List of mathematical shapes explained

Following is a list of some mathematically well-defined shapes.

Algebraic curves

See main article: List of 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

See main article: List of surfaces.

Minimal surfaces

Non-orientable surfaces

Quadrics

Pseudospherical surfaces

Algebraic surfaces

See the list of algebraic surfaces.

Miscellaneous surfaces

Fractals

See main article: List of fractals by Hausdorff dimension.

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 segment010001
2∞ polygons∞ star polygons1100∞
35 Platonic solids4 Kepler–Poinsot solids3 tilings∞∞∞∞
46 convex polychora10 Schläfli–Hess polychora1 honeycomb4 011∞
5 3 convex 5-polytopes03 tetracombs5 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.

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

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

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

Cross-polytope
5-orthoplex
6-orthoplex
7-orthoplex
8-orthoplex
9-orthoplex
10-orthoplex
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

See main article: Uniform polyhedron.

Duals of uniform polyhedra

Johnson solids

See main article: Johnson solid.

Other nonuniform polyhedra

Spherical polyhedra

See main article: spherical polyhedron.

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.

    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

    Ten dimensions

    10-polytope

    Dimensional families

    Regular polytope and List of regular polytopes
    Uniform polytope
    Honeycombs

    Geometry

    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¿ CurvesCubic Plane Curve
    Quartic Plane Curve
    Rational CurvesConic Section(s)
    Unit Circle
    Unit Hyperbola
    Degree 3Folium 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 4Ampersand Curve
    Bean Curve
    Bicorn
    Bow Curve
    Bullet-Nose Curve
    Cruciform Curve

    Notes and References

    1. Web site: Courbe a Réaction Constante, Quintique De L'Hospital . Constant Reaction Curve, Quintic of l'Hospital.
    2. Web site: https://web.archive.org/web/20041114002246/http://www.mathcurve.com/courbes2d/isochron/isochrone%20leibniz . Isochrone de Leibniz . dead. 14 November 2004.
    3. Web site: https://web.archive.org/web/20041113201905/http://www.mathcurve.com/courbes2d/isochron/isochrone%20varignon . Isochrone de Varignon . dead. 13 November 2004.
    4. Web site: Spirale de Galilée. Robert. Ferreol. www.mathcurve.com.
    5. Web site: Seiffert's Spherical Spiral. Eric W. . Weisstein. mathworld.wolfram.com.
    6. Web site: Slinky. Eric W. . Weisstein. mathworld.wolfram.com.
    7. Web site: Monkeys tree fractal curve . https://archive.today/20020921135308/http://www.coaauw.org/boulder-eyh/eyh_fractal.html . 21 September 2002 .
    8. Web site: Self-Avoiding Random Walks - Wolfram Demonstrations Project . WOLFRAM Demonstrations Project . 14 June 2019.
    9. Web site: Hedgehog . Weisstein . Eric W. . mathworld.wolfram.com.
    10. Web site: Courbe De Ribaucour . Ribaucour curve . mathworld.wolfram.com.

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