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list of mathematical shapes

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Following is a list of shapes studied in mathematics.

[[Algebraic curve]]s

{{Main|List of curves}}

=Rational curves=

==Degree 2==

==Degree 3==

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==Degree 4==

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==Degree 5==

  • Quintic of l'Hospital{{Cite web |url=http://www.mathcurve.com/courbes2d/quintique%20de%20l'hospital/quintique%20de%20l'hospital.shtml |title=Courbe a Réaction Constante, Quintique De L'Hospital |trans-title=Constant Reaction Curve, Quintic of l'Hospital}}

==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

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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{{cite web|url=http://mathworld.wolfram.com/SeiffertsSphericalSpiral.html|title=Seiffert's Spherical Spiral|first=Eric W. |last=Weisstein|website=mathworld.wolfram.com}}
  • Slinky spiral{{cite web|url=http://mathworld.wolfram.com/Slinky.html|title=Slinky|first=Eric W. |last=Weisstein|website=mathworld.wolfram.com}}
  • Twisted cubic
  • Viviani's curve

Surfaces in 3-space

{{main|List of surfaces}}

[[Minimal surface]]s

[[orientability|Non-orientable]] surfaces

[[Quadric]]s

Pseudospherical surfaces

[[Algebraic surface]]s

See the list of algebraic surfaces.

Miscellaneous surfaces

Fractals

{{main|List of fractals by Hausdorff dimension}}

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=Random fractals=

{{More citations needed section|small=yes|talk=Placing 'mathematical shapes' that do not have a wikipedia article|date=April 2018}}

Regular polytopes

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

class="wikitable"
Dimension

!Convex

!Nonconvex

!Convex
Euclidean
tessellations

!Convex
hyperbolic
tessellations

!Nonconvex
hyperbolic
tessellations

!Hyperbolic Tessellations
with infinite cells
and/or vertex figures

!Abstract
Polytopes

align=center

|1

1 line segment010001
align=center

|2

∞ polygons∞ star polygons1100∞
align=center

|3

5 Platonic solids4 Kepler–Poinsot solids3 tilings∞∞∞∞
align=center

|4

6 convex polychora10 Schläfli–Hess polychora1 honeycomb4011∞
align=center

|5

3 convex 5-polytopes03 tetracombs542∞
align=center

|6

3 convex 6-polytopes01 pentacombs005∞
align=center

|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=

=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 =

SimplexHypercubeCross-polytope
5-simplex5-cube5-orthoplex
6-simplex6-cube6-orthoplex
7-simplex7-cube7-orthoplex
8-simplex8-cube8-orthoplex
9-simplex9-cube9-orthoplex
10-simplex10-cube10-orthoplex
11-simplex11-cube11-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

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=Tilings=

=Uniform polyhedra=

{{main|Uniform polyhedron}}

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=Duals of uniform polyhedra=

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=Johnson solids=

{{main|Johnson solid}}

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=Other nonuniform polyhedra=

=Spherical polyhedra=

{{main|spherical polyhedron}}

=Honeycombs=

;Convex uniform honeycomb

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;Dual uniform honeycomb

;Others

;Convex uniform honeycombs in hyperbolic space

=Other=

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=Regular and uniform compound polyhedra=

;Polyhedral compound and Uniform polyhedron compound

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;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|reason=See Talk: Placing 'mathematical shapes' that do not have a wikipedia article|date=April 2018}}

=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

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Geometry and other areas of mathematics

Image:Ford circles.svgs]]

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Glyphs and symbols

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Table of all the Shapes

This is a table of all the shapes above.

class="wikitable"

|+Table of Shapes

!Section

!Sub-Section

!Sup-Section

!Name

rowspan="24" |Algebraic Curves

| rowspan="2" |¿ Curves

| rowspan="2" |¿ Curves

|Cubic Plane Curve

Quartic Plane Curve
rowspan="22" |Rational Curves

| rowspan="3" |Degree 2

|Conic Section(s)

Unit Circle
Unit Hyperbola
rowspan="10" |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
rowspan="9" |Degree 4

|Ampersand Curve

Bean Curve
Bicorn
Bow Curve
Bullet-Nose Curve
Cruciform Curve

References

{{Reflist}}

{{DEFAULTSORT:Mathematical shapes}}

Mathematical