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In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol. [1]

82 relations: Ball (mathematics), Binary tetrahedral group, Circumference, Complex polytope, Convex hull, Convex polytope, Cube, Cubic pyramid, Cuboctahedron, Dihedral angle, Dodecagon, Dual polyhedron, Duality (mathematics), Elongated hexagonal bipyramid, Euclidean space, F4 (mathematics), Face (geometry), Geometry, Golden ratio, Grand stellated 120-cell, Great 120-cell, Great circle, Harold Scott MacDonald Coxeter, Hexagonal antiprism, Hopf fibration, Hurwitz quaternion, Hyperplane, Hypersphere, Icosahedron, Isogonal figure, Isohedral figure, Isotoxal figure, Kissing number problem, Net (polyhedron), Norman Johnson (mathematician), North Pole, Octacube (sculpture), Octahedron, Permutation, Perspective (graphical), Platonic solid, Polygon, Polyhedral combinatorics, Projective cover, Quaternion, Rectification (geometry), Rectified 24-cell, Reflection (mathematics), Regular 4-polytope, Rhombic dodecahedron, ..., Ring (mathematics), Root system, Runcinated tesseracts, Schläfli symbol, Schlegel diagram, Simple Lie group, Simplex, Snub 24-cell, SO(8), Solvable group, South Pole, Sphere packing, Stereographic projection, Stereoscopy, Symmetry group, Tessellation, Tesseract, Tesseractic honeycomb, Tetrakis hexahedron, Triangle, Truncated 24-cells, Truncation (geometry), Uniform 4-polytope, Unit (ring theory), Vertex figure, Voronoi diagram, Weyl group, 120-cell, 16-cell, 24-cell, 24-cell honeycomb, 3-sphere. Expand index (32 more) »

Ball (mathematics)

In mathematics, a ball is the space bounded by a sphere.

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Binary tetrahedral group

In mathematics, the binary tetrahedral group, denoted 2T or 2,3,3 is a certain nonabelian group of order 24.

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In geometry, the circumference (from Latin circumferentia, meaning "carrying around") of a circle is the (linear) distance around it.

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Complex polytope

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

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Convex hull

In mathematics, the convex hull or convex envelope or convex closure of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X., p. 3.

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Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.

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

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Cubic pyramid

In 4-dimensional geometry, the cubic pyramid is bounded by one cube on the base and 6 square pyramid cells which meet at the apex.

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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces.

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Dihedral angle

A dihedral angle is the angle between two intersecting planes.

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In geometry, a dodecagon or 12-gon is any twelve-sided polygon.

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Dual polyhedron

In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.

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Duality (mathematics)

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself.

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Elongated hexagonal bipyramid

In geometry, the elongated hexagonal bipyramid is constructed by elongating a hexagonal bipyramid (by inserting a hexagonal prism between its congruent halves).

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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F4 (mathematics)

In mathematics, F4 is the name of a Lie group and also its Lie algebra f4.

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Face (geometry)

In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.

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Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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Golden ratio

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

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Grand stellated 120-cell

In geometry, the grand stellated 120-cell or grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol.

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Great 120-cell

In geometry, the great 120-cell or great polydodecahedron is a regular star 4-polytope with Schläfli symbol.

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Great circle

A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere.

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Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.

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Hexagonal antiprism

In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.

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Hopf fibration

In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere.

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Hurwitz quaternion

In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of an odd integer; a mixture of integers and half-integers is excluded).

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In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

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In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its center.

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In geometry, an icosahedron is a polyhedron with 20 faces.

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Isogonal figure

In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure.

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Isohedral figure

In geometry, a polytope of dimension 3 (a polyhedron) or higher is isohedral or face-transitive when all its faces are the same.

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Isotoxal figure

In geometry, a polytope (for example, a polygon or a polyhedron), or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges.

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Kissing number problem

In geometry, a kissing number is defined as the number of non-overlapping unit spheres that can be arranged such that they each touch another given unit sphere.

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Net (polyhedron)

In geometry a net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron.

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Norman Johnson (mathematician)

Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was a mathematician, previously at Wheaton College, Norton, Massachusetts.

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North Pole

The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is (subject to the caveats explained below) defined as the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface.

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Octacube (sculpture)

The Octacube is a large, steel sculpture of a mathematical object: the 24-cell or "octacube".

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In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.

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In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

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Perspective (graphical)

Perspective (from perspicere "to see through") in the graphic arts is an approximate representation, generally on a flat surface (such as paper), of an image as it is seen by the eye.

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Platonic solid

In three-dimensional space, a Platonic solid is a regular, convex polyhedron.

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In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit.

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Polyhedral combinatorics

Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.

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Projective cover

In the branch of abstract mathematics called category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual of injective envelopes.

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In mathematics, the quaternions are a number system that extends the complex numbers.

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Rectification (geometry)

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.

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Rectified 24-cell

In geometry, the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra.

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Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.

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Regular 4-polytope

In mathematics, a regular 4-polytope is a regular four-dimensional polytope.

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Rhombic dodecahedron

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces.

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Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.

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Runcinated tesseracts

In four-dimensional geometry, a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract.

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Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

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Schlegel diagram

In geometry, a Schlegel diagram is a projection of a polytope from R^d into R^ through a point beyond one of its facets or faces.

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Simple Lie group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.

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In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

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Snub 24-cell

In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells.

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In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space.

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Solvable group

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions.

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South Pole

The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where the Earth's axis of rotation intersects its surface.

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Sphere packing

In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space.

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Stereographic projection

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

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Stereoscopy (also called stereoscopics, or stereo imaging) is a technique for creating or enhancing the illusion of depth in an image by means of stereopsis for binocular vision.

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Symmetry group

In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.

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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.

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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.

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Tesseractic honeycomb

In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol, and constructed by a 4-dimensional packing of tesseract facets.

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Tetrakis hexahedron

In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron, tetrakis cube, and kiscube) is a Catalan solid.

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A triangle is a polygon with three edges and three vertices.

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Truncated 24-cells

In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell.

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Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.

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Uniform 4-polytope

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

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Unit (ring theory)

In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.

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Vertex figure

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

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Voronoi diagram

In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane.

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Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system.

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In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol.

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In four-dimensional geometry, a 16-cell is a regular convex 4-polytope.

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In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

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24-cell honeycomb

In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells.

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In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere.

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[1] https://en.wikipedia.org/wiki/24-cell

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