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List of mathematical shapes

Summary

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

Algebraic curves edit

Rational curves edit

Degree 2 edit

Degree 3 edit

Degree 4 edit

Degree 5 edit

  • Quintic of l'Hospital[1]

Degree 6 edit

Families of variable degree edit

Curves of genus one edit

Curves with genus greater than one edit

Curve families with variable genus edit

Transcendental curves edit

Piecewise constructions edit

Curves generated by other curves edit

Space curves edit

Surfaces in 3-space edit

Minimal surfaces edit

Non-orientable surfaces edit

Quadrics edit

Pseudospherical surfaces edit

Algebraic surfaces edit

See the list of algebraic surfaces.

Miscellaneous surfaces edit

Fractals edit

Random fractals edit

Regular polytopes edit

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 edit

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 edit

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 edit

One-dimensional regular polytope edit

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 edit

Convex edit

Degenerate (spherical) edit

Non-convex edit

Tessellation edit

Three-dimensional regular polytopes edit

Convex edit

Degenerate (spherical) edit

Non-convex edit

Tessellations edit

Euclidean tilings edit
Hyperbolic tilings edit
Hyperbolic star-tilings edit

Four-dimensional regular polytopes edit

Degenerate (spherical) edit

Non-convex edit

Tessellations of Euclidean 3-space edit

Degenerate tessellations of Euclidean 3-space edit

Tessellations of hyperbolic 3-space edit

Five-dimensional regular polytopes and higher edit

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 edit

Tessellations of Euclidean 5-space and higher edit

Tessellations of hyperbolic 4-space edit

Tessellations of hyperbolic 5-space edit

Apeirotopes edit

Abstract polytopes edit

2D with 1D surface edit

Polygons named for their number of sides

Tilings edit

Uniform polyhedra edit

Duals of uniform polyhedra edit

Johnson solids edit

Other nonuniform polyhedra edit

Spherical polyhedra edit

Honeycombs edit

Convex uniform honeycomb
Dual uniform honeycomb
Others
Convex uniform honeycombs in hyperbolic space

Other edit

Regular and uniform compound polyhedra edit

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 edit

5D with 4D surfaces edit

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.[citation needed]

Honeycombs edit

Six dimensions edit

Six-dimensional space, 6-polytope and uniform 6-polytope

Honeycombs edit

Seven dimensions edit

Seven-dimensional space, uniform 7-polytope

Honeycombs edit

Eight dimension edit

Eight-dimensional space, uniform 8-polytope

Honeycombs edit

Nine dimensions edit

9-polytope

Hyperbolic honeycombs edit

Ten dimensions edit

10-polytope

Dimensional families edit

Regular polytope and List of regular polytopes
Uniform polytope
Honeycombs

Geometry edit

Geometry and other areas of mathematics edit

 
Ford circles

Glyphs and symbols edit

References edit

  1. ^ "Courbe a Réaction Constante, Quintique De L'Hospital" [Constant Reaction Curve, Quintic of l'Hospital].
  2. ^ "Isochrone de Leibniz". Archived from the original on 14 November 2004.
  3. ^ "Isochrone de Varignon". Archived from the original on 13 November 2004.
  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. ^ "Self-Avoiding Random Walks - Wolfram Demonstrations Project". WOLFRAM Demonstrations Project. Retrieved 14 June 2019.
  9. ^ Weisstein, Eric W. "Hedgehog". mathworld.wolfram.com.
  10. ^ "Courbe De Ribaucour" [Ribaucour curve]. mathworld.wolfram.com.

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