List of mathematical shapes

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

Algebraic 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

Minimal surfaces

Non-orientable surfaces

Quadrics

Pseudospherical surfaces

Algebraic 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.

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
2polygonsstar 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

Euclidean tilings
Hyperbolic tilings
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

2D with 1D surface

Polygons named for their number of sides

Tilings

Uniform polyhedra

Duals of uniform polyhedra

Johnson solids

Other nonuniform polyhedra

Spherical polyhedra

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
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

Honeycombs

Seven dimensions

Seven-dimensional space, uniform 7-polytope

Honeycombs

Eight dimension

Eight-dimensional space, uniform 8-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

Glyphs and symbols

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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