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List of mathematical shapes

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

Algebraic curves

Cubic plane curve
Quartic plane curve

Rational curves

Degree 2

Conic sections
Unit circle
Unit hyperbola

Degree 3

Degree 4

Degree 5

Quintic of l'Hospital

Degree 6

Astroid
Atriphtaloid
Nephroid
Quadrifolium

Families of variable degree

Epicycloid
Epispiral
Epitrochoid
Hypocycloid
Lissajous curve
Poinsot's spirals
Rational normal curve
Rose curve

Curves of genus one

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

Curves with genus greater than one

Butterfly curve
Elkies trinomial curves
Hyperelliptic curve
Klein quartic
Classical modular curve
Bolza surface
Macbeath surface

Curve families with variable genus

Polynomial lemniscate
Fermat curve
Sinusoidal spiral
Superellipse
Hurwitz surface

Transcendental curves

Bowditch curve
Brachistochrone
Butterfly curve
Catenary
Clélies
Cochleoid
Cycloid
Horopter
Isochrone
- Isochrone of Huygens (Tautochrone)
- Isochrone of Leibniz
- Isochrone of Varignon
Lamé curve
Pursuit curve
Rhumb line
Spirals
- Archimedean spiral
- Cornu spiral
- Cotes' spiral
- Fermat's spiral
- Galileo's spiral
- Hyperbolic spiral
- Lituus
- Logarithmic spiral
- Nielsen's spiral
Syntractrix
Tractrix
Trochoid

Piecewise constructions

Bézier curve
Splines
- B-spline
- Nonuniform rational B-spline
Ogee
Loess curve
Lowess
Polygonal curve
- Maurer rose
Reuleaux triangle
Bézier triangle

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
Slinky spiral
Twisted cubic
Viviani's curve

Surfaces in 3-space

Plane
Quadric surfaces
- Cone
- Cylinder
- Ellipsoid
  - Spheroid
    - Sphere
- Hyperboloid
- Paraboloid
Bicylinder
Tricylinder
Möbius strip
Torus

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

Klein bottle
Real projective plane
- Cross-cap
- Roman surface
- Boy's surface

Quadrics

Sphere
Spheroid
- Oblate spheroid
Cone
Ellipsoid
Hyperboloid of one sheet
Hyperboloid of two sheets
Hyperbolic paraboloid (a ruled surface)
Paraboloid
Sphericon
Oloid

Pseudospherical surfaces

Dini's surface
Pseudosphere

Algebraic surfaces

See the list of algebraic surfaces.

Cayley cubic
Barth sextic
Clebsch cubic
Monkey saddle (saddle-like surface for 3 legs.)
Torus
Dupin cyclide (inversion of a torus)
Whitney umbrella

Miscellaneous surfaces

Right conoid (a ruled surface)

Fractals

Random fractals

von Koch curve with random interval
von Koch curve with random orientation
polymer shapes
diffusion-limited aggregation
Self-avoiding random walk
Brownian motion
Lichtenberg figure
Percolation theory
Multiplicative cascade

Regular polytopes

This table shows a summary of regular polytope counts by dimension.

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

Point

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

Polygon
Equilateral
Cyclic polygon

Convex polygon
Star polygon
Pentagram

Convex

Regular polygon
Equilateral triangle
Simplex
Square
Cross-polytope
Hypercube
Pentagon
Hexagon
Heptagon
Octagon
Enneagon
Decagon
Hendecagon
Dodecagon
Tridecagon
Tetradecagon
Pentadecagon
Hexadecagon
Heptadecagon
Octadecagon
Enneadecagon
Icosagon
Hectogon
Chiliagon
Regular polygon

Degenerate (spherical)

Monogon
Digon

Non-convex

star polygon
Pentagram
Heptagram
Octagram
Enneagram
Decagram

Tessellation

Apeirogon

Three-dimensional regular polytopes

polyhedron

Convex

Platonic solid
- Tetrahedron, the 3-space Simplex
- Cube, the 3-space hypercube
- Octahedron, the 3-space Cross-polytope
- Dodecahedron
- Icosahedron

Degenerate (spherical)

hosohedron
dihedron
Henagon#In spherical geometry

Non-convex

Kepler–Poinsot polyhedra
- Small stellated dodecahedron
- Great dodecahedron
- Great stellated dodecahedron
- Great icosahedron

Tessellations

Euclidean tilings

Square tiling
Triangular tiling
Hexagonal tiling
Apeirogon
Dihedron

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

Star or (Schläfli–Hess) regular 4-polytope
- Icosahedral 120-cell
- Small stellated 120-cell
- Great 120-cell
- Grand 120-cell
- Great stellated 120-cell
- Grand stellated 120-cell
- Great grand 120-cell
- Great icosahedral 120-cell
- Grand 600-cell
- Great grand stellated 120-cell

Tessellations of Euclidean 3-space

Honeycomb
Cubic honeycomb

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

Order-4 dodecahedral honeycomb
Order-5 dodecahedral honeycomb
Order-5 cubic honeycomb
Icosahedral honeycomb
Order-3 icosahedral honeycomb
Order-4 octahedral honeycomb
Triangular tiling honeycomb
Square tiling honeycomb
Order-4 square tiling honeycomb
Order-6 tetrahedral honeycomb
Order-6 cubic honeycomb
Order-6 dodecahedral honeycomb
Hexagonal tiling honeycomb
Order-4 hexagonal tiling honeycomb
Order-5 hexagonal tiling honeycomb
Order-6 hexagonal tiling honeycomb

Five-dimensional regular polytopes and higher

5-polytope
Honeycomb
Tetracomb

Tessellations of Euclidean 4-space

honeycombs
Tesseractic honeycomb
16-cell honeycomb
24-cell honeycomb

Tessellations of Euclidean 5-space and higher

Hypercubic honeycomb
Hypercube
Square tiling
Cubic honeycomb
Tesseractic honeycomb
5-cube honeycomb
6-cube honeycomb
7-cube honeycomb
8-cube honeycomb
Hypercubic honeycomb

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

Apeirotope
- Apeirogon
- Apeirohedron
- Regular skew polyhedron

Abstract polytopes

Abstract polytope
- 11-cell
- 57-cell

2D with 1D surface

Convex polygon
Concave polygon
Constructible polygon
Cyclic polygon
Equiangular polygon
Equilateral polygon
Regular polygon
Penrose tile
Polyform
Balbis
Gnomon
Golygon
Star without crossing lines
Star polygon
- Hexagram
  - Star of David
- Heptagram
- Octagram
  - Star of Lakshmi
- Decagram
- Pentagram

Polygons named for their number of sides

Tilings

List of uniform tilings
Uniform tilings in hyperbolic plane
Archimedean tiling
- Square tiling
- Triangular tiling
- Hexagonal tiling
- Truncated square tiling
- Snub square tiling
- Trihexagonal tiling
- Truncated hexagonal tiling
- Rhombitrihexagonal tiling
- Truncated trihexagonal tiling
- Snub hexagonal tiling
- Elongated triangular tiling

Uniform polyhedra

Regular polyhedron
- Platonic solid
  - Tetrahedron
  - Cube
  - Octahedron
  - Dodecahedron
  - Icosahedron
- Kepler–Poinsot polyhedron (regular star polyhedra)
  - Great icosahedron
  - Small stellated dodecahedron
  - Great dodecahedron
  - Great stellated dodecahedron
- Abstract regular polyhedra (Projective polyhedron)
  - Hemicube
  - Hemi-octahedron
  - Hemi-dodecahedron
  - Hemi-icosahedron
Archimedean solid
- Truncated tetrahedron
- Cuboctahedron
- Truncated cube
- Truncated octahedron
- Rhombicuboctahedron
- Truncated cuboctahedron
- Snub cube
- Icosidodecahedron
- Truncated dodecahedron
- Truncated icosahedron
- Rhombicosidodecahedron
- Truncated icosidodecahedron
- Snub dodecahedron
Prismatic uniform polyhedron
- Prism
- Antiprism

Uniform star polyhedron

Duals of uniform polyhedra

Catalan solid
- Triakis tetrahedron
- Rhombic dodecahedron
- Triakis octahedron
- Tetrakis hexahedron
- Deltoidal icositetrahedron
- Disdyakis dodecahedron
- Pentagonal icositetrahedron
- Rhombic triacontahedron
- Triakis icosahedron
- Pentakis dodecahedron
- Deltoidal hexecontahedron
- Disdyakis triacontahedron
- Pentagonal hexecontahedron

Johnson solids

Other nonuniform polyhedra

Pyramid
Bipyramid
Disphenoid
Parallelepiped
Cuboid
Rhombohedron
Trapezohedron
Frustum
Trapezo-rhombic dodecahedron
Rhombo-hexagonal dodecahedron
Truncated trapezohedron
Deltahedron
Zonohedron
Prismatoid
Cupola
Bicupola

Spherical polyhedra

Dihedron
Hosohedron

Honeycombs

Convex uniform honeycomb

Dual uniform honeycomb

Disphenoid tetrahedral honeycomb
Rhombic dodecahedral honeycomb

Others

Trapezo-rhombic dodecahedral honeycomb
Weaire–Phelan structure

Convex uniform honeycombs in hyperbolic space

Order-4 dodecahedral honeycomb
Order-5 cubic honeycomb
Order-5 dodecahedral honeycomb
Icosahedral honeycomb

Other

Regular and uniform compound polyhedra

Polyhedral compound and Uniform polyhedron compound

4-polytope
- Hecatonicosachoron
- Hexacosichoron
- Hexadecachoron
- Icositetrachoron
- Pentachoron
- Tesseract
Hypercone

Convex regular 4-polytope

5-cell, Tesseract, 16-cell, 24-cell, 120-cell, 600-cell

Abstract regular polytope

11-cell, 57-cell

Schläfli–Hess 4-polytope (Regular star 4-polytope)

Icosahedral 120-cell, Small stellated 120-cell, Great 120-cell, Grand 120-cell, Great stellated 120-cell, Grand stellated 120-cell, Great grand 120-cell, Great icosahedral 120-cell, Grand 600-cell, Great grand stellated 120-cell

Uniform 4-polytope

Rectified 5-cell, Truncated 5-cell, Cantellated 5-cell, Runcinated 5-cell
Rectified tesseract, Truncated tesseract, Cantellated tesseract, Runcinated tesseract
Rectified 16-cell, Truncated 16-cell
Rectified 24-cell, Truncated 24-cell, Cantellated 24-cell, Runcinated 24-cell, Snub 24-cell
Rectified 120-cell, Truncated 120-cell, Cantellated 120-cell, Runcinated 120-cell
Rectified 600-cell, Truncated 600-cell, Cantellated 600-cell

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
- 5-dimensional cross-polytope
- 5-dimensional hypercube
- 5-dimensional simplex

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

Simplex
Hypercube
Cross-polytope

Uniform polytope

Demihypercube
Uniform 1_k2 polytope
Uniform 2_k1 polytope
Uniform k₂₁ polytope

Honeycombs

Hypercubic honeycomb
Alternated hypercubic honeycomb

Geometry

Hyperplexicons

Glowvoid
Warith's void
Warith's hyperplexicon shape
Gaxxoid
Gyroid
Hyperplexicon Gyroid
Planetium
Epyoid
Xenroid
Xenoshape
Xenoid
Emperoids
- Hypervoid
- Hyperoid
- Warith-Nathaniyal mixbox
- Mixbox
- Forcoid
Corporoid
Primoid
Oppan's gyroid
Zahian's Hyperplexicon
Nathaniyal's object
Hyperplexicon

Geometry and other areas of mathematics

Glyphs and symbols

Table of all the Shapes

This is a table of all the shapes above.

References

Text submitted to CC-BY-SA license. Source: List of mathematical shapes by Wikipedia (Historical)

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