Following is a list of some mathematically well-defined shapes .
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.
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 elementEdge, a 1-dimensional element Face, a 2-dimensional element Cell, a 3-dimensional element Hypercell or Teron, a 4-dimensional elementFacet , an (n -1)-dimensional elementRidge , an (n -2)-dimensional elementPeak, 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 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 Rectified 5-cell , Truncated 5-cell , Cantellated 5-cell , Runcinated 5-cell Rectified tesseract , Truncated tesseract , Cantellated tesseract , Runcinated tesseract Rectified 16-cell , Truncated 16-cell Rectified 24-cell , Truncated 24-cell , Cantellated 24-cell , Runcinated 24-cell , Snub 24-cell Rectified 120-cell , Truncated 120-cell , Cantellated 120-cell , Runcinated 120-cell Rectified 600-cell , Truncated 600-cell , Cantellated 600-cell Prismatic uniform polychoron Grand antiprism Duoprism Tetrahedral prism , Truncated tetrahedral prism Truncated cubic prism , Truncated octahedral prism , Cuboctahedral prism , Rhombicuboctahedral prism , Truncated cuboctahedral prism , Snub cubic prism Truncated dodecahedral prism , Truncated icosahedral prism , Icosidodecahedral prism , Rhombicosidodecahedral prism , Truncated icosidodecahedral prism , Snub dodecahedral prism Uniform antiprismatic prism Honeycombs 5D with 4D surfaces Five-dimensional space , 5-polytope and uniform 5-polytope 5-simplex , Rectified 5-simplex , Truncated 5-simplex , Cantellated 5-simplex , Runcinated 5-simplex , Stericated 5-simplex 5-demicube , Truncated 5-demicube , Cantellated 5-demicube , Runcinated 5-demicube 5-cube , Rectified 5-cube , 5-cube , Truncated 5-cube , Cantellated 5-cube , Runcinated 5-cube , Stericated 5-cube 5-orthoplex , Rectified 5-orthoplex , Truncated 5-orthoplex , Cantellated 5-orthoplex , Runcinated 5-orthoplex 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 6-simplex , Rectified 6-simplex , Truncated 6-simplex , Cantellated 6-simplex , Runcinated 6-simplex , Stericated 6-simplex , Pentellated 6-simplex 6-demicube , Truncated 6-demicube , Cantellated 6-demicube , Runcinated 6-demicube , Stericated 6-demicube 6-cube , Rectified 6-cube , 6-cube , Truncated 6-cube , Cantellated 6-cube , Runcinated 6-cube , Stericated 6-cube , Pentellated 6-cube 6-orthoplex , Rectified 6-orthoplex , Truncated 6-orthoplex , Cantellated 6-orthoplex , Runcinated 6-orthoplex , Stericated 6-orthoplex 122 polytope , 221 polytope Honeycombs Seven dimensions Seven-dimensional space , uniform 7-polytope 7-simplex , Rectified 7-simplex , Truncated 7-simplex , Cantellated 7-simplex , Runcinated 7-simplex , Stericated 7-simplex , Pentellated 7-simplex , Hexicated 7-simplex 7-demicube , Truncated 7-demicube , Cantellated 7-demicube , Runcinated 7-demicube , Stericated 7-demicube , Pentellated 7-demicube 7-cube , Rectified 7-cube , 7-cube , Truncated 7-cube , Cantellated 7-cube , Runcinated 7-cube , Stericated 7-cube , Pentellated 7-cube , Hexicated 7-cube 7-orthoplex , Rectified 7-orthoplex , Truncated 7-orthoplex , Cantellated 7-orthoplex , Runcinated 7-orthoplex , Stericated 7-orthoplex , Pentellated 7-orthoplex 132 polytope , 231 polytope , 321 polytope Honeycombs Eight dimension Eight-dimensional space , uniform 8-polytope 8-simplex , Rectified 8-simplex , Truncated 8-simplex , Cantellated 8-simplex , Runcinated 8-simplex , Stericated 8-simplex , Pentellated 8-simplex , Hexicated 8-simplex , Heptellated 8-simplex 8-orthoplex , Rectified 8-orthoplex , Truncated 8-orthoplex , Cantellated 8-orthoplex , Runcinated 8-orthoplex, Stericated 8-orthoplex, Pentellated 8-orthoplex, Hexicated 8-orthoplex[citation needed ] 8-cube , Rectified 8-cube , Truncated 8-cube , Cantellated 8-cube, Runcinated 8-cube, Stericated 8-cube, Pentellated 8-cube, Hexicated 8-cube, Heptellated 8-cube[citation needed ] 8-demicube , Truncated 8-demicube , Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube[citation needed ] 142 polytope , 241 polytope , 421 polytope , Truncated 421 polytope, Truncated 241 polytope, Truncated 142 polytope, Cantellated 421 polytope, Cantellated 241 polytope, Runcinated 421 polytope[citation needed ] 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 Glowvoid Warith's void Warith's hyperplexicon shape Gaxxoid Gyroid Hyperplexicon Gyroid Planetium Epyoid Xenroid Xenoshape Xenoid EmperoidsHypervoid Hyperoid Warith-Nathaniyal mixbox Mixbox Forcoid Corporoid Primoid Oppan's gyroid Zahian's Hyperplexicon Nathaniyal's object Hyperplexicon Geometry and other areas of mathematics Ford circles Glyphs and symbols Table of all the Shapes This is a table of all the shapes above.
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 .