# List of mathematical shapes

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

## Algebraic surfaces

See the list of algebraic surfaces.

## 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.

### 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 {}.

### 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

## 2D with 1D surface

Polygons named for their number of sides

### Honeycombs

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

### 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

## 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.[citation needed]

## Six dimensions

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

## Seven dimensions

Seven-dimensional space, uniform 7-polytope

## Eight dimension

Eight-dimensional space, uniform 8-polytope

9-polytope

10-polytope

## Dimensional families

Regular polytope and List of regular polytopes
Uniform polytope
Honeycombs

## 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.