List of mathematical shapes

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

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

Convex

Degenerate (spherical)

Non-convex

Degenerate (spherical)

Non-convex

Kepler–Poinsot polyhedra

Tessellations

Euclidean tilings

Hyperbolic tilings

Hyperbolic star-tilings

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)

Non-convex

Star or (Schläfli–Hess) regular 4-polytope

Tessellations of Euclidean 3-space

Degenerate tessellations of Euclidean 3-space

Tessellations of hyperbolic 3-space

Five-dimensional regular polytopes and higher

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

Tessellations of Euclidean 5-space and higher

Tessellations of hyperbolic 4-space

Tessellations of hyperbolic 5-space

Apeirotopes

Apeirotope

Abstract polytopes

Abstract polytope
- 11-cell
- 57-cell

2D with 1D surface

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
- 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

Uniform polyhedra

Regular polyhedron
- Platonic solid
- Kepler–Poinsot polyhedron (regular star polyhedra)
- Abstract regular polyhedra (Projective polyhedron)
Archimedean solid
Prismatic uniform polyhedron
- Prism
- Antiprism

Uniform star polyhedron

Duals of uniform polyhedra

Catalan solid

non-convex

Johnson solids

Other nonuniform polyhedra

Spherical polyhedra

Honeycombs

Convex uniform honeycomb

Dual uniform honeycomb

Others

Convex uniform honeycombs in hyperbolic space

Other

Regular and uniform compound polyhedra

Polyhedral compound and Uniform polyhedron compound

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

Prismatic uniform polychoron

Honeycombs

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

Prismatic uniform 5-polytope: For each polytope of dimension n, there is a prism of dimension n+1.^{[citation needed]}

Honeycombs

Six dimensions

Six-dimensional space, 6-polytope and uniform 6-polytope

Honeycombs

Seven dimensions

Seven-dimensional space, uniform 7-polytope

Honeycombs

Eight dimension

Eight-dimensional space, uniform 8-polytope

Honeycombs

Nine dimensions

9-polytope

Hyperbolic honeycombs

E₉ honeycomb

Ten dimensions

10-polytope

Dimensional families

Regular polytope and List of regular polytopes

Uniform polytope

Honeycombs

Geometry

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

Hyperplexicons

Geometry and other areas of mathematics

Glyphs and symbols

Table of all the Shapes

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
	¿ Curves	¿ Curves	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

^ "Courbe a Réaction Constante, Quintique De L'Hospital" [Constant Reaction Curve, Quintic of l'Hospital].
^ "Isochrone de Leibniz". Archived from the original on 14 November 2004.
^ "Isochrone de Varignon". Archived from the original on 13 November 2004.
^ Ferreol, Robert. "Spirale de Galilée". www.mathcurve.com.
^ Weisstein, Eric W. "Seiffert's Spherical Spiral". mathworld.wolfram.com.
^ Weisstein, Eric W. "Slinky". mathworld.wolfram.com.
^ "Monkeys tree fractal curve". Archived from the original on 21 September 2002.
^ "Self-Avoiding Random Walks - Wolfram Demonstrations Project". WOLFRAM Demonstrations Project. Retrieved 14 June 2019.
^ Weisstein, Eric W. "Hedgehog". mathworld.wolfram.com.
^ "Courbe De Ribaucour" [Ribaucour curve]. mathworld.wolfram.com.

[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.

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Algebraic curves

Rational curves

Degree 2

Degree 3

Degree 4

Degree 5

Degree 6

Families of variable degree

Curves of genus one

Curves with genus greater than one

Curve families with variable genus

Transcendental curves

Piecewise constructions

Curves generated by other curves

Space curves

Surfaces in 3-space

Minimal surfaces

Non-orientable surfaces

Quadrics

Pseudospherical surfaces

Algebraic surfaces

Miscellaneous surfaces

Fractals

Random fractals

Regular polytopes

Polytope elements

Tessellations

Zero dimension

One-dimensional regular polytope

Two-dimensional regular polytopes

Convex

Degenerate (spherical)

Non-convex

Tessellation

Three-dimensional regular polytopes

Convex

Degenerate (spherical)

Non-convex

Tessellations

Euclidean tilings

Hyperbolic tilings

Hyperbolic star-tilings

Four-dimensional regular polytopes

Degenerate (spherical)

Non-convex

Tessellations of Euclidean 3-space

Degenerate tessellations of Euclidean 3-space

Tessellations of hyperbolic 3-space

Five-dimensional regular polytopes and higher

Tessellations of Euclidean 4-space

Tessellations of Euclidean 5-space and higher

Tessellations of hyperbolic 4-space

Tessellations of hyperbolic 5-space

Apeirotopes

Abstract polytopes

2D with 1D surface

Tilings

Uniform polyhedra

Duals of uniform polyhedra

Johnson solids

Other nonuniform polyhedra

Spherical polyhedra

Honeycombs

Other

Regular and uniform compound polyhedra

Honeycombs

5D with 4D surfaces

Honeycombs

Six dimensions

Honeycombs

Seven dimensions

Honeycombs

Eight dimension

Honeycombs

Nine dimensions

Hyperbolic honeycombs

Ten dimensions

Dimensional families

Geometry

Hyperplexicons