Polychoron
Encyclopedia
4-simplex (5-cell) |
4-orthoplex (16-cell) |
4-cube (8-cell, Tesseract) |
24-cell |
120-cell |
600-cell |
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 polychoron or 4-polytope is a four-dimensional polytope
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
, edges
Edge (geometry)
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
, faces
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
(polygons), and cells (polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
). Each face is shared by exactly two cells.
The two-dimensional analogue of a polychoron is a polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
, and the three-dimensional analogue is a polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
.
The term polychoron (plural polychora), from the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
roots poly ('many') and choros ('room' or 'space') and has been advocated by Norman Johnson and George Olshevsky
George Olshevsky
George Olshevsky is a freelance editor, writer, publisher, amateur paleontologist, and mathematician living in San Diego, California.Olshevsky maintains the comprehensive online Dinosaur Genera List...
, but it is little known in general polytope theory. Other names for polychoron include: polyhedroid and polycell.
Topologically 4-polytopes are closely related to the uniform honeycombs
Convex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...
, such as the cubic honeycomb
Cubic honeycomb
The cubic honeycomb is the only regular space-filling tessellation 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....
, which tessellate 3-space; similarly the 3D cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space.
Definition
Polychora are closed four-dimensional figures. We can describe them further only through analogy with such three dimensionDimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
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, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
.
The most familiar example of a polychoron is the tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
or hypercube, the 4d analogue of the cube. A tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
has vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
, edges
Edge (geometry)
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. 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 that describes the corners or intersections of geometric shapes.-Of an angle:...
is a point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
where four or more edges meet. An edge
Edge (geometry)
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. 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 points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
where three or more faces meet, and a face is a polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, 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
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
. Furthermore, the following requirements must be met:
- Each face must join exactly two cells.
- Adjacent cells are not in the same three-dimensional hyperplaneHyperplaneA hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
. - The figure is not a compound of other figures which meet the requirements.
Euler characteristic
The Euler characteristicEuler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
for 4-polytopes that are topological 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...
s (including all convex 4-polytopes) is zero. χ=V-E+F-C=0.
For example, the convex regular 4-polytope
Convex regular 4-polytope
In mathematics, a convex regular 4-polytope is a 4-dimensional polytope that is both regular and convex. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
s:
Name | Schläfli symbol | Vertices | Edges | Faces | Cells | χ |
---|---|---|---|---|---|---|
5-cell | {3,3,3} | 5 | 10 | 10 | 5 | 0 |
16-cell 16-cell In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.... |
{3,3,4} | 8 | 24 | 32 | 16 | 0 |
tesseract Tesseract In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8... |
{4,3,3} | 16 | 32 | 24 | 8 | 0 |
24-cell | {3,4,3} | 24 | 96 | 96 | 24 | 0 |
120-cell | {5,3,3} | 600 | 1200 | 720 | 120 | 0 |
600-cell | {3,3,5} | 120 | 720 | 1200 | 600 | 0 |
Classification
Polychora may be classified based on properties like "convexityConvex 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...
" 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 convexConvex polytopeA 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...
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 groupSymmetry groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
under which all vertices are equivalent, and its cells are uniform polyhedraUniform polyhedronA uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
. The edges of a uniform polychoron must be equal in length.
- A uniform polychoron is semi-regular if its cells are regular polyhedraRegular polyhedronA regular polyhedron is a polyhedron whose faces are congruent 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. it is transitive on its flags...
. 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 polyhedronRegular polyhedronA regular polyhedron is a polyhedron whose faces are congruent 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. it is transitive on its flags...
for examples.
- A regular polychoron which is also a convex polychoron is said to be a convex regular polychoronConvex regular 4-polytopeIn mathematics, a convex regular 4-polytope is a 4-dimensional polytope that is both regular and convex. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
.
- A polychoron is prismatic if it is the Cartesian productCartesian productIn mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of two lower-dimensional polytopes. A prismatic polychoron is uniform if its factors are uniform. The hypercubeTesseractIn geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
is prismatic (product of two squareSquare (geometry)In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
s, or of a cubeCubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
and line segmentLine segmentIn geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
), but is considered separately because it has symmetries other than those inherited from its factors.
- A 3-space tessellationTessellationA tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
is the division of three-dimensional Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
into a regular gridGrid (spatial index)In the context of a spatial index, a grid is a regular tessellation of a manifold or 2-D surface that divides it into a series of contiguous cells, which can then be assigned unique identifiers and used for spatial indexing purposes...
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 groupSpace groupIn mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
and whose cells are uniform polyhedraUniform polyhedronA uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
.
Categories
The following lists the various categories of polychora classified according to the criteria above:Uniform polychora (vertex-transitive
Vertex-transitive
In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same...
):
- Convex uniform polychora (64, plus two infinite families)
- 47 non-prismatic convex uniform polychora including:
- 6 convex regular polychoraConvex regular 4-polytopeIn mathematics, a convex regular 4-polytope is a 4-dimensional polytope that is both regular and convex. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
- 6 convex regular polychora
- Prismatic uniform polychora:
- {} x {p,q} : 18 polyhedral hyperprisms (including cubic hyperprism, the regular hypercubeHypercubeIn geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
) - hyperprisms built on antiprisms (infinite family)
- {p} x {q} : DuoprismDuoprismIn geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...
s (infinite family)
- {} x {p,q} : 18 polyhedral hyperprisms (including cubic hyperprism, the regular hypercube
- 47 non-prismatic convex uniform polychora including:
- Non-convex uniform polychora (10 + unknown)
- 10 (regular) Schläfli-Hess polychora
- 57 hyperprisms built on nonconvex uniform polyhedraNonconvex uniform polyhedronIn geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting...
- Unknown total number of nonconvex uniform polychora: The Uniform Polychora Project now counts 1849 known cases.
- Infinite uniform polychora of Euclidean 3-spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
(uniform tessellations of convex uniform cells)- 28 convex uniform honeycombConvex uniform honeycombIn geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...
s: uniform convex polyhedral tessellations, including:- 1 regular tessellation, cubic honeycombCubic honeycombThe cubic honeycomb is the only regular space-filling tessellation 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....
: {4,3,4}
- 1 regular tessellation, cubic honeycomb
- 28 convex uniform honeycomb
- Infinite uniform polychora of hyperbolic 3-spaceHyperbolic spaceIn mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
(uniform tessellations of convex uniform cells)- 76 Wythoffian convex uniform honeycombs in hyperbolic spaceConvex uniform honeycombs in hyperbolic spaceIn geometry, a convex uniform honeycomb is a tessellation of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter–Dynkin...
, including:- 4 regular tessellation of hyperbolic space: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}
- 76 Wythoffian convex uniform honeycombs in hyperbolic space
Dual uniform polychora (cell-transitive):
- 41 unique dual convex uniform 4-polytopes
- 17 unique dual convex uniform polyhedral prisms
- infinite family of dual convex uniform duoprisms (irregular tetrahedral cells)
- 27 unique convex dual uniform honeycombs, including:
- Rhombic dodecahedral honeycombRhombic dodecahedral honeycombThe rhombic dodecahedra honeycomb is a space-filling tessellation in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which is believed to be the densest possible packing of equal spheres in ordinary space .It consists of copies of a single cell, the rhombic...
- Disphenoid tetrahedral honeycombDisphenoid tetrahedral honeycombThe disphenoid tetrahedral honeycomb is a space-filling tessellation in Euclidean 3-space made up of identical nonregular tetrahedral cells....
- Rhombic dodecahedral honeycomb
Others:
- Weaire-Phelan structureWeaire-Phelan structureIn geometry, the Weaire–Phelan structure is a complex 3-dimensional structure representing an idealised foam of equal-sized bubbles. In 1993, Trinity College Dublin physicist Denis Weaire and his student Robert Phelan found that in computer simulations of foam, this structure was a better...
periodic space-filling honeycomb with irregular cells
Abstract regular polychora
Abstract polytope
In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc...
:
- 11-cell11-cellIn mathematics, the 11-cell is a self-dual abstract regular 4-polytope . Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has660 symmetries...
- 57-cell
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
- Convex regular 4-polytopeConvex regular 4-polytopeIn mathematics, a convex regular 4-polytope is a 4-dimensional polytope that is both regular and convex. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
- The 3-sphere3-sphereIn 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 is a figure in 4-dimensional space related to the duoprismDuoprismIn geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...
s. It is also not a polychoron because its bounding volumes are not polyhedral.
External links
- Uniform Polychora, Jonathan Bowers
- Uniform polychoron Viewer - Java3D Applet with sources
- Dr. R. Klitzing, polychora