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
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 Deltoid curve Devil's curve Hippopede Kampyle of Eudoxus Kappa curve Lemniscate of Booth Lemniscate of Gerono Lemniscate of Bernoulli Limaçon Cardioid Limaçon trisectrix Trifolium curve
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 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
Caustic including Catacaustic and Diacaustic Cissoid Conchoid Evolute Glissette Inverse curve Involute Isoptic including Orthoptic Orthotomic Negative pedal curve Pedal curve Parallel curve Radial curve Roulette Strophoid
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
Apollonian gasket Apollonian sphere packing Blancmange curve Cantor dust Cantor set Cantor tesseract Circle inversion fractal De Rham curve Douady rabbit Dragon curve Fibonacci word fractal Flame fractal Fractal curve Gosper curve Gosper island H-fractal Hénon map Hexaflake Hilbert curve Ikeda map attractor Iterated function system Jerusalem cube Julia set Koch curve Koch snowflake L-system Lévy C curve Feigenbaum attractor Lorenz attractor Lyapunov fractal Mandelbrot set Mandelbrot tree Mandelbulb Menger sponge Monkeys tree Moore curve N-flake Pascal triangle Peano curve Penrose tiling Pinwheel tiling Pythagoras tree Rauzy fractal Rössler attractor Sierpiński arrowhead curve Sierpinski carpet Sierpiński curve Sierpinski triangle Smith–Volterra–Cantor set T-square Takagi or Blancmange curve Triflake Vicsek fractal von Koch curve Weierstrass function Z-order curve
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. DimensionConvexNonconvexConvexEuclideantessellationsConvexhyperbolictessellationsNonconvexhyperbolictessellationsHyperbolic Tessellationswith infinite cellsand/or vertex figuresAbstractPolytopes11 line segment0100012∞ polygons∞ star polygons1100∞35 Platonic solids4 Kepler–Poinsot solids3 tilings∞ ∞∞∞46 convex polychora10 Schläfli–Hess polychora1 honeycomb4 011∞5 3 convex 5-polytopes0 3 tetracombs 5 42∞63 convex 6-polytopes01 pentacombs005∞7+301000∞ 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
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 Simplex Hypercube Cross-polytope 5-simplex 5-cube 5-orthoplex 6-simplex 6-cube 6-orthoplex
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
Monogon — 1 sided Digon — 2 sided Triangle Acute triangle Equilateral triangle Isosceles triangle Obtuse triangle Rational triangle Right triangle 30-60-90 triangle Isosceles right triangle Kepler triangle Scalene triangle Quadrilateral Cyclic quadrilateral square kite Parallelogram Rhombus (equilateral parallelogram) Lozenge Rhomboid Rectangle square (regular quadrilateral) Tangential quadrilateral Trapezoid or trapezium Isosceles trapezoid Pentagon Regular pentagon Hexagon Lemoine hexagon Heptagon Octagon Regular octagon Nonagon Decagon Regular decagon Hendecagon Dodecagon Triskaidecagon Tetradecagon Pentadecagon Hexadecagon Heptadecagon Octadecagon Enneadecagon Icosagon Triacontagon Tetracontagon Pentacontagon Hexacontagon Heptacontagon Octacontagon Enneacontagon Hectogon 257-gon Chiliagon Myriagon 65537-gon Megagon Apeirogon
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)