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List of mathematical shapes

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

Algebraic curves

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Main article: List of curves

Rational curves

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Degree 2

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Degree 3

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Degree 4

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Degree 5

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Degree 6

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Families of variable degree

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Curves of genus one

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Curves with genus greater than one

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Curve families with variable genus

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Transcendental curves

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Piecewise constructions

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Curves generated by other curves

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Space curves

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Surfaces in 3-space

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Main article: List of surfaces

Minimal surfaces

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Non-orientable surfaces

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Quadrics

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Pseudospherical surfaces

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Algebraic surfaces

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See the list of algebraic surfaces.

Miscellaneous surfaces

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Fractals

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Main article: List of fractals by Hausdorff dimension

Random fractals

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Regular polytopes

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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

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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

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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

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One-dimensional regular polytope

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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

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Convex

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Degenerate (spherical)
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Non-convex

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Tessellation

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Three-dimensional regular polytopes

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Convex

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Degenerate (spherical)

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Non-convex

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Tessellations

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Euclidean tilings
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Hyperbolic tilings
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Hyperbolic star-tilings
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Four-dimensional regular polytopes

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Degenerate (spherical)

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Non-convex

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Tessellations of Euclidean 3-space

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Degenerate tessellations of Euclidean 3-space

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Tessellations of hyperbolic 3-space

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Five-dimensional regular polytopes and higher

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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

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Tessellations of Euclidean 5-space and higher

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Tessellations of hyperbolic 4-space

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Tessellations of hyperbolic 5-space

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Apeirotopes

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Abstract polytopes

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2D with 1D surface

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Polygons named for their number of sides

Tilings

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Uniform polyhedra

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Main article: Uniform polyhedron

Duals of uniform polyhedra

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Johnson solids

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Main article: Johnson solid

Other nonuniform polyhedra

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Spherical polyhedra

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Main article: spherical polyhedron

Honeycombs

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

Other

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Regular and uniform compound polyhedra

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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

Honeycombs

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5D with 4D surfaces

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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]

Honeycombs

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Six dimensions

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

Honeycombs

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Seven dimensions

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Seven-dimensional space, uniform 7-polytope

Honeycombs

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Eight dimension

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Eight-dimensional space, uniform 8-polytope

Honeycombs

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Nine dimensions

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9-polytope

Hyperbolic honeycombs

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Ten dimensions

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10-polytope

Dimensional families

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Regular polytope and List of regular polytopes
Uniform polytope
Honeycombs

Geometry

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Hyperplexicons

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Geometry and other areas of mathematics

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Ford circles

Glyphs and symbols

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Table of all the Shapes

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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

References

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  1. ^ "Courbe a Réaction Constante, Quintique De L'Hospital" [Constant Reaction Curve, Quintic of l'Hospital].
  2. ^ "Isochrone de Leibniz". Archived from the original on 14 November 2004.
  3. ^ "Isochrone de Varignon". Archived from the original on 13 November 2004.
  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. ^ "Self-Avoiding Random Walks - Wolfram Demonstrations Project". WOLFRAM Demonstrations Project. Retrieved 14 June 2019.
  9. ^ Weisstein, Eric W. "Hedgehog". mathworld.wolfram.com.
  10. ^ "Courbe De Ribaucour" [Ribaucour curve]. mathworld.wolfram.com.
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