List of mathematical shapes

From Infogalactic: the planetary knowledge core
Jump to: navigation, search

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

Contents

Algebraic curves

Rational curves

Degree 2

Degree 3

<templatestyles src="Div col/styles.css"/>

Degree 4

<templatestyles src="Div col/styles.css"/>

Degree 5

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

<templatestyles src="Div col/styles.css"/>

Space curves

Surfaces in 3-space

<templatestyles src="Module:Hatnote/styles.css"></templatestyles>

Main article: List of surfaces

Minimal surfaces

Non-orientable surfaces

Quadrics

Pseudospherical surfaces

Algebraic surfaces

See the list of algebraic surfaces.

Miscellaneous surfaces

Fractals

<templatestyles src="Module:Hatnote/styles.css"></templatestyles>

Main article: List of fractals by Hausdorff dimension

<templatestyles src="Div col/styles.css"/>

Random fractals

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.

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

Degenerate (spherical)

Non-convex

Tessellation

Three-dimensional regular polytopes

Convex

Degenerate (spherical)

Non-convex

Tessellations

Euclidean tilings
Hyperbolic tilings
Hyperbolic star-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

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

Tessellations of Euclidean 5-space and higher

Tessellations of hyperbolic 4-space

Tessellations of hyperbolic 5-space

Apeirotopes

Abstract polytopes

Non-regular polytopes

Lua error in package.lua at line 80: module 'strict' not found.

2D with 1D surface

Polygons named for their number of sides

<templatestyles src="Div col/styles.css"/>

Tilings

Uniform polyhedra

<templatestyles src="Module:Hatnote/styles.css"></templatestyles>

Main article: Uniform polyhedron

<templatestyles src="Div col/styles.css"/>

Duals of uniform polyhedra

  • non-convex

<templatestyles src="Div col/styles.css"/>

Johnson solids

<templatestyles src="Module:Hatnote/styles.css"></templatestyles>

Main article: Johnson solid

<templatestyles src="Div col/styles.css"/>

Other nonuniform polyhedra

Spherical polyhedra

<templatestyles src="Module:Hatnote/styles.css"></templatestyles>

Main article: spherical polyhedron

Honeycombs

Honeycombs

Convex uniform honeycomb

<templatestyles src="Div col/styles.css"/>

Dual uniform honeycomb
Others
Convex uniform honeycombs in hyperbolic space

Other

<templatestyles src="Div col/styles.css"/>

Regular and uniform compound polyhedra

Polyhedral compound and Uniform polyhedron compound

<templatestyles src="Div col/styles.css"/>

Convex regular 4-polytope
Abstract regular polytope
Schläfli–Hess 4-polytope (Regular star 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 dimension

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

<templatestyles src="Div col/styles.css"/>

Geometry and other areas of mathematics

Ford circles

<templatestyles src="Div col/styles.css"/>

Glyphs and symbols

<templatestyles src="Div col/styles.css"/>

References

<templatestyles src="Reflist/styles.css" />

Cite error: Invalid <references> tag; parameter "group" is allowed only.

Use <references />, or <references group="..." />
Retrieved from "https://infogalactic.com/w/index.php?title=List_of_mathematical_shapes&oldid=165560"