List of mathematical shapes

Following is a list of some mathematically well-defined shapes.

Algebraic curves

Rational curves

Degree 2

Degree 3

Degree 4

Degree 5
  • Quintic of l'Hospital[1]

Degree 6

Families of variable degree

Curves of genus one

Curves with genus greater than one

Curve families with variable genus

Transcendental curves

Piecewise constructions

Curves generated by other curves

Space curves

Surfaces in 3-space
Main article: List of surfaces

Minimal surfaces

Non-orientable surfaces

Quadrics

Pseudospherical surfaces

Algebraic surfaces

See the list of algebraic surfaces.

Miscellaneous surfaces

Fractals
Main article: List of fractals by Hausdorff dimension

Random fractals

Regular polytopes

This table shows a summary of regular polytope counts by dimension.

Dimension Convex Nonconvex Convex
Euclidean
tessellations		 Convex
hyperbolic
tessellations		 Nonconvex
hyperbolic
tessellations		 Hyperbolic Tessellations
with infinite cells
and/or vertex figures		 Abstract
Polytopes
11 line segment010001
2∞ polygons∞ star polygons1100∞
35 Platonic solids4 Kepler–Poinsot solids3 tilings∞∞∞∞
46 convex polychora10 Schläfli–Hess polychora1 honeycomb4011∞
53 convex 5-polytopes03 tetracombs542∞
63 convex 6-polytopes01 pentacombs005∞
7+301000∞

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 element
  • Edge, a 1-dimensional element
  • Face, a 2-dimensional element
  • Cell, a 3-dimensional element
  • Hypercell or Teron, a 4-dimensional element
  • Facet, an (n-1)-dimensional element
  • Ridge, an (n-2)-dimensional element
  • Peak, an (n-3)-dimensional element

For 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
Degenerate (spherical)

Non-convex

Tessellation

Three-dimensional regular polytopes

Convex

Degenerate (spherical)

Non-convex

Tessellations
Euclidean tilings
Hyperbolic tilings
  • Lobachevski plane
  • Hyperbolic tiling
Hyperbolic star-tilings

Four-dimensional regular polytopes

Degenerate (spherical)

Non-convex

Tessellations of Euclidean 3-space

Degenerate tessellations of Euclidean 3-space

Tessellations of hyperbolic 3-space

Five-dimensional regular polytopes and higher
SimplexHypercubeCross-polytope
5-simplex5-cube5-orthoplex
6-simplex6-cube6-orthoplex
7-simplex7-cube7-orthoplex
8-simplex8-cube8-orthoplex
9-simplex9-cube9-orthoplex
10-simplex10-cube10-orthoplex
11-simplex11-cube11-orthoplex

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

Non-regular polytopes

2D with 1D surface

Polygons named for their number of sides

Tilings

Uniform polyhedra
Main article: Uniform polyhedron

Duals of uniform polyhedra

Johnson solids
Main article: Johnson solid

Other nonuniform polyhedra

Spherical polyhedra
Main article: spherical polyhedron

Honeycombs
Convex uniform honeycomb
  • Cubic honeycomb
  • Truncated cubic honeycomb
  • Bitruncated cubic honeycomb
  • Cantellated cubic honeycomb
  • Cantitruncated cubic honeycomb
  • Rectified cubic honeycomb
  • Runcitruncated cubic honeycomb
  • Omnitruncated cubic honeycomb
  • Tetrahedral-octahedral honeycomb
  • Truncated alternated cubic honeycomb
  • Cantitruncated alternated cubic honeycomb
  • Runcinated alternated cubic honeycomb
  • Quarter cubic honeycomb
  • Gyrated tetrahedral-octahedral honeycomb
  • Gyrated triangular prismatic honeycomb
  • Gyroelongated alternated cubic honeycomb
  • Gyroelongated triangular prismatic honeycomb
  • Elongated triangular prismatic honeycomb
  • Elongated alternated cubic honeycomb
  • Hexagonal prismatic honeycomb
  • Triangular prismatic honeycomb
  • Triangular-hexagonal prismatic honeycomb
  • Truncated hexagonal prismatic honeycomb
  • Truncated square prismatic honeycomb
  • Rhombitriangular-hexagonal prismatic honeycomb
  • Omnitruncated triangular-hexagonal prismatic honeycomb
  • Snub triangular-hexagonal prismatic honeycomb
  • Snub square prismatic honeycomb
Dual uniform honeycomb
Others
Convex uniform honeycombs in hyperbolic space

Other

Regular and uniform compound polyhedra
Polyhedral compound and Uniform polyhedron compound
Convex regular 4-polytope
Abstract regular polytope
Schläfli–Hess 4-polytope (Regular star 4-polytope)
Uniform 4-polytope
Prismatic uniform polychoron

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.

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
  • 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
  • 8-demicube, Truncated 8-demicube, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube
  • 142 polytope, 241 polytope, 421 polytope, Truncated 421 polytope, Truncated 241 polytope, Truncated 142 polytope, Cantellated 421 polytope, Cantellated 241 polytope, Runcinated 421 polytope

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
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References
  1. "Courbe a Réaction Constante, Quintique De L'Hospital" [Constant Reaction Curve, Quintic of l'Hospital].
  2. https://web.archive.org/web/20041114002246/http://www.mathcurve.com/courbes2d/isochron/isochrone%20leibniz. Archived from the original on 14 November 2004. Missing or empty |title= (help)
  3. https://web.archive.org/web/20041113201905/http://www.mathcurve.com/courbes2d/isochron/isochrone%20varignon. Archived from the original on 13 November 2004. Missing or empty |title= (help)
  4. Ferreol, Robert. "Spirale de Galilée". www.mathcurve.com.
  5. Weisstein, Eric W. "Seiffert's Spherical Spiral". mathworld.wolfram.com.
  6. Weisstein, Eric W. "Slinky". mathworld.wolfram.com.
  7. "Monkeys tree fractal curve". Archived from the original on 21 September 2002.
  8. WOLFRAM Demonstrations Project http://demonstrations.wolfram.com/SelfAvoidingRandomWalks/#more. Retrieved 14 June 2019. Missing or empty |title= (help)
  9. Weisstein, Eric W. "Hedgehog". mathworld.wolfram.com.
  10. "Courbe De Ribaucour" [Ribaucour curve]. mathworld.wolfram.com.
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