List of mathematical shapes

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


Algebraic curvesEdit

Transcendental curvesEdit

Piecewise constructionsEdit

Curves generated by other curvesEdit

Space curvesEdit

Surfaces in 3-spaceEdit

Minimal surfacesEdit

Non-orientable surfacesEdit


Pseudospherical surfacesEdit

Algebraic surfacesEdit

See the list of algebraic surfaces.

Miscellaneous surfacesEdit


Random fractalsEdit

Regular PolytopesEdit

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

Dimension Convex Nonconvex Convex
Hyperbolic Tessellations
with infinite cells
and/or vertex figures
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 elementsEdit

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.


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 dimensionEdit

One-dimensional regular polytopeEdit

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 polytopesEdit


Degenerate (spherical)Edit



Three-dimensional regular polytopesEdit


Degenerate (spherical)Edit



Euclidean tilingsEdit
Hyperbolic tilingsEdit
Hyperbolic star-tilingsEdit

Four-dimensional regular polytopesEdit

Degenerate (spherical)Edit


Tessellations of Euclidean 3-spaceEdit

Degenerate tessellations of Euclidean 3-spaceEdit

Tessellations of hyperbolic 3-spaceEdit

Five-dimensional regular polytopes and higherEdit

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-spaceEdit

Tessellations of Euclidean 5-space and higherEdit

Tessellations of hyperbolic 4-spaceEdit

Tessellations of hyperbolic 5-spaceEdit


Abstract polytopesEdit

Non-regular polytopesEdit

2D with 1D surfaceEdit

Polygons named for their number of sides


Uniform polyhedraEdit

Duals of uniform polyhedraEdit

Johnson solidsEdit

Other nonuniform polyhedraEdit

Spherical polyhedraEdit



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


Regular and uniform compound polyhedraEdit

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


5D with 4D surfacesEdit

Six dimensionsEdit

Seven dimensionsEdit

Eight dimensionEdit

Nine dimensionsEdit

Ten dimensionsEdit

Dimensional familiesEdit


Geometry and other areas of mathematicsEdit

Glyphs and symbolsEdit


  1. ^ "Courbe a Réaction Constante, Quintique De L'Hospital" [Constant Reaction Curve, Quintic of l'Hospital].
  2. ^ Archived from the original on 14 November 2004. Missing or empty |title= (help)
  3. ^ Archived from the original on 13 November 2004. Missing or empty |title= (help)
  4. ^ Ferreol, Robert. "Spirale de Galilée".
  5. ^ Weisstein, Eric W. "Seiffert's Spherical Spiral".
  6. ^ Weisstein, Eric W. "Slinky".
  7. ^ "Monkeys tree fractal curve". Archived from the original on 21 September 2002.
  8. ^ WOLFRAM Demonstrations Project Retrieved 14 June 2019. Missing or empty |title= (help)
  9. ^ Weisstein, Eric W. "Hedgehog".
  10. ^ "Courbe De Ribaucour" [Ribaucour curve].