List of mathematical shapes

From Wikipedia, the free encyclopedia

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]

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.