Following is a list of shapes studied in mathematics.
Algebraic curves
Cubic plane curve Quartic plane curve Fractal
Rational curves
Degree 2 Conic sections Unit circle Unit hyperbola
Degree 3
Degree 4
Degree 5 Quintic of l'Hospital CURVE WITH CONSTANT REACTION, L'HOSPITAL QUINTIC
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 Golden 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
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 dimensions 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 1k2 polytope Uniform 2k1 polytope Uniform k21 polytope Honeycombs Hypercubic honeycomb Alternated hypercubic honeycomb
Geometry
Geometry and other areas of mathematics
Glyphs and symbols
Table of all the Shapes This is a table of all the shapes above.
References