered cubic (a checkerboard in which the red 4-cubes have a central vertex but the black 4-cubes do not). The tesseract can make a regular tessellation
Tesseractic_honeycomb
square tiling honeycomb, is the same thing as the rectified square tiling honeycomb, . It has cube and square tiling facets, with a triangular prism vertex
Order-4 square tiling honeycomb
Order-4_square_tiling_honeycomb
F4 [12] B4 [8] B3 [6] B2 [4] Octahedron centered Dual octahedron centered 1 24-cell (rectified 16-cell) = {3,4,3} = r{3,3,4} 2 rectified 24-cell (cantellated
List_of_F4_polytopes
Type of uniform 4-polytope in four-dimensional geography
prismatic prism - - 4 cubes and 4 cubes - (same as 4-4 duoprism and same as tesseract) Pentagonal prismatic prism - - 5 cubes and 4 pentagonal prisms
Prismatic_uniform_4-polytope
forms, the 5-orthoplex, and 5-cube with 10 and 32 vertices respectively. The 5-demicube is added as an alternation of the 5-cube. They can be visualized as
B5_polytope
Uniform 9-polytope
demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite
9-demicube
symmetry. There are two regular forms, the tesseract and 16-cell, with 16 and 8 vertices respectively. They can be visualized as symmetric orthographic projections
B4_polytope
9-polytopes with BC9 symmetry. The rectified 9-orthoplex is the vertex figure for the demienneractic honeycomb. or Rectified enneacross (Acronym: riv) (Jonathan
Rectified_9-orthoplexes
stericated 6-orthoplex. A second construction in 6-space, from the center of a rectified 6-orthoplex is given by coordinate permutations of: (1,-1,0,0,0,0) The
Stericated_5-simplexes
Coxeter-Dynkin diagram, shown as . Rectified heptapeton (Acronym: ril) (Jonathan Bowers) The vertices of the rectified 6-simplex can be most simply positioned
Rectified_6-simplexes
Archimedean solid with 32 faces
contributing two radii and an edge. The icosidodecahedron is a rectified dodecahedron and also a rectified icosahedron, existing as the full-edge truncation between
Icosidodecahedron
Geometric object
) The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell are included at the beginning for completeness. The demipenteract
Uniform_k_21_polytope
Geometric operation applied to a polyhedron
, r } {\displaystyle s{\begin{Bmatrix}p,q,r\end{Bmatrix}}} , and . A rectified polychoron { p q , r } {\displaystyle {\begin{Bmatrix}p\\q,r\end{Bmatrix}}}
Snub_(geometry)
Uniform 10-polytope
10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices removed. It is part of a dimensionally infinite
10-demicube
Cube capped by two square pyramids
antiprisms), formed by superimposing octahedra into the cuboctahedra of the rectified cubic honeycomb. Both honeycombs have a symmetry of [ 4 , 3 , 4 ] {\displaystyle
Elongated_square_bipyramid
6-dimensional geometric object
6-simplex honeycomb, . The 6-demicube honeycomb, , vertex figure is a rectified 6-orthoplex and facets are the 6-orthoplex and 6-demicube. The uniform
6-polytope
Uniform polytope in 8 dimensional geometry
also shown. The rectified 241 is a rectification of the 241 polytope, with vertices positioned at the mid-edges of the 241. Rectified
2_41_polytope
5-cell centers of the 10-simplex. The rectified 10-simplex is the vertex figure of the 11-demicube. Rectified hendecaxennon (acronym: ru) (Jonathan Bowers)
Rectified_10-simplexes
Alternate reality game created by Mind Candy
they did not fit on the print layout. While these problems have been rectified, Mind Candy has no plans to replace or reprint these cards in the future
Perplex_City
RECTIFIED 8-CUBES
RECTIFIED 8-CUBES
Surname or Lastname
English, Scottish, French, German, Spanish, Portuguese, and Jewish
English, Scottish, French, German, Spanish, Portuguese, and Jewish : from the Hebrew personal name Gavriel ‘God has given me strength’. This was borne by an archangel in the Bible (Daniel 8:16 and 9:21), who in the New Testament announced the impending birth of Jesus to the Virgin Mary (Luke 1:26–38). It has been a comparatively popular personal name in all parts of Europe, among both Christians and Jews, during the Middle Ages and since. Compare Michael and Raphael.
Female
Greek
(Κανδάκη) Greek name of foreign origin, KANDAKE means "prince of servants." In Acts 8:27 of the New Testament bible, a queen of Ethiopia is referred to by this name. But it was not actually a personal name, but the name of a dynasty of Ethiopian queens.Â
Male
Japanese
(1-晋, 2-信, 3-紳, 4-心, 5-慎, 6-新, 7-進, 8-真) Japanese name SHIN means 1) "advancing," 2) "belief," 3) "gentleman," 4) "heart," 5) "humble," 6) "new," 7) "progressive," and 8) "true." Compare with another form of Shin.
Surname or Lastname
Scottish (common in the Northern Isles)
Scottish (common in the Northern Isles) : patronymic from the personal name Magnus.English : patronymic from the Middle English nickname or byname Mann.Jewish (Ashkenazic) : patronymic from Man 8.
Surname or Lastname
English
English : from the Middle English vernacular form, Maudeleyn, of the New Testament Greek personal name Magdalēnē. This is a byname, meaning ‘woman from Magdala’ (a village on the Sea of Galilee, deriving its name from Hebrew migdal ‘tower’), denoting the woman cured of evil spirits by Jesus (Luke 8:2), who later became a faithful follower. In Christian folk belief she was generally identified with the repentant sinner who washed Christ’s feet with her tears in Luke 7; hence the name came to be used as a byname for a prostitute, also a tearful woman. The popularity of the personal name increased with the supposed discovery of her relics in the 13th century.
Male
Japanese
(1-æš, 2-æ‚Ÿ, 3-è¡, 4-知, 5-覚, 6-è«, 7-了, 8-智) Japanese name SATORU means 1) "daybreak," 2) "enlightened," 3) "fast learner," 4) "knowledgeable," 5) "perceptive," 6) "persuasive," 7) "understanding," or 8) "wise."
Girl/Female
Hindu, Indian
Collection of 8
Male
Hebrew
 (×¢Ö²×–Ö¸×זֵל): Hebrew word (not name), AZA'ZEL means "entire removal" and "scapegoat." In the bible, this word is found in the law of the day of atonement (Leviticus 16:8, 10, 26). It refers to a goat used for sacrifice for the sins of the people. In modern times, Azazel was interpreted as a Satanic, goat-like demon. The name has even been used for the "Angel of Death."
Male
English
Anglicized form of Hebrew Aza'zel, AZAZEL means "entire removal" and "scapegoat." In the bible, this word is found in the law of the day of atonement (Leviticus 16:8, 10, 26). It refers to a goat used for sacrifice for the sins of the people. In modern times, Azazel was interpreted as a Satanic, goat-like demon. The name has even been used for the "Angel of Death."
Female
English
Latin form of Greek Kandake, which is of foreign origin, CANDACE means "prince of servants." In Acts 8:27 of the New Testament bible, a queen of Ethiopia is referred to by this name. But it was not actually a personal name, but the name of a dynasty of Ethiopian queens.Â
Surname or Lastname
English
English : from the personal name Horace, Latin Horatius, a Roman family name of unknown origin, associated chiefly with the name of the poet Quintus Horatius Flaccus (65–8 bc).
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