List of mathematical shapes

Following is a list of shapes studied in mathematics.

## Algebraic curves

[edit]

Main article: List of curves

* Cubic plane curve
* Quartic plane curve
* Fractal

### Rational curves

[edit]

#### Degree 2

[edit]

* Conic sections
* Unit circle
* Unit hyperbola

#### Degree 3

[edit]

* 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

[edit]

* 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[citation needed]

#### Degree 5

[edit]

* Quintic of l'Hospital[1]

#### Degree 6

[edit]

* Astroid
* Atriphtaloid
* Nephroid
* Quadrifolium

#### Families of variable degree

[edit]

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

### Curves of genus one

[edit]

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

### Curves with genus greater than one

[edit]

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

### Curve families with variable genus

[edit]

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

## Transcendental curves

[edit]

* 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
* Golden spiral
* Syntractrix
* Tractrix
* Trochoid

## Piecewise constructions

[edit]

* 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

[edit]

* 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

[edit]

* 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

[edit]

Main article: List of surfaces

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

## Minimal surfaces

[edit]

* 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

[edit]

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

## Quadrics

[edit]

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

## Pseudospherical surfaces

[edit]

* Dini's surface
* Pseudosphere

## Algebraic surfaces

[edit]

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

[edit]

* Right conoid (a ruled surface)

## Fractals

[edit]

Main article: List of fractals by Hausdorff dimension

* Apollonian gasket
* Apollonian sphere packing
* Blancmange curve
* Cantor dust
* Cantor set
* Cantor tesseract[citation needed]
* 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[7]
* 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[citation needed]
* Vicsek fractal
* von Koch curve
* Weierstrass function
* Z-order curve

### Random fractals

[edit]

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* 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

## Regular polytopes

[edit]

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

[edit]

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

[edit]

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

[edit]

* Point

### One-dimensional regular polytope

[edit]

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

[edit]

* Polygon
* Equilateral
* Cyclic polygon

* Convex polygon
* Star polygon
* Pentagram

#### Convex

[edit]

* 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)

[edit]

* Monogon
* Digon

#### Non-convex

[edit]

* star polygon
* Pentagram
* Heptagram
* Octagram
* Enneagram
* Decagram

#### Tessellation

[edit]

* Apeirogon

### Three-dimensional regular polytopes

[edit]

* polyhedron

#### Convex

[edit]

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

#### Degenerate (spherical)

[edit]

* hosohedron
* dihedron
* Henagon#In spherical geometry

#### Non-convex

[edit]

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

#### Tessellations

[edit]

##### Euclidean tilings

[edit]

* Square tiling
* Triangular tiling
* Hexagonal tiling
* Apeirogon
* Dihedron

##### Hyperbolic tilings

[edit]

* Lobachevski plane
* Hyperbolic tiling

##### Hyperbolic star-tilings

[edit]

* Order-7 heptagrammic tiling
* Heptagrammic-order heptagonal tiling
* Order-9 enneagrammic tiling[citation needed]
* Enneagrammic-order enneagonal tiling[citation needed]

### Four-dimensional regular polytopes

[edit]

* 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)

[edit]

* Ditope
* Hosotope
* 3-sphere

#### Non-convex

[edit]

* 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

[edit]

* Honeycomb
* Cubic honeycomb

#### Degenerate tessellations of Euclidean 3-space

[edit]

* 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

#### Tessellations of hyperbolic 3-space

[edit]

* 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

[edit]

* 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

#### Tessellations of Euclidean 4-space

[edit]

* honeycombs
* Tesseractic honeycomb
* 16-cell honeycomb
* 24-cell honeycomb

#### Tessellations of Euclidean 5-space and higher

[edit]

* 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

[edit]

* 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

[edit]

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

### Apeirotopes

[edit]

* Apeirotope
* Apeirogon
* Apeirohedron
* Regular skew polyhedron

### Abstract polytopes

[edit]

* Abstract polytope
* 11-cell
* 57-cell

## 2D with 1D surface

[edit]

* 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

[edit]

* 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

[edit]

Main article: Uniform polyhedron

* 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

*     * Cubitruncated cuboctahedron
* Cubohemioctahedron
* Ditrigonal dodecadodecahedron
* Dodecadodecahedron
* Great cubicuboctahedron
* Great dirhombicosidodecahedron
* Great disnub dirhombidodecahedron
* Great ditrigonal dodecicosidodecahedron
* Great ditrigonal icosidodecahedron
* Great dodecahemicosahedron
* Great dodecahemidodecahedron
* Great dodecicosahedron
* Great dodecicosidodecahedron
* Great icosicosidodecahedron
* Great icosidodecahedron
* Great icosihemidodecahedron
* Great inverted snub icosidodecahedron
* Great retrosnub icosidodecahedron
* Great rhombidodecahedron
* Great rhombihexahedron
* Great snub dodecicosidodecahedron
* Great snub icosidodecahedron
* Great stellated truncated dodecahedron
* Great truncated cuboctahedron
* Great truncated icosidodecahedron
* Icosidodecadodecahedron
* Icositruncated dodecadodecahedron
* Inverted snub dodecadodecahedron
* Nonconvex great rhombicosidodecahedron
* Nonconvex great rhombicuboctahedron
* Octahemioctahedron
* Rhombicosahedron
* Rhombidodecadodecahedron
* Small cubicuboctahedron
* Small ditrigonal dodecicosidodecahedron
* Small ditrigonal icosidodecahedron
* Small dodecahemicosahedron
* Small dodecahemidodecahedron
* Small dodecicosahedron
* Small dodecicosidodecahedron
* Small icosicosidodecahedron
* Small icosihemidodecahedron
* Small retrosnub icosicosidodecahedron
* Small rhombidodecahedron
* Small rhombihexahedron
* Small snub icosicosidodecahedron
* Small stellated truncated dodecahedron
* Snub dodecadodecahedron
* Snub icosidodecadodecahedron
* Stellated truncated hexahedron
* Tetrahemihexahedron
* Truncated dodecadodecahedron
* Truncated great dodecahedron
* Truncated great icosahedron

### Duals of uniform polyhedra

[edit]

* 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

* 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

[edit]

Main article: Johnson solid

* Augmented dodecahedron
* Augmented hexagonal prism
* Augmented pentagonal prism
* Augmented sphenocorona
* Augmented triangular prism
* Augmented tridiminished icosahedron
* Augmented truncated cube
* Augmented truncated dodecahedron
* Augmented truncated tetrahedron
* Biaugmented pentagonal prism
* Biaugmented triangular prism
* Biaugmented truncated cube
* Bigyrate diminished rhombicosidodecahedron
* Bilunabirotunda
* Diminished rhombicosidodecahedron
* Disphenocingulum
* Elongated pentagonal bipyramid
* Elongated pentagonal cupola
* Elongated pentagonal gyrobicupola
* Elongated pentagonal gyrobirotunda
* Elongated pentagonal gyrocupolarotunda
* Elongated pentagonal orthobicupola
* Elongated pentagonal orthobirotunda
* Elongated pentagonal orthocupolarotunda
* Elongated pentagonal pyramid
* Elongated pentagonal rotunda
* Elongated square bipyramid
* Elongated square cupola
* Elongated square gyrobicupola
* Elongated square pyramid
* Elongated triangular bipyramid
* Elongated triangular cupola
* Elongated triangular gyrobicupola
* Elongated triangular orthobicupola
* Elongated triangular pyramid
* Gyrate bidiminished rhombicosidodecahedron
* Gyrate rhombicosidodecahedron
* Gyrobifastigium
* Gyroelongated pentagonal bicupola
* Gyroelongated pentagonal birotunda
* Gyroelongated pentagonal cupola
* Gyroelongated pentagonal cupolarotunda
* Gyroelongated pentagonal pyramid
* Gyroelongated pentagonal rotunda
* Gyroelongated square bicupola
* Gyroelongated square bipyramid
* Gyroelongated square cupola
* Gyroelongated square pyramid
* Gyroelongated triangular bicupola
* Gyroelongated triangular cupola
* Hebesphenomegacorona
* Metabiaugmented dodecahedron
* Metabiaugmented hexagonal prism
* Metabiaugmented truncated dodecahedron
* Metabidiminished icosahedron
* Metabidiminished rhombicosidodecahedron
* Metabigyrate rhombicosidodecahedron
* Metagyrate diminished rhombicosidodecahedron
* Parabiaugmented dodecahedron
* Parabiaugmented hexagonal prism
* Parabiaugmented truncated dodecahedron
* Parabidiminished rhombicosidodecahedron
* Parabigyrate rhombicosidodecahedron
* Paragyrate diminished rhombicosidodecahedron
* Pentagonal bipyramid
* Pentagonal cupola
* Pentagonal gyrobicupola
* Pentagonal gyrocupolarotunda
* Pentagonal orthobicupola
* Pentagonal orthobirotunda
* Pentagonal orthocupolarotunda
* Pentagonal pyramid
* Pentagonal rotunda
* Snub disphenoid
* Snub square antiprism
* Sphenocorona
* Sphenomegacorona
* Square cupola
* Square gyrobicupola
* Square orthobicupola
* Square pyramid
* Triangular bipyramid
* Triangular cupola
* Triangular hebesphenorotunda
* Triangular orthobicupola
* Triaugmented dodecahedron
* Triaugmented hexagonal prism
* Triaugmented triangular prism
* Triaugmented truncated dodecahedron
* Tridiminished icosahedron
* Tridiminished rhombicosidodecahedron
* Trigyrate rhombicosidodecahedron

### Other nonuniform polyhedra

[edit]

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

### Spherical polyhedra

[edit]

Main article: spherical polyhedron

* Dihedron
* Hosohedron

### Honeycombs

[edit]

Convex uniform honeycomb

* Cubic honeycomb
* Truncated cubic honeycomb
* Bitruncated cubic honeycomb
* Cantellated cubic honeycomb
* Cantitruncated cubic honeycomb
* Rectified cubic honeycomb
* Runcitruncated cubic honeycomb
* Omnitruncated cubic honeycomb
* Tetrahedral-octahedral honeycomb
* Truncated alternated cubic honeycomb
* Cantitruncated alternated cubic honeycomb
* Runcinated alternated cubic honeycomb
* Quarter cubic honeycomb
* Gyrated tetrahedral-octahedral honeycomb
* Gyrated triangular prismatic honeycomb
* Gyroelongated alternated cubic honeycomb
* Gyroelongated triangular prismatic honeycomb
* Elongated triangular prismatic honeycomb
* Elongated alternated cubic honeycomb
* Hexagonal prismatic honeycomb
* Triangular prismatic honeycomb
* Triangular-hexagonal prismatic honeycomb
* Truncated hexagonal prismatic honeycomb
* Truncated square prismatic honeycomb
* Rhombitriangular-hexagonal prismatic honeycomb
* Omnitruncated triangular-hexagonal prismatic honeycomb
* Snub triangular-hexagonal prismatic honeycomb
* Snub square prismatic 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

[edit]

* Apeirogonal prism
* Apeirohedron
* Bicupola
* Cupola
* Bifrustum
* Boerdijk–Coxeter helix
* Császár polyhedron
* Flexible polyhedron
* Gyroelongated square dipyramid
* Heronian tetrahedron
* Hexagonal bifrustum
* Hexagonal truncated trapezohedron
* Hill tetrahedron
* Holyhedron
* Infinite skew polyhedron
* Jessen's icosahedron
* Near-miss Johnson solid
* Parallelepiped
* Pentagonal bifrustum
* Polytetrahedron
* Pyritohedron
* Rhombic enneacontahedron
* Rhombic icosahedron
* Rhombo-hexagonal dodecahedron
* Rhombohedron
* Scalenohedron
* Schönhardt polyhedron
* Square bifrustum
* Square truncated trapezohedron
* Szilassi polyhedron
* Tetradecahedron
* Tetradyakis hexahedron
* Tetrated dodecahedron
* Triangular bifrustum
* Triaugmented triangular prism
* Truncated rhombic dodecahedron
* Truncated trapezohedron
* Truncated triakis tetrahedron
* Tridyakis icosahedron
* Trigonal trapezohedron
* Regular skew polyhedron
* Waterman polyhedron
* Wedge

### Regular and uniform compound polyhedra

[edit]

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 prisms
* Compound of six pentagrammic crossed antiprisms
* Compound of six pentagrammic prisms
* 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 tetrahedra
* Compound of twelve pentagonal antiprisms with rotational freedom
* Compound of twelve pentagonal prisms
* 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
* 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

[edit]

* 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

[edit]

* 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]

### Honeycombs

[edit]

* 5-cubic honeycomb
* 5-simplex honeycomb
* Truncated 5-simplex honeycomb
* 5-demicubic honeycomb

## Six dimensions

[edit]

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

[edit]

* 6-cubic honeycomb
* 6-simplex honeycomb
* 6-demicubic honeycomb
* 222 honeycomb

## Seven dimensions

[edit]

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

[edit]

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

## Eight dimension

[edit]

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]

### Honeycombs

[edit]

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

## Nine dimensions

[edit]

9-polytope

* 9-cube
* 9-demicube
* 9-orthoplex
* 9-simplex

### Hyperbolic honeycombs

[edit]

* E9 honeycomb

## Ten dimensions

[edit]

10-polytope

* 10-cube
* 10-demicube
* 10-orthoplex
* 10-simplex

## Dimensional families

[edit]

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

[edit]

* Triangle
* Automedian triangle
* Delaunay triangulation
* Equilateral triangle
* Golden triangle
* Hyperbolic triangle (non-Euclidean geometry)
* Isosceles triangle
* Kepler triangle
* Reuleaux triangle
* Right triangle
* Sierpinski triangle (fractal geometry)
* Special right triangles
* Spiral of Theodorus
* Thomson cubic
* Triangular bipyramid
* Triangular prism
* Triangular pyramid
* Triangular tiling

## Geometry and other areas of mathematics

[edit]

Ford circles

* Annulus
* Apollonian circles
* Apollonian gasket
* Arbelos
* Borromean rings
* Circle
* Circular sector
* Circular segment
* Cyclic quadrilateral
* Cycloid
* Epitrochoid
* Epicycloid
* Cardioid
* Nephroid
* Deferent and epicycle
* Ex-tangential quadrilateral
* Horocycle
* Hypotrochoid
* Hypocycloid
* Astroid
* Deltoid curve
* Lune
* Pappus chain
* Peaucellier–Lipkin linkage
* Robbins pentagon
* Salinon
* Semicircle
* Squircle
* Steiner chain
* Tangential quadrilateral
* Bicentric quadrilateral

## Glyphs and symbols

[edit]

* Borromean rings
* Crescent
* Vesica piscis
* Arc
* Caustic
* Cissoid
* Conchoid
* Cubic Hermite curve
* Curve of constant width
* hedgehog[9]
* Parametric curve
* Bézier curve
* Spline
* Hermite spline
* Beta spline[citation needed]
* B-spline
* Higher-order spline[citation needed]
* NURBS
* Ray
* Reuleaux triangle
* Ribaucour curve[10]

## Table of all the Shapes

[edit]

This is a table of all the shapes above.

Table of Shapes  Section  | Sub-Section  | Sup-Section  | Name

|
|
|

Algebraic Curves | ¿ Curves | ¿ Curves | Cubic Plane Curve
Quartic Plane Curve
Rational Curves | Degree 2 | Conic Section(s)
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

## References

[edit]

1. ^ "Courbe a Réaction Constante, Quintique De L'Hospital" [Constant Reaction Curve, Quintic of l'Hospital].
2. ^ "Isochrone de Leibniz". Archived from the original on 14 November 2004.
3. ^ "Isochrone de Varignon". 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. ^ "Self-Avoiding Random Walks - Wolfram Demonstrations Project". WOLFRAM Demonstrations Project. Retrieved 14 June 2019.
9. ^ Weisstein, Eric W. "Hedgehog". mathworld.wolfram.com.
10. ^ "Courbe De Ribaucour" [Ribaucour curve]. mathworld.wolfram.com.