List of mathematical shapes
Following is a list of shapes studied in mathematics .
Rational curves
Degree 2
Degree 3
Degree 4
Degree 5
Degree 6
Families of variable degree
Curves of genus one
Curves with genus greater than one
Curve families with variable genus
Transcendental curves
Curves generated by other curves
Space curves
Surfaces in 3-space
Pseudospherical surfaces
See the list of algebraic surfaces .
Miscellaneous surfaces
Fractals
Random fractals
Regular polytopes
This table shows a summary of regular polytope counts by dimension.
There are no nonconvex Euclidean regular tessellations in any number of dimensions.
Polytope elements
The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body.
Vertex , a 0-dimensional elementEdge , a 1-dimensional elementFace , a 2-dimensional elementCell , a 3-dimensional elementHypercell or Teron, a 4-dimensional elementFacet , an (n -1)-dimensional elementRidge , an (n -2)-dimensional elementPeak , an (n -3)-dimensional elementFor example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak.
Vertex figure : not itself an element of a polytope, but a diagram showing how the elements meet.
Tessellations
The classical convex polytopes may be considered tessellations , or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
Zero dimension
One-dimensional regular polytope
There is only one polytope in 1 dimension, whose boundaries are the two endpoints of a line segment , represented by the empty Schläfli symbol {}.
Two-dimensional regular polytopes
Convex
Non-convex
Tessellation
Three-dimensional regular polytopes
Convex
Non-convex
Tessellations
Euclidean tilings
Hyperbolic tilings
Hyperbolic star-tilings
Four-dimensional regular polytopes
Non-convex
Tessellations of Euclidean 3-space
Degenerate tessellations of Euclidean 3-space
Tessellations of hyperbolic 3-space
Five-dimensional regular polytopes and higher
Tessellations of Euclidean 4-space
Tessellations of Euclidean 5-space and higher
Tessellations of hyperbolic 4-space
Tessellations of hyperbolic 5-space
Apeirotopes
Abstract polytopes
2D with 1D surface
Polygons named for their number of sides
Tilings
Johnson solids
Spherical polyhedra
Honeycombs
Convex uniform honeycomb Dual uniform honeycomb Others Convex uniform honeycombs in hyperbolic space
Other
Polyhedral compound and Uniform polyhedron compound Convex regular 4-polytope Abstract regular polytope Schläfli–Hess 4-polytope (Regular star 4-polytope)Uniform 4-polytope Rectified 5-cell , Truncated 5-cell , Cantellated 5-cell , Runcinated 5-cell Rectified tesseract , Truncated tesseract , Cantellated tesseract , Runcinated tesseract Rectified 16-cell , Truncated 16-cell Rectified 24-cell , Truncated 24-cell , Cantellated 24-cell , Runcinated 24-cell , Snub 24-cell Rectified 120-cell , Truncated 120-cell , Cantellated 120-cell , Runcinated 120-cell Rectified 600-cell , Truncated 600-cell , Cantellated 600-cell Prismatic uniform polychoron Grand antiprism Duoprism Tetrahedral prism , Truncated tetrahedral prism Truncated cubic prism , Truncated octahedral prism , Cuboctahedral prism , Rhombicuboctahedral prism , Truncated cuboctahedral prism , Snub cubic prism Truncated dodecahedral prism , Truncated icosahedral prism , Icosidodecahedral prism , Rhombicosidodecahedral prism , Truncated icosidodecahedral prism , Snub dodecahedral prism Uniform antiprismatic prism
Honeycombs
5D with 4D surfaces
Five-dimensional space , 5-polytope and uniform 5-polytope 5-simplex , Rectified 5-simplex , Truncated 5-simplex , Cantellated 5-simplex , Runcinated 5-simplex , Stericated 5-simplex 5-demicube , Truncated 5-demicube , Cantellated 5-demicube , Runcinated 5-demicube 5-cube , Rectified 5-cube , 5-cube , Truncated 5-cube , Cantellated 5-cube , Runcinated 5-cube , Stericated 5-cube 5-orthoplex , Rectified 5-orthoplex , Truncated 5-orthoplex , Cantellated 5-orthoplex , Runcinated 5-orthoplex Prismatic uniform 5-polytope For each polytope of dimension n , there is a prism of dimension n +1.[citation needed
Honeycombs
Six dimensions
Six-dimensional space , 6-polytope and uniform 6-polytope 6-simplex , Rectified 6-simplex , Truncated 6-simplex , Cantellated 6-simplex , Runcinated 6-simplex , Stericated 6-simplex , Pentellated 6-simplex 6-demicube , Truncated 6-demicube , Cantellated 6-demicube , Runcinated 6-demicube , Stericated 6-demicube 6-cube , Rectified 6-cube , 6-cube , Truncated 6-cube , Cantellated 6-cube , Runcinated 6-cube , Stericated 6-cube , Pentellated 6-cube 6-orthoplex , Rectified 6-orthoplex , Truncated 6-orthoplex , Cantellated 6-orthoplex , Runcinated 6-orthoplex , Stericated 6-orthoplex 122 polytope , 221 polytope
Honeycombs
Seven dimensions
Seven-dimensional space , uniform 7-polytope 7-simplex , Rectified 7-simplex , Truncated 7-simplex , Cantellated 7-simplex , Runcinated 7-simplex , Stericated 7-simplex , Pentellated 7-simplex , Hexicated 7-simplex 7-demicube , Truncated 7-demicube , Cantellated 7-demicube , Runcinated 7-demicube , Stericated 7-demicube , Pentellated 7-demicube 7-cube , Rectified 7-cube , 7-cube , Truncated 7-cube , Cantellated 7-cube , Runcinated 7-cube , Stericated 7-cube , Pentellated 7-cube , Hexicated 7-cube 7-orthoplex , Rectified 7-orthoplex , Truncated 7-orthoplex , Cantellated 7-orthoplex , Runcinated 7-orthoplex , Stericated 7-orthoplex , Pentellated 7-orthoplex 132 polytope , 231 polytope , 321 polytope
Honeycombs
Eight dimension
Eight-dimensional space , uniform 8-polytope 8-simplex , Rectified 8-simplex , Truncated 8-simplex , Cantellated 8-simplex , Runcinated 8-simplex , Stericated 8-simplex , Pentellated 8-simplex , Hexicated 8-simplex , Heptellated 8-simplex 8-orthoplex , Rectified 8-orthoplex , Truncated 8-orthoplex , Cantellated 8-orthoplex , Runcinated 8-orthoplex , Stericated 8-orthoplex , Pentellated 8-orthoplex , Hexicated 8-orthoplex [citation needed 8-cube , Rectified 8-cube , Truncated 8-cube , Cantellated 8-cube , Runcinated 8-cube , Stericated 8-cube , Pentellated 8-cube , Hexicated 8-cube , Heptellated 8-cube [citation needed 8-demicube , Truncated 8-demicube , Cantellated 8-demicube , Runcinated 8-demicube , Stericated 8-demicube , Pentellated 8-demicube , Hexicated 8-demicube [citation needed 142 polytope , 241 polytope , 421 polytope , Truncated 421 polytope , Truncated 241 polytope , Truncated 142 polytope , Cantellated 421 polytope , Cantellated 241 polytope , Runcinated 421 polytope [citation needed
Honeycombs
Nine dimensions
9-polytope
Hyperbolic honeycombs
Ten dimensions
10-polytope
Dimensional families
Regular polytope and List of regular polytopes Uniform polytope Honeycombs
Geometry
Geometry and other areas of mathematics
Ford circles
Glyphs and symbols
Table of all the Shapes
This is a table of all the shapes above.
References
