Wikipedia

List of mathematical shapes

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

Algebraic curves

Rational curves

Degree 2

Degree 3

Degree 4

Degree 5

  • Quintic of l'Hospital[1]

Degree 6

Families of variable degree

Curves of genus one

  • Bicuspid curve
  • Cassini oval
  • Cassinoide
  • Cubic curve
  • Elliptic curve
  • Watt's curve

Curves with genus greater than one

Curve families with variable genus

Transcendental curves

Piecewise constructions

Curves generated by other curves

Space curves

  • Conchospiral
  • Helix
    • Tendril perversion (a transition between back-to-back helices)
    • Hemihelix, a quasi-helical shape characterized by multiple tendril perversions
  • Seiffert's spiral[5]
  • Slinky spiral[6]
  • Twisted cubic
  • Viviani's curve

Surfaces in 3-space

Main article: List of surfaces

Minimal surfaces

  • Catalan's minimal surface
  • Costa's minimal surface
  • Catenoid
  • Enneper surface
  • Gyroid
  • Helicoid
  • Lidinoid
  • Riemann's minimal surface
  • Saddle tower
  • Scherk surface
  • Schwarz minimal surface
  • Triply periodic minimal surface

Non-orientable surfaces

Quadrics

Pseudospherical surfaces

Algebraic surfaces

See the list of algebraic surfaces.

Miscellaneous surfaces

Fractals

Main article: List of fractals by Hausdorff dimension

Random fractals

  • von Koch curve with random interval
  • von Koch curve with random orientation
  • Boundary of Brownian motion
  • 2D polymer
  • Percolation front in 2D, Corrosion front in 2D
  • diffusion-limited aggregation
  • Random walk with no self-intersection[8]
  • 3D polymer
  • 2D percolation cluster hull
  • 2D percolation cluster
  • Brownian motion
  • Lichtenberg figure
  • Percolation theory
  • Multiplicative cascade

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
  • Lobachevski plane
  • Hyperbolic tiling
Hyperbolic star-tilings
  • Order-7 heptagrammic tiling
  • Heptagrammic-order heptagonal tiling
  • Order-9 enneagrammic tiling
  • Enneagrammic-order enneagonal tiling

Four-dimensional regular polytopes

  • convex regular 4-polytope
    • 5-cell, the 4-space Simplex
    • 8-cell, the 4-space Hypercube
    • 16-cell, the 4-space Cross-polytope
    • 24-cell
    • 120-cell
    • 600-cell

Degenerate (spherical)

  • Ditope
  • Hosotope
  • 3-sphere

Non-convex

Tessellations of Euclidean 3-space

Degenerate tessellations of Euclidean 3-space

  • Hosohedron
  • Dihedron
  • Order-2 apeirogonal tiling
  • Apeirogonal hosohedron
  • Order-4 square hosohedral honeycomb
  • Order-6 triangular hosohedral honeycomb
  • Hexagonal hosohedral honeycomb
  • Order-2 square tiling honeycomb
  • Order-2 triangular tiling honeycomb
  • Order-2 hexagonal tiling honeycomb

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

  • honeycombs
  • Tesseractic honeycomb
  • 16-cell honeycomb
  • 24-cell honeycomb

Tessellations of Euclidean 5-space and higher

Tessellations of hyperbolic 4-space

  • honeycombs
  • Order-5 5-cell honeycomb
  • 120-cell honeycomb
  • Order-5 tesseractic honeycomb
  • Order-4 120-cell honeycomb
  • Order-5 120-cell honeycomb
  • Order-4 24-cell honeycomb
  • Cubic honeycomb honeycomb
  • Small stellated 120-cell honeycomb
  • Pentagrammic-order 600-cell honeycomb
  • Order-5 icosahedral 120-cell honeycomb
  • Great 120-cell honeycomb

Tessellations of hyperbolic 5-space

  • 5-orthoplex honeycomb
  • 24-cell honeycomb honeycomb
  • 16-cell honeycomb honeycomb
  • Order-4 24-cell honeycomb honeycomb
  • Tesseractic honeycomb honeycomb

Apeirotopes

Abstract polytopes

2D with 1D surface

Polygons named for their number of sides

Tilings

Uniform polyhedra

Main article: Uniform polyhedron
  • Uniform star polyhedron

Duals of uniform polyhedra

  • non-convex
    • Great complex icosidodecahedron
    • Great deltoidal hexecontahedron
    • Great deltoidal icositetrahedron
    • Great dirhombicosidodecacron
    • Great dirhombicosidodecahedron
    • Great disdyakis dodecahedron
    • Great disdyakis triacontahedron
    • Great disnub dirhombidodecacron
    • Great ditrigonal dodecacronic hexecontahedron
    • Great dodecacronic hexecontahedron
    • Great dodecahemicosacron
    • Great dodecicosacron
    • Great hexacronic icositetrahedron
    • Great hexagonal hexecontahedron
    • Great icosacronic hexecontahedron
    • Great icosihemidodecacron
    • Great inverted pentagonal hexecontahedron
    • Great pentagonal hexecontahedron
    • Great pentagrammic hexecontahedron
    • Great pentakis dodecahedron
    • Great rhombic triacontahedron
    • Great rhombidodecacron
    • Great rhombihexacron
    • Great stellapentakis dodecahedron
    • Great triakis icosahedron
    • Great triakis octahedron
    • Great triambic icosahedron
    • Medial deltoidal hexecontahedron
    • Medial disdyakis triacontahedron
    • Medial hexagonal hexecontahedron
    • Medial icosacronic hexecontahedron
    • Medial inverted pentagonal hexecontahedron
    • Medial pentagonal hexecontahedron
    • Medial rhombic triacontahedron
    • Hexahemioctacron
    • Hemipolyhedron
    • Octahemioctacron
    • Rhombicosacron
    • Small complex icosidodecahedron
    • Small ditrigonal dodecacronic hexecontahedron
    • Small dodecacronic hexecontahedron
    • Small dodecahemicosacron
    • Small dodecahemidodecacron
    • Small dodecicosacron
    • Small hexacronic icositetrahedron
    • Small hexagonal hexecontahedron
    • Small hexagrammic hexecontahedron
    • Small icosacronic hexecontahedron
    • Small icosihemidodecacron
    • Small rhombidodecacron
    • Small rhombihexacron
    • Small stellapentakis dodecahedron
    • Small triambic icosahedron
    • Tetrahemihexacron

Johnson solids

Main article: Johnson solid

Other nonuniform polyhedra

Spherical polyhedra

Main article: spherical polyhedron

Honeycombs

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

Other

Regular and uniform compound polyhedra

Polyhedral compound and Uniform polyhedron compound
  • Compound of cube and octahedron
  • Compound of dodecahedron and icosahedron
  • Compound of eight octahedra with rotational freedom
  • Compound of eight triangular prisms
  • Compound of five cubes
  • Compound of five cuboctahedra
  • Compound of five cubohemioctahedra
  • Compound of five great cubicuboctahedra
  • Compound of five great dodecahedra
  • Compound of five great icosahedra
  • Compound of five great rhombihexahedra
  • Compound of five icosahedra
  • Compound of five octahedra
  • Compound of five octahemioctahedra
  • Compound of five small cubicuboctahedra
  • Compound of five small rhombicuboctahedra
  • Compound of five small rhombihexahedra
  • Compound of five small stellated dodecahedra
  • Compound of five stellated truncated cubes
  • Compound of five tetrahedra
  • Compound of five tetrahemihexahedra
  • Compound of five truncated cubes
  • Compound of five truncated tetrahedra
  • Compound of five uniform great rhombicuboctahedra
  • Compound of four hexagonal prisms
  • Compound of four octahedra
  • Compound of four octahedra with rotational freedom
  • Compound of four tetrahedra
  • Compound of four triangular prisms
  • Compound of great icosahedron and great stellated dodecahedron
  • Compound of six cubes with rotational freedom
  • Compound of six decagonal prisms
  • Compound of six decagrammic prisms
  • Compound of six pentagonal antiprisms
  • Compound of six pentagonal prisms
  • Compound of six pentagrammic antiprisms
  • Compound of six pentagrammic crossed antiprisms
  • Compound of six pentagrammic prisms
  • Compound of six square antiprisms
  • Compound of six tetrahedra
  • Compound of six tetrahedra with rotational freedom
  • Compound of small stellated dodecahedron and great dodecahedron
  • Compound of ten hexagonal prisms
  • Compound of ten octahedra
  • Compound of ten tetrahedra
  • Compound of ten triangular prisms
  • Compound of ten truncated tetrahedra
  • Compound of three cubes
  • Compound of three square antiprisms
  • Compound of three tetrahedra
  • Compound of twelve pentagonal antiprisms with rotational freedom
  • Compound of twelve pentagonal prisms
  • Compound of twelve pentagrammic antiprisms
  • Compound of twelve pentagrammic crossed antiprisms with rotational freedom
  • Compound of twelve pentagrammic prisms
  • Compound of twelve tetrahedra with rotational freedom
  • Compound of twenty octahedra
  • Compound of twenty octahedra with rotational freedom
  • Compound of twenty tetrahemihexahedra
  • Compound of twenty triangular prisms
  • Compound of two great dodecahedra
  • Compound of two great icosahedra
  • Compound of two great inverted snub icosidodecahedra
  • Compound of two great retrosnub icosidodecahedra
  • Compound of two great snub icosidodecahedra
  • Compound of two icosahedra
  • Compound of two inverted snub dodecadodecahedra
  • Compound of two small stellated dodecahedra
  • Compound of two snub cubes
  • Compound of two snub dodecadodecahedra
  • Compound of two snub dodecahedra
  • Compound of two snub icosidodecadodecahedra
  • Compound of two truncated tetrahedra
  • Prismatic compound of antiprisms
  • Prismatic compound of antiprisms with rotational freedom
  • Prismatic compound of prisms
  • Prismatic compound of prisms with rotational freedom
  • 4-polytope
  • Spherical cone
Convex regular 4-polytope
  • 5-cell, Tesseract, 16-cell, 24-cell, 120-cell, 600-cell
Abstract regular polytope
Schläfli–Hess 4-polytope (Regular star 4-polytope)
Uniform 4-polytope
Prismatic uniform polychoron
  • Grand antiprism
  • Duoprism
  • Tetrahedral prism, Truncated tetrahedral prism
  • Truncated cubic prism, Truncated octahedral prism, Cuboctahedral prism, Rhombicuboctahedral prism, Truncated cuboctahedral prism, Snub cubic prism
  • Truncated dodecahedral prism, Truncated icosahedral prism, Icosidodecahedral prism, Rhombicosidodecahedral prism, Truncated icosidodecahedral prism, Snub dodecahedral prism
  • Uniform antiprismatic prism

Honeycombs

  • Tesseractic honeycomb
  • 24-cell honeycomb
  • Snub 24-cell honeycomb
  • Rectified 24-cell honeycomb
  • Truncated 24-cell honeycomb
  • 16-cell honeycomb
  • 5-cell honeycomb
  • Omnitruncated 5-cell honeycomb
  • Truncated 5-cell honeycomb
  • Omnitruncated 5-simplex honeycomb

5D with 4D surfaces

  • regular 5-polytope
Five-dimensional space, 5-polytope and uniform 5-polytope
  • 5-simplex, Rectified 5-simplex, Truncated 5-simplex, Cantellated 5-simplex, Runcinated 5-simplex, Stericated 5-simplex
  • 5-demicube, Truncated 5-demicube, Cantellated 5-demicube, Runcinated 5-demicube
  • 5-cube, Rectified 5-cube, 5-cube, Truncated 5-cube, Cantellated 5-cube, Runcinated 5-cube, Stericated 5-cube
  • 5-orthoplex, Rectified 5-orthoplex, Truncated 5-orthoplex, Cantellated 5-orthoplex, Runcinated 5-orthoplex
Prismatic uniform 5-polytope
For each polytope of dimension n, there is a prism of dimension n+1.

Honeycombs

  • 5-cubic honeycomb
  • 5-simplex honeycomb
  • Truncated 5-simplex honeycomb
  • 5-demicubic honeycomb

Six dimensions

Six-dimensional space, 6-polytope and uniform 6-polytope
  • 6-simplex, Rectified 6-simplex, Truncated 6-simplex, Cantellated 6-simplex, Runcinated 6-simplex, Stericated 6-simplex, Pentellated 6-simplex
  • 6-demicube, Truncated 6-demicube, Cantellated 6-demicube, Runcinated 6-demicube, Stericated 6-demicube
  • 6-cube, Rectified 6-cube, 6-cube, Truncated 6-cube, Cantellated 6-cube, Runcinated 6-cube, Stericated 6-cube, Pentellated 6-cube
  • 6-orthoplex, Rectified 6-orthoplex, Truncated 6-orthoplex, Cantellated 6-orthoplex, Runcinated 6-orthoplex, Stericated 6-orthoplex
  • 122 polytope, 221 polytope

Honeycombs

  • 6-cubic honeycomb
  • 6-simplex honeycomb
  • 6-demicubic honeycomb
  • 222 honeycomb

Seven dimensions

Seven-dimensional space, uniform 7-polytope
  • 7-simplex, Rectified 7-simplex, Truncated 7-simplex, Cantellated 7-simplex, Runcinated 7-simplex, Stericated 7-simplex, Pentellated 7-simplex, Hexicated 7-simplex
  • 7-demicube, Truncated 7-demicube, Cantellated 7-demicube, Runcinated 7-demicube, Stericated 7-demicube, Pentellated 7-demicube
  • 7-cube, Rectified 7-cube, 7-cube, Truncated 7-cube, Cantellated 7-cube, Runcinated 7-cube, Stericated 7-cube, Pentellated 7-cube, Hexicated 7-cube
  • 7-orthoplex, Rectified 7-orthoplex, Truncated 7-orthoplex, Cantellated 7-orthoplex, Runcinated 7-orthoplex, Stericated 7-orthoplex, Pentellated 7-orthoplex
  • 132 polytope, 231 polytope, 321 polytope

Honeycombs

  • 7-cubic honeycomb
  • 7-demicubic honeycomb
  • 331 honeycomb, 133 honeycomb

Eight dimension

Eight-dimensional space, uniform 8-polytope
  • 8-simplex, Rectified 8-simplex, Truncated 8-simplex, Cantellated 8-simplex, Runcinated 8-simplex, Stericated 8-simplex, Pentellated 8-simplex, Hexicated 8-simplex, Heptellated 8-simplex
  • 8-orthoplex, Rectified 8-orthoplex, Truncated 8-orthoplex, Cantellated 8-orthoplex, Runcinated 8-orthoplex, Stericated 8-orthoplex, Pentellated 8-orthoplex, Hexicated 8-orthoplex
  • 8-cube, Rectified 8-cube, Truncated 8-cube, Cantellated 8-cube, Runcinated 8-cube, Stericated 8-cube, Pentellated 8-cube, Hexicated 8-cube, Heptellated 8-cube
  • 8-demicube, Truncated 8-demicube, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube
  • 142 polytope, 241 polytope, 421 polytope, Truncated 421 polytope, Truncated 241 polytope, Truncated 142 polytope, Cantellated 421 polytope, Cantellated 241 polytope, Runcinated 421 polytope

Honeycombs

  • 8-cubic honeycomb
  • 8-demicubic honeycomb
  • 521 honeycomb, 251 honeycomb, 152 honeycomb

Nine dimensions

9-polytope
  • 9-cube
  • 9-demicube
  • 9-orthoplex
  • 9-simplex

Hyperbolic honeycombs

  • E9 honeycomb

Ten dimensions

10-polytope
  • 10-cube
  • 10-demicube
  • 10-orthoplex
  • 10-simplex

Dimensional families

Regular polytope and List of regular polytopes
Uniform polytope
  • Demihypercube
  • Uniform 1k2 polytope
  • Uniform 2k1 polytope
  • Uniform k21 polytope
Honeycombs

Geometry

Geometry and other areas of mathematics

Ford circles

Glyphs and symbols

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.
  3. ^ https://web.archive.org/web/20041113201905/http://www.mathcurve.com/courbes2d/isochron/isochrone%20varignon. 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. ^ WOLFRAM Demonstrations Project http://demonstrations.wolfram.com/SelfAvoidingRandomWalks/#more. 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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