List of mathematical shapes

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

• Cubic plane curve
• Quartic plane curve

• Conic sections
• Unit circle
• Unit hyperbola

• Quintic of l'Hospital^[1]

• Astroid
• Atriphtaloid
• Nephroid
• Quadrifolium

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

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

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

• Polynomial lemniscate
• Fermat curve
• Sinusoidal spiral
• Superellipse
• Hurwitz surface

• Bowditch curve
• Brachistochrone
• Butterfly curve
• Catenary
• Clélies
• Cochleoid
• Cycloid
• Horopter
• Isochrone
  • Isochrone of Huygens (Tautochrone)
  • Isochrone of Leibniz^[2]
  • Isochrone of Varignon^[3]
• Lamé curve
• Pursuit curve
• Rhumb line
• Spirals
  • Archimedean spiral
  • Cornu spiral
  • Cotes' spiral
  • Fermat's spiral
  • Galileo's spiral^[4]
  • Hyperbolic spiral
  • Lituus
  • Logarithmic spiral
  • Nielsen's spiral
• Syntractrix
• Tractrix
• Trochoid

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

• 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

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

• 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

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

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

• Dini's surface
• Pseudosphere

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

• Right conoid (a ruled surface)

• von Koch curve with random interval
• von Koch curve with random orientation
• polymer shapes
• diffusion-limited aggregation
• Self-avoiding random walk^[8]
• Brownian motion
• Lichtenberg figure
• Percolation theory
• Multiplicative cascade

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.

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.

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.

• Point

There is only one polytope in 1 dimension, whose boundaries are the two endpoints of a line segment, represented by the empty Schläfli symbol {}.

• Polygon
• Equilateral
• Cyclic polygon

• Convex polygon
• Star polygon
• Pentagram

• 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

• Monogon
• Digon

• star polygon
• Pentagram
• Heptagram
• Octagram
• Enneagram
• Decagram

• Apeirogon

• polyhedron

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

• hosohedron
• dihedron
• Henagon#In spherical geometry

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

• Square tiling
• Triangular tiling
• Hexagonal tiling
• Apeirogon
• Dihedron

• Lobachevski plane
• Hyperbolic tiling

• Order-7 heptagrammic tiling
• Heptagrammic-order heptagonal tiling
• Order-9 enneagrammic tiling^{[citation needed]}
• Enneagrammic-order enneagonal tiling^{[citation needed]}

• 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

• Ditope
• Hosotope
• 3-sphere

• 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

• Honeycomb
• Cubic honeycomb

• Hosohedron
• Dihedron
• Order-2 apeirogonal tiling
• Apeirogonal hosohedron
• Order-4 square hosohedral honeycomb
• Order-6 triangular hosohedral honeycomb
• Hexagonal hosohedral honeycomb^{[citation needed]}
• Order-2 square tiling honeycomb
• Order-2 triangular tiling honeycomb
• Order-2 hexagonal tiling honeycomb

• 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

• 5-polytope
• Honeycomb
• Tetracomb

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

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

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

• 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

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

• Apeirotope
  • Apeirogon
  • Apeirohedron
  • Regular skew polyhedron

• Abstract polytope
  • 11-cell
  • 57-cell

• 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

• 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

• 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

• 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

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

• Dihedron
• Hosohedron

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

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

• 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

• 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.^{[citation needed]}

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

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

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

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

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

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^{[citation needed]}
• 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^{[citation needed]}
• 8-demicube, Truncated 8-demicube, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube^{[citation needed]}
• 1₄₂ polytope, 2₄₁ polytope, 4₂₁ polytope, Truncated 4₂₁ polytope, Truncated 2₄₁ polytope, Truncated 1₄₂ polytope, Cantellated 4₂₁ polytope, Cantellated 2₄₁ polytope, Runcinated 4₂₁ polytope^{[citation needed]}

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

9-polytope

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

• E₉ honeycomb

10-polytope

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

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

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