List of mathematical shapes

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

Algebraic curves

Main page: List of 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 page: List of surfaces

Minimal surfaces

Non-orientable surfaces

Quadrics

Pseudospherical surfaces

Algebraic surfaces

See the list of algebraic surfaces.

Miscellaneous surfaces

Fractals

Main page: 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
1 1 line segment 0 1 0 0 0 1
2 ∞ polygons ∞ star polygons 1 1 0 0 ∞
3 5 Platonic solids 4 Kepler–Poinsot solids 3 tilings ∞ ∞ ∞ ∞
4 6 convex polychora 10 Schläfli–Hess polychora 1 honeycomb 4 0 11 ∞
5 3 convex 5-polytopes 0 3 tetracombs 5 4 2 ∞
6 3 convex 6-polytopes 0 1 pentacombs 0 0 5 ∞
7+ 3 0 1 0 0 0 ∞

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

Simplex Hypercube Cross-polytope
5-simplex 5-cube 5-orthoplex
6-simplex 6-cube 6-orthoplex
7-simplex 7-cube 7-orthoplex
8-simplex 8-cube 8-orthoplex
9-simplex 9-cube 9-orthoplex
10-simplex 10-cube 10-orthoplex
11-simplex 11-cube 11-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

2D with 1D surface

Polygons named for their number of sides


Tilings

Uniform polyhedra

Main page: Uniform polyhedron


Duals of uniform polyhedra

  • non-convex
    • Great complex icosidodecahedron
    • Great deltoidal hexecontahedron
    • Great deltoidal icositetrahedron
    • Great dirhombicosidodecacron
    • Great dirhombicosidodecahedron
    • Great disdyakis dodecahedron
    • Great disdyakis triacontahedron
    • Great disnub dirhombidodecacron
    • Great ditrigonal dodecacronic hexecontahedron
    • Great dodecacronic hexecontahedron
    • Great dodecahemicosacron
    • Great dodecicosacron
    • Great hexacronic icositetrahedron
    • Great hexagonal hexecontahedron
    • Great icosacronic hexecontahedron
    • Great icosihemidodecacron
    • Great inverted pentagonal hexecontahedron
    • Great pentagonal hexecontahedron
    • Great pentagrammic hexecontahedron
    • Great pentakis dodecahedron
    • Great rhombic triacontahedron
    • Great rhombidodecacron
    • Great rhombihexacron
    • Great stellapentakis dodecahedron
    • Great triakis icosahedron
    • Great triakis octahedron
    • Great triambic icosahedron
    • Medial deltoidal hexecontahedron
    • Medial disdyakis triacontahedron
    • Medial hexagonal hexecontahedron
    • Medial icosacronic hexecontahedron
    • Medial inverted pentagonal hexecontahedron
    • Medial pentagonal hexecontahedron
    • Medial rhombic triacontahedron
    • Hexahemioctacron
    • Hemipolyhedron
    • Octahemioctacron
    • Rhombicosacron
    • Small complex icosidodecahedron
    • Small ditrigonal dodecacronic hexecontahedron
    • Small dodecacronic hexecontahedron
    • Small dodecahemicosacron
    • Small dodecahemidodecacron
    • Small dodecicosacron
    • Small hexacronic icositetrahedron
    • Small hexagonal hexecontahedron
    • Small hexagrammic hexecontahedron
    • Small icosacronic hexecontahedron
    • Small icosihemidodecacron
    • Small rhombidodecacron
    • Small rhombihexacron
    • Small stellapentakis dodecahedron
    • Small triambic icosahedron
    • Tetrahemihexacron


Johnson solids

Main page: Johnson solid


Other nonuniform polyhedra

Spherical polyhedra

Main page: Spherical polyhedron

Honeycombs

Convex uniform 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.[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


References

  1. ↑ "Courbe a Réaction Constante, Quintique De L'Hospital". http://www.mathcurve.com/courbes2d/quintique%20de%20l'hospital/quintique%20de%20l'hospital.shtml. 
  2. ↑ "Isochrone de Leibniz". Archived on 14 November 2004. Error: If you specify |archivedate=, you must also specify |archiveurl=. http://www.mathcurve.com/courbes2d/isochron/isochrone%20leibniz. 
  3. ↑ "Isochrone de Varignon". Archived on 13 November 2004. Error: If you specify |archivedate=, you must also specify |archiveurl=. http://www.mathcurve.com/courbes2d/isochron/isochrone%20varignon. 
  4. ↑ Ferreol, Robert. "Spirale de Galilée". http://www.mathcurve.com/courbes2d/galilee/galilee.shtml. 
  5. ↑ Weisstein, Eric W.. "Seiffert's Spherical Spiral". http://mathworld.wolfram.com/SeiffertsSphericalSpiral.html. 
  6. ↑ Weisstein, Eric W.. "Slinky". http://mathworld.wolfram.com/Slinky.html. 
  7. ↑ "Monkeys tree fractal curve". http://www.coaauw.org/boulder-eyh/eyh_fractal.html. 
  8. ↑ "Self-Avoiding Random Walks - Wolfram Demonstrations Project". http://demonstrations.wolfram.com/SelfAvoidingRandomWalks/#more. Retrieved 14 June 2019. 
  9. ↑ Weisstein, Eric W.. "Hedgehog". http://mathworld.wolfram.com/Hedgehog.html. 
  10. ↑ "Courbe De Ribaucour". http://www.mathcurve.com/courbes2d/ribaucour/ribaucour.shtml. 



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