List of mathematical shapes


Following is a list of shapes studied in mathematics.

[Algebraic curve]s

Rational curves

Degree 2

Degree 3

Degree 4

Degree 5

  • Quintic of l'Hospital

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 surface]s

Non-orientable">orientability">Non-orientable surfaces

[Quadric]s

Pseudospherical surfaces

[Algebraic surface]s

See the list of algebraic surfaces.

Miscellaneous surfaces

Fractals

Random fractals

Regular polytopes

This table shows a summary of regular polytope counts by dimension.
DimensionConvexNonconvexConvex
Euclidean
tessellations		Convex
hyperbolic
tessellations		Nonconvex
hyperbolic
tessellations		Hyperbolic Tessellations
with infinite cells
and/or vertex figures		Abstract
Polytopes
11 line segment010001
2∞ polygons∞ star polygons1100∞
35 [|Platonic solids]4 Kepler–Poinsot solids3 [|tilings]∞∞∞∞
46 [|convex polychora]10 [|Schläfli–Hess polychora]1 [|honeycomb]4011∞
53 [|convex 5-polytopes]03 [|tetracombs]542∞
63 [|convex 6-polytopes]01 [|pentacombs]005∞
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 -dimensional element
  • Ridge, an -dimensional element
  • Peak, an -dimensional element
For example, in a polyhedron, 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 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

Non-convex

Tessellation

Three-dimensional regular polytopes

Convex

Degenerate (spherical)

Non-convex

Tessellations

Euclidean 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

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
;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
;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 dimensions

;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

Table of all the Shapes

This is a table of all the shapes above.
SectionSub-SectionSup-SectionName
Algebraic Curves¿ Curves¿ CurvesCubic Plane Curve
Algebraic Curves¿ Curves¿ CurvesQuartic Plane Curve
Algebraic CurvesRational CurvesDegree 2Conic Section
Algebraic CurvesRational CurvesDegree 2Unit Circle
Algebraic CurvesRational CurvesDegree 2Unit Hyperbola
Algebraic CurvesRational CurvesDegree 3Folium of Descartes
Algebraic CurvesRational CurvesDegree 3Cissoid of Diocles
Algebraic CurvesRational CurvesDegree 3Conchoid of de Sluze
Algebraic CurvesRational CurvesDegree 3Right Strophoid
Algebraic CurvesRational CurvesDegree 3Semicubical Parabola
Algebraic CurvesRational CurvesDegree 3Serpentine Curve
Algebraic CurvesRational CurvesDegree 3Trident Curve
Algebraic CurvesRational CurvesDegree 3Trisectrix of Maclaurin
Algebraic CurvesRational CurvesDegree 3Tschirnhausen Cubic
Algebraic CurvesRational CurvesDegree 3Witch of Agnesi
Algebraic CurvesRational CurvesDegree 4Ampersand Curve
Algebraic CurvesRational CurvesDegree 4Bean Curve
Algebraic CurvesRational CurvesDegree 4Bicorn
Algebraic CurvesRational CurvesDegree 4Bow Curve
Algebraic CurvesRational CurvesDegree 4Bullet-Nose Curve
Algebraic CurvesRational CurvesDegree 4Cruciform Curve
Algebraic CurvesRational CurvesDegree 4
Algebraic CurvesRational CurvesDegree 4
Algebraic CurvesRational CurvesDegree 4