## Rectified 8-cubes

In 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, the zeroth being the 8-cube, and the 7th and last being the 8-orthoplex. ...

## Rectified 8-orthoplexes

In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex. There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being th ...

## Rectified 8-simplexes

In 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 of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are loca ...

## Runcinated 8-simplexes

In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations of the regular 8-simplex. There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation ...

## Stericated 8-simplexes

In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and r ...

## Truncated 8-cubes

In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube. There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edg ...

## Truncated 8-orthoplexes

In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex. There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the e ...

## Truncated 8-simplexes

In 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 of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge o ...

## 1 52 honeycomb

In geometry, the 1 52 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. It contains 1 42 and 51 facets, in a birectified 8-simplex vertex figure. It is the final figure in the 1 k2 polytope family.

## 2 51 honeycomb

In 8-dimensional geometry, the 2 51 honeycomb is a space-filling uniform tessellation. It is composed of 2 41 polytope and 8-simplex facets arranged in an 8-demicube vertex figure. It is the final figure in the 2 k1 family.

## 5 21 honeycomb

In geometry, the 5 21 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5 21 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram. This honeycomb was first studied by Gosset who call ...

## 9-cube

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. It can be named ...

## 9-demicube

In geometry, a 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 family of uniform polytopes called demihypercubes. E. L. Elte identified it i ...

## 9-orthoplex

In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 ...

## 9-simplex

In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces. ...

## Rectified 9-cubes

In nine-dimensional 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 is the 9-cube itself, and the 8th is the dual 9-orthoplex. Vertices ...

## Rectified 9-orthoplexes

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex. There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-cen ...

## Rectified 9-simplexes

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex. These polytopes are part of a family of 271 uniform 9-polytopes with A 9 symmetry. There are unique 4 degrees of r ...

## Uniform 9-polytope

In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets. A uniform 9-polytope is one which is vertex-transitive, and ...

## 10-cube

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-cu ...

## 10-demicube

In geometry, a 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 family of uniform polytopes called demihypercubes. E. L. Elte identified it ...

## 10-orthoplex

In geometry, a 10-orthoplex or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells 4-faces, 13440 5-faces, 15360 6-faces, 11520 7-faces, 5120 8-faces, and 1024 9-faces. ...

## 10-simplex

In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, ...

## E9 honeycomb

In geometry, an E 9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. T ¯ 9 {\displaystyle {\bar {T}}_{9}}, also is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded. E 10 is ...

## Rectified 10-cubes

In ten-dimensional 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 the zeroth being the 10-cube itself. Vertices of the rectified 10-cub ...

## Rectified 10-orthoplexes

In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex. There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the e ...

## Rectified 10-simplexes

In ten-dimensional geometry, a rectified 10-simplex is a convex uniform 10-polytope, being a rectification of the regular 10-simplex. These polytopes are part of a family of 527 uniform 10-polytopes with A 10 symmetry. There are unique 5 degrees ...

## Uniform 10-polytope

In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge. A uniform 10-polytope is one which is vertex-transitive, and construct ...

## Biregular graph

In graph-theoretic mathematics, a biregular graph or semiregular bipartite graph is a bipartite graph G = {\displaystyle G=} for which every two vertices on the same side of the given bipartition have the same degree as each other. If the degree ...

## Intersection graph

In the mathematical area of graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can b ...

## Polygon-circle graph

In the mathematical discipline of graph theory, a polygon-circle graph is an intersection graph of a set of convex polygons all of whose vertices lie on a common circle. These graphs have also been called spider graphs. This class of graphs was f ...

## Barbell graph

In the mathematical discipline of graph theory, the n -barbell graph is a special type of undirected graph consisting of two non-overlapping n -vertex cliques together with a single edge that has an endpoint in each clique.

## Bouquet graph

In mathematics, a bouquet graph B m {\displaystyle B_{m}}, for an integer parameter m {\displaystyle m}, is an undirected graph with one vertex and m {\displaystyle m} edges, all of which are self-loops. It is the graph-theoretic analogue of the ...

## Cocktail party graph

## Complete bipartite graph

In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set. Graph theory itself is typically dated as b ...

## Half graph

In graph theory, a branch of mathematics, a half graph is a special type of bipartite graph. These graphs are called the half graphs because they have approximately half of the edges of a complete bipartite graph on the same vertices. The name wa ...

## Pancake graph

In the mathematical field of graph theory, the pancake graph P n or n -pancake graph is a graph whose vertices are the permutations of n symbols from 1 to n and its edges are given between permutations transitive by prefix reversals. Pancake sort ...

## Shannon multigraph

In the mathematical discipline of graph theory, Shannon multigraphs, named after Claude Shannon by Vizing, are a special type of triangle graphs, which are used in the field of edge coloring in particular. A Shannon multigraph is multigraph with ...

## Godel operation

In mathematical set theory, a set of Godel operations is a finite collection of operations on sets that can be used to construct the constructible sets from ordinals. Godel introduced the original set of 8 Godel operations 1., 8 under the name fu ...

## Jensen hierarchy

In set theory, a mathematical discipline, the Jensen hierarchy or J-hierarchy is a modification of Godels constructible hierarchy, L, that circumvents certain technical difficulties that exist in the constructible hierarchy. The J-Hierarchy figur ...

## Minimal model (set theory)

In set theory, a branch of mathematics, the minimal model is the minimal standard model of ZFC. The minimal model was introduced by Shepherdson and rediscovered by Cohen. The existence of a minimal model cannot be proved in ZFC, even assuming tha ...

## Silver machine

In set theory, Silver machines are devices used for bypassing the use of fine structure in proofs of statements holding in L. They were invented by set theorist Jack Silver as a means of proving global square holds in the constructible universe.

## Pythagorean interval

In musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 and the perfect fourth with ratio 4/3 a ...

## List of intervals in 5-limit just intonation

The intervals of 5-limit just intonation are ratios involving only the powers of 2, 3, and 5. The fundamental intervals are the superparticular ratios 2/1, 3/2 and 5/4. That is, the notes of the major triad are in the ratio 1:5/4:3/2 or 4:5:6. In ...

## Five-limit tuning

Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning, is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note by products of integer powers of 2, 3, or 5, ...

## 7-limit tuning

7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14:11 is ...

## Undecimal minor sixth

## Tridecimal minor third

## Tridecimal neutral seventh

## Tridecimal neutral third 