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List of mathematical shapes

Article Discussion

Template:Short description Following is a list of shapes studied in mathematics.

Algebraic curvesEdit

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Rational curvesEdit

Degree 2Edit

Degree 3Edit

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Degree 4Edit

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Degree 5Edit

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Degree 6Edit

Families of variable degreeEdit

Curves of genus oneEdit

Curves with genus greater than oneEdit

Curve families with variable genusEdit

Transcendental curvesEdit

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Piecewise constructionsEdit

Curves generated by other curvesEdit

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Space curvesEdit

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Surfaces in 3-spaceEdit

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Minimal surfacesEdit

Non-orientable surfacesEdit

QuadricsEdit

Pseudospherical surfacesEdit

Algebraic surfacesEdit

See the list of algebraic surfaces.

Miscellaneous surfacesEdit

FractalsEdit

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Random fractalsEdit

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Regular polytopesEdit

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 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.

TessellationsEdit

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

ConvexEdit

Degenerate (spherical)Edit

Non-convexEdit

TessellationEdit

Three-dimensional regular polytopesEdit

ConvexEdit

Degenerate (spherical)Edit

Non-convexEdit

TessellationsEdit

Euclidean tilingsEdit
Hyperbolic tilingsEdit
Hyperbolic star-tilingsEdit

Four-dimensional regular polytopesEdit

Degenerate (spherical)Edit

Non-convexEdit

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

ApeirotopesEdit

Abstract polytopesEdit

2D with 1D surfaceEdit

Polygons named for their number of sides Template:Div col

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TilingsEdit

Uniform polyhedraEdit

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Duals of uniform polyhedraEdit

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Johnson solidsEdit

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Other nonuniform polyhedraEdit

Spherical polyhedraEdit

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HoneycombsEdit

Convex uniform honeycomb

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Dual uniform honeycomb
Others
Convex uniform honeycombs in hyperbolic space

OtherEdit

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Regular and uniform compound polyhedraEdit

Polyhedral compound and Uniform polyhedron compound

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Convex regular 4-polytope
Abstract regular polytope
Schläfli–Hess 4-polytope (Regular star 4-polytope)
Uniform 4-polytope
Prismatic uniform polychoron

HoneycombsEdit

5D with 4D surfacesEdit

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.Template:Citation needed

HoneycombsEdit

Six dimensionsEdit

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

HoneycombsEdit

Seven dimensionsEdit

Seven-dimensional space, uniform 7-polytope

HoneycombsEdit

Eight dimensionEdit

Eight-dimensional space, uniform 8-polytope

HoneycombsEdit

Nine dimensionsEdit

9-polytope

Hyperbolic honeycombsEdit

Ten dimensionsEdit

10-polytope

Dimensional familiesEdit

Regular polytope and List of regular polytopes
Uniform polytope
Honeycombs

GeometryEdit

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Geometry and other areas of mathematicsEdit

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Ford circles

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Glyphs and symbolsEdit

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Table of all the ShapesEdit

This is a table of all the shapes above.

Table of Shapes
Section Sub-Section Sup-Section Name
Algebraic Curves ¿ Curves ¿ Curves Cubic Plane Curve
Quartic Plane Curve
Rational Curves Degree 2 Conic Section(s)
Unit Circle
Unit Hyperbola
Degree 3 Folium of Descartes
Cissoid of Diocles
Conchoid of de Sluze
Right Strophoid
Semicubical Parabola
Serpentine Curve
Trident Curve
Trisectrix of Maclaurin
Tschirnhausen Cubic
Witch of Agnesi
Degree 4 Ampersand Curve
Bean Curve
Bicorn
Bow Curve
Bullet-Nose Curve
Cruciform Curve

ReferencesEdit

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