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Polychoron
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a four-dimensional polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
 is sometimes called a polychoron (plural: polychora), from the Greek
Greek language

Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European....
 root poly, meaning "many", and choros meaning "room" or "space". It is also called a 4-polytope or polyhedroid.






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Encyclopedia


Complete Graph K5

4-simplex
(5-cell)
4-orthoplex
(16-cell)
4-cube
(Tesseract)

24-cell
24-cell

In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....

120-cell
120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....

600-cell
600-cell

In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....
Graphs of six convex regular 4-polytope
Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
.
Hypercube
In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a four-dimensional polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
 is sometimes called a polychoron (plural: polychora), from the Greek
Greek language

Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European....
 root poly, meaning "many", and choros meaning "room" or "space". It is also called a 4-polytope or polyhedroid. The two-dimensional analogue of a polychoron is a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
, and the three-dimensional analogue is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
.

(Note that the term polychoron is a recent invention and has limited usage at present. It has been advocated by Norman Johnson and George Olshevsky
George Olshevsky

George Olshevsky is a freelance editing, writer, publisher, paleontologist, and mathematician living in San Diego, California.Olshevsky maintains the comprehensive online Dinosaur Genera List....
—see the Uniform Polychora Project
Uniform Polychora Project

The Uniform Polychora Project is a collaborative effort in geometry to recognize and standardize terms used to describe objects in higher-dimensional spaces....
—but it is little known in general polytope theory.)

Definition


Polychora are closed four-dimension
Fourth dimension

In physics and mathematics, a vector of n real number can be understood as a Coordinate system in an n-dimensional Euclidean space. When n = 4, the set of all such locations is called 4-dimensional Euclidean space....
al figures. We can describe them further only through analogy with such three dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al polyhedron counterparts as pyramids and cubes
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
.

The most familiar example of a polychoron is the tesseract
Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
 or hypercube, the 4d analogue of the cube. A tesseract
Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
 has vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
, edges
Edge (geometry)

In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertex in a polytope. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
, faces, and cells. A vertex
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
 is a point
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
 where four or more edges meet. An edge
Edge (geometry)

In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertex in a polytope. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
 is a line segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
 where three or more faces meet, and a face is a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
 where two cells meet. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
. Furthermore, the following requirements must be met:
  1. Each face must join exactly two cells.
  2. Adjacent cells are not in the same three-dimensional hyperplane
    Hyperplane

    A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
    .
  3. The figure is not a compound of other figures which meet the requirements.


Classification


Polychora may be classified based on properties like "convexity" and "symmetry
Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
".

  • A polychoron is convex
    Convex set

    In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
     if its boundary (including its cells, faces and edges) does not intersect itself and the line segment joining any two points of the polychoron is contained in the polychoron or its interior; otherwise, it is non-convex. Self-intersecting polychora are also known as star polychora, from analogy with the star-like shapes of the non-convex Kepler-Poinsot polyhedra.


  • A polychoron is uniform if it has a symmetry group
    Symmetry group

    The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
     under which all vertices are equivalent, and its cells are uniform polyhedra
    Uniform polyhedron

    A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
    . The edges of a uniform polychoron must be equal in length.


  • A uniform polychoron is semi-regular if its cells are regular polyhedra
    Regular polyhedron

    A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
    . The cells may be of two or more kinds, provided that they have the same kind of face.


  • A semi-regular polychoron is said to be regular if its cells are all of the same kind of regular polyhedron; see regular polyhedron
    Regular polyhedron

    A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
     for examples.


  • A regular polychoron which is also a convex polychoron is said to be a convex regular polychoron
    Convex regular 4-polytope

    In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
    .


  • A polychoron is prismatic if it is the Cartesian product
    Cartesian product

    In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after Ren? Descartes, whose formulation of analytic geometry gave rise to this concept....
     of two lower-dimensional polytopes. A prismatic polychoron is uniform if its factors are uniform. The hypercube
    Tesseract

    In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
     is prismatic (product of two square
    Square (geometry)

    In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
    s, or of a cube
    Cube

    A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
     and line segment
    Line segment

    In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
    ), but is considered separately because it has symmetries other than those inherited from its factors.


  • A 3-space tessellation
    Tessellation

    A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
     is the division of three-dimensional Euclidean space
    Euclidean space

    Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
     into a regular grid
    Grid

    'Grid' may refer to:In 'entertainment and media':* The Grid * The Grid * Grid , the eighth original album by the Japanese band m.o.v.e.* ...
     of polyhedral cells. Strictly speaking, tessellations are not polychora as they do not bound a "4D" volume, but we include them here for the sake of completeness because they are similar in many ways to polychora. A uniform 3-space tessellation is one whose vertices are related by a space group
    Space group

    The space group of a crystal or crystallographic group is a mathematical description of the symmetry inherent in the structure. The word 'group' in the name comes from the group , which is used to build the set of space groups....
     and whose cells are uniform polyhedra
    Uniform polyhedron

    A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
    .


Categories


The following lists the various categories of polychora classified according to the criteria above:

Uniform polychora
  • Convex uniform polychora (64, plus two infinite families)
    • 47 non-prismatic convex uniform polychora including:
      • 6 convex regular polychora
        Convex regular 4-polytope

        In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
    • Prismatic uniform polychora:
      • x : 18 polyhedral hyperprisms (including cubic hyperprism, the regular hypercube
        Hypercube

        In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
        )
      • hyperprisms built on antiprisms (infinite family)
      • x : Duoprism
        Duoprism

        In geometry, a duoprism is a polytope resulting from the Cartesian product of two polytopes of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an -polytope, where n and m are 2 or higher....
        s (infinite family)
  • Non-convex uniform polychora (10 + unknown)
    • 10 Schläfli-Hess polychora
      Schläfli-Hess polychoron

      In four dimensional geometry, Schl?fli-Hess polychora are the complete set of 10 Regular polytope self-intersecting Star polytope . They are named in honor of their discoverers: Ludwig Schl?fli and Edmund Hess....
    • 57 hyperprisms built on nonconvex uniform polyhedra
    • Unknown total number of nonconvex uniform polychora: The Uniform Polychora Project
      Uniform Polychora Project

      The Uniform Polychora Project is a collaborative effort in geometry to recognize and standardize terms used to describe objects in higher-dimensional spaces....
       now counts 1849 known cases.


  • Infinite uniform polychora of Euclidean 3-space
    Euclidean space

    Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
     (uniform tessellations of convex uniform cells)
    • 28 convex uniform honeycomb
      Convex uniform honeycomb

      In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
      s: uniform convex polyhedral tessellations, including:
      • 1 regular tessellation: (cubic honeycomb
        Cubic honeycomb

        The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs....
        )


  • Infinite uniform polychora of hyperbolic 3-space
    Hyperbolic space

    In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1....
     (uniform tessellations of convex uniform cells)
    • 33 uniform convex polyhedral tessellations including:
      • 4 regular tessellation of hyperbolic space
        List of regular polytopes

        This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....
    • Unknown others


  • Abstract regular polychora
    Abstract polytope

    In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorics properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths etc....
    • There are two special abstract regular polychora: 11-cell
      11-cell

      In mathematics, the 11-cell is a duality abstract polytope . Its 11 cells are hemi-icosahedron. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has...
      , and 57-cell
      57-cell

      In mathematics, the 57-cell is a duality abstract polytope . Its 57 Cell s are hemi-dodecahedron. It also has 57 vertices, 171 edges and 171 faces....
      .


These categories include only the polychora that exhibit a high degree of symmetry. Many other polychora are possible, but they have not been studied as extensively as the ones included in these categories.

See also

  • The 3-sphere
    3-sphere

    In mathematics, a '3-sphere' is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space....
     (or glome) is another commonly discussed figure that resides in 4-dimensional space. This is not a polychoron, since it is not bounded by polyhedral cells.
  • The duocylinder
    Duocylinder

    The duocylinder, or double cylinder, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disk s of radius r:...
     is a figure in 4-dimensional space related to the duoprism
    Duoprism

    In geometry, a duoprism is a polytope resulting from the Cartesian product of two polytopes of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an -polytope, where n and m are 2 or higher....
    s. It is also not a polychoron because its bounding volumes are not polyhedral.
  • The Klein bottle
    Klein bottle

    In mathematics, the Klein bottle is a certain non-orientability surface, i.e., a surface with no distinct "inner" and "outer" sides. Other related non-orientable objects include the M?bius strip and the real projective plane....
     is the generalization of the Möbius strip
    Möbius strip

    The M?bius strip or M?bius band is a surface with only one side and only one boundary component. The M?bius strip has the mathematical property of being orientability....
     to 4D. It is also not a polychoron because it is a 2-manifold
    Manifold

    In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
     (2-dimensional surface, not bounded by 3-dimensional volumes).


External links

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