5-polytope
Encyclopedia
5-simplex (hexateron) |
5-orthoplex, 211 (Pentacross) |
 5-cube (Penteract) |
 5-demicube. 121 (Demipenteract) |
In five-dimensional 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 5-polytope is a 5-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...
, bounded by (4-polytope) facets. Each polyhedral
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
cell being shared by exactly two polychoron
Polychoron
In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...
facets. A proposed name for 5-polytopes is polyteron.
Definition
A 5-polytope is a closed five-dimensional figure with verticesVertex (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...
, and cells, and hypercell
Hypercell
In geometry, a hypercell is a descriptive term for an element of a polytope or tessellation, usually representing an element one dimension higher than a cell. The most generally accepted term is 4-face because it contains a 4-dimensional interior...
s. A vertex 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 five or more edges meet. An edge 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 four 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 three or more cells meet. A cell is a polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
, and a hypercell is a polychoron
Polychoron
In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...
. Furthermore, the following requirements must be met:
- Each cell must join exactly two hypercells.
- Adjacent hypercells are not in the same four-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.
Regular 5-polytopes
Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} polychoral facetsFacet (mathematics)
A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
around each face
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...
.
There are exactly three such convex regular 5-polytopes:
- {3,3,3,3} - HexateronHexateronIn five dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has 6 vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 pentachoron facets. It has a dihedral angle of cos−1, or approximately 78.46°.- Alternate names :...
(5-simplexSimplexIn geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
) - {4,3,3,3} - PenteractPenteractIn five dimensional geometry, a 5-cube is a name for a five dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract hypercells....
(5-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...
) - {3,3,3,4} - PentacrossPentacrossIn five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell hypercells....
(5-orthoplex)
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 5-polytopes that are topological 4-spheres (including all convex 5-polytopes) is two. χ=V-E+F-C+H=2.
For the 3 convex regular 5-polytopes and two semiregular 5-polytope, their elements are:
| Name | Schläfli symbol | Vertices | Edges | Faces | Cells | 4-faces | χ |
|---|---|---|---|---|---|---|---|
| 5-simplex | {3,3,3,3} | 6 | 15 | 20 | 15 | 6 | 2 |
| 5-orthoplex | {3,3,3,4} | 10 | 40 | 80 | 80 | 32 | 2 |
| 5-demicube | {31,2,1} | 16 | 80 | 160 | 120 | 26 | 2 |
| 5-cube | {4,3,3,3} | 32 | 80 | 80 | 40 | 10 | 2 |
| Rectified pentacross Rectified pentacross In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the... |
t1{3,3,3,4} | 40 | 240 | 400 | 240 | 42 | 2 |
Classification
5-polytopes 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 5-polytope 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 5-polytope is contained in the 5-polytope or its interior; otherwise, it is non-convex. Self-intersecting 5-polytopes are also known as star polytopes, from analogy with the star-like shapes of the non-convex Kepler-Poinsot polyhedra.
- A uniform 5-polytope 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 facets are uniform polychoraUniform polychoronIn geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
. The edges of a uniform 5-polytope must be equal in length.
- A semi-regular 5-polytope contains two or more types of regular polychoral facets. There is only one such figure, called a demipenteractDemipenteractIn five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices deleted.It was discovered by Thorold Gosset...
.
- A regular 5-polytope has all identical regular polychoron facets. All regular polytera are convex.
- A prismatic 5-polytope is constructed by a 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 5-polytope is uniform if its factors are uniform. The hypercubePenteractIn five dimensional geometry, a 5-cube is a name for a five dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract hypercells....
is prismatic (product of a squareSquare (geometry)In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
s and 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...
), but is considered separately because it has symmetries other than those inherited from its factors.
- A 4-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 four-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 grid of polychoral facets. Strictly speaking, tessellations are not polytera as they do not bound a "5D" volume, but we include them here for the sake of completeness because they are similar in many ways to polytera. A uniform 4-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 facets are uniform polychoraUniform polychoronIn geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
.
Pyramids
Pyramidal 5-polytopes, or 5-pyramids, can be generated by a polychoronPolychoron
In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...
base in a 4-space hyperplane connected to a point off the hyperplane. The 5-simplex is the simplest example with a 4-simplex base.
External links
- Polytopes of Various Dimensions, Jonathan Bowers
- Uniform Polytera, Jonathan Bowers
- Multi-dimensional Glossary, Garrett Jones