List of mathematical shapes

Last updated

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

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
2polygons 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 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

Hyperplexicons

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

Related Research Articles

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

<span class="mw-page-title-main">Runcinated 5-cell</span> Four-dimensional geometrical object

In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination of the regular 5-cell.

<span class="mw-page-title-main">Runcinated tesseracts</span>

In four-dimensional geometry, a runcinated tesseract is a convex uniform 4-polytope, being a runcination of the regular tesseract.

<span class="mw-page-title-main">Cubic honeycomb</span> Only regular space-filling tessellation of the cube

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

<span class="mw-page-title-main">Tetrahedral-octahedral honeycomb</span> Quasiregular space-filling tesselation

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

<span class="mw-page-title-main">Order-4 dodecahedral honeycomb</span> Regular tiling of hyperbolic 3-space

In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations of hyperbolic 3-space. With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.

<span class="mw-page-title-main">Order-5 dodecahedral honeycomb</span> Regular tiling of hyperbolic 3-space

In hyperbolic geometry, the order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.

<span class="mw-page-title-main">Order-5 cubic honeycomb</span> Regular tiling of hyperbolic 3-space

In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

<span class="mw-page-title-main">Icosahedral honeycomb</span> Regular tiling of hyperbolic 3-space

In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.

<span class="mw-page-title-main">Alternation (geometry)</span> Removal of alternate vertices

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

<span class="mw-page-title-main">Runcinated 24-cells</span>

In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination of the regular 24-cell.

<span class="mw-page-title-main">Simplicial polytope</span> Polytope whose facets are all simplices

In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a simplicial polyhedron in three dimensions contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph.

<span class="mw-page-title-main">Uniform honeycombs in hyperbolic space</span> Tiling of hyperbolic 3-space by uniform polyhedra

In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.

<span class="mw-page-title-main">Order-4 hexagonal tiling honeycomb</span>

In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

<span class="mw-page-title-main">Order-6 cubic honeycomb</span>

The order-6 cubic honeycomb is a paracompact regular space-filling tessellation in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

<span class="mw-page-title-main">Order-6 dodecahedral honeycomb</span> Regular geometrical object in hyperbolic space

The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.

<span class="mw-page-title-main">Order-5 hexagonal tiling honeycomb</span>

In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere, a flat plane in hyperbolic space that approaches a single ideal point at infinity.

<span class="mw-page-title-main">Square tiling honeycomb</span>

In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.

<span class="mw-page-title-main">Order-4 octahedral honeycomb</span>

The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.

References

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  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.
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