| Compound of four triangular prisms | |
|---|---|
 | |
| Type | Uniform compound |
| Index | UC30 |
| Polyhedra | 4 triangular prisms |
| Faces | 8 triangles, 12 squares |
| Edges | 36 |
| Vertices | 24 |
| Symmetry group | chiral octahedral (O) |
| Subgroup restricting to one constituent | 3-fold dihedral (D3) |
This uniform polyhedron compound is a chiral symmetric arrangement of 4 triangular prisms, aligned with the axes of three-fold rotational symmetry of an octahedron.
Cartesian coordinates for the vertices of this compound are all the even permutations of
with an even number of minuses in the '±' choices, together with all the odd permutations with an odd number of minuses in the '±' choices.
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.
In mathematics, when X is a finite set with at least two elements, the permutations of X fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity of a permutation of X can be defined as the parity of the number of inversions for σ, i.e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y).
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it isn't. For example, −4, 0, 82 are even because
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges. It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length.
In geometry, a pentagonal icositetrahedron or pentagonal icosikaitetrahedron is a Catalan solid which is the dual of the snub cube. In crystallography it is also called a gyroid.
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations.
In number theory, Zolotarev's lemma states that the Legendre symbol
In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U40. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol sr{5⁄2,5}, as a snub great dodecahedron.
In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U74. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol sr{3/2,5/3}.
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative, or it may be considered both positive and negative. Whenever not specifically mentioned, this article adheres to the first convention.
The compound of five truncated tetrahedra is a uniform polyhedron compound. It's composed of 5 truncated tetrahedra rotated around a common axis. It may be formed by truncating each of the tetrahedra in the compound of five tetrahedra. A far-enough truncation creates the compound of five octahedra. Its convex hull is a nonuniform snub dodecahedron.
This uniform polyhedron compound is a symmetric arrangement of 3 square antiprisms, aligned with the three axes of 4-fold rotational symmetry of a cube.
This uniform polyhedron compound is a chiral symmetric arrangement of 6 pentagrammic antiprisms, aligned with the axes of fivefold rotational symmetry of a dodecahedron.
In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
In graph theory, a cycle decomposition is a decomposition into cycles. Every vertex in a graph that has a cycle decomposition must have even degree.
In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope.
The original Rubik's cube was a mechanical 3×3×3 cube puzzle invented in 1974 by the Hungarian sculptor and professor of architecture Ernő Rubik. Extensions of the Rubik's cube have been around for a long time and come in both hardware and software forms. The major extension have been the availability of cubes of larger size and the availability of the more complex cubes with marked centres. The properties of Rubik’s family cubes of any size together with some special attention to software cubes is the main focus of this article. Many properties are mathematical in nature and are functions of the cube size variable.
Heap's algorithm generates all possible permutations of n objects. It was first proposed by B. R. Heap in 1963. The algorithm minimizes movement: it generates each permutation from the previous one by interchanging a single pair of elements; the other n−2 elements are not disturbed. In a 1977 review of permutation-generating algorithms, Robert Sedgewick concluded that it was at that time the most effective algorithm for generating permutations by computer.
In quantum information, the gnu code refers to a particular family of quantum error correcting codes, with the special property of being invariant under permutations of the qubits. Given integers g, n, and m, the two codewords are
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