List of mathematical shapes

Last updated

Following is a list of shapes studied in mathematics.

Contents

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

See the list of algebraic surfaces.

Miscellaneous surfaces

Fractals

Random fractals

Regular polytopes

This table shows a summary of regular polytope counts by dimension.

DimensionConvexNonconvexConvex
Euclidean
tessellations		Convex
hyperbolic
tessellations		Nonconvex
hyperbolic
tessellations		Hyperbolic Tessellations
with infinite cells
and/or vertex figures		Abstract
Polytopes
11 line segment 010001
2∞ polygons ∞ star polygons 1100∞
35 Platonic solids 4 Kepler–Poinsot solids 3 tilings ∞∞∞∞
46 convex polychora 10 Schläfli–Hess polychora 1 honeycomb 4011∞
53 convex 5-polytopes 03 tetracombs 542∞
63 convex 6-polytopes 01 pentacombs 005∞
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.

For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak.

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

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

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

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

Abstract polytopes

2D with 1D surface

Polygons named for their number of sides

Tilings

Uniform polyhedra

Duals of uniform polyhedra

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
Abstract regular polytope
Schläfli–Hess 4-polytope (Regular star 4-polytope)
Uniform 4-polytope
Prismatic uniform polychoron

Honeycombs

5D with 4D surfaces

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 dimensions

Eight-dimensional space, uniform 8-polytope

Honeycombs

Nine dimensions

9-polytope

Hyperbolic honeycombs

Ten dimensions

10-polytope

Dimensional families

Regular polytope and List of regular polytopes
Uniform polytope
Honeycombs

Geometry

Geometry and other areas of mathematics

Ford circles Ford circles.svg
Ford circles

Glyphs and symbols

Table of all the Shapes

This is a table of all the shapes above.

Table of Shapes
SectionSub-SectionSup-SectionName
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

  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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