Cubic plane curveQuartic plane curveConic sectionsUnit circleUnit hyperbolaQuintic of l'Hospital [1]AstroidAtriphtaloidNephroidQuadrifoliumEpicycloidEpispiralEpitrochoidHypocycloidLissajous curvePoinsot's spiralsRational normal curveRose curveBicuspid curveCassini ovalCassinoideCubic curveElliptic curveWatt's curveButterfly curveElkies trinomial curvesHyperelliptic curveKlein quarticClassical modular curveBolza surfaceMacbeath surfacePolynomial lemniscateFermat curveSinusoidal spiralSuperellipseHurwitz surfaceBowditch curveBrachistochroneButterfly curveCatenaryCléliesCochleoidCycloidHoropterInterscendental curve: A curve defined using irrational exponents, such as y=x√2. See Salmon (1879, p. 206)IsochroneIsochrone of Huygens (Tautochrone)Isochrone of Leibniz[2]Isochrone of Varignon[3]Lamé curvePursuit curveRhumb lineSpiralsArchimedean spiralCornu spiralCotes' spiralFermat's spiralGalileo's spiral [4]Hyperbolic spiralLituusLogarithmic spiralNielsen's spiralSyntractrixTractrixTrochoidBézier curveSplinesB-splineNonuniform rational B-splineOgeeLoess curveLowessPolygonal curveMaurer roseReuleaux triangleBézier triangleConchospiralHelixTendril perversion (a transition between back-to-back helices)Hemihelix, a quasi-helical shape characterized by multiple tendril perversionsSeiffert's spiral [5]Slinky spiral [6]Twisted cubicViviani's curvePlaneQuadric surfacesConeCylinderEllipsoidSpheroidSphereHyperboloidParaboloidMöbius stripTorusCatalan's minimal surfaceCosta's minimal surfaceCatenoidEnneper surfaceGyroidHelicoidLidinoidRiemann's minimal surfaceSaddle towerScherk surfaceSchwarz minimal surfaceTriply periodic minimal surfaceKlein bottleReal projective planeCross-capRoman surfaceBoy's surfaceSphereSpheroidOblate spheroidConeEllipsoidHyperboloid of one sheetHyperboloid of two sheetsHyperbolic paraboloid (a ruled surface)ParaboloidSphericonOloidDini's surfacePseudosphereSee the list of algebraic surfaces.
Cayley cubicBarth sexticClebsch cubicMonkey saddle (saddle-like surface for 3 legs.)TorusDupin cyclide (inversion of a torus)Whitney umbrellaRight conoid (a ruled surface)Random fractals
von Koch curve with random intervalvon Koch curve with random orientationBoundary of Brownian motion2D polymerPercolation front in 2D, Corrosion front in 2Ddiffusion-limited aggregationRandom walk with no self-intersection3D polymer2D percolation cluster hull2D percolation clusterBrownian motionLichtenberg figureBrownian motionPercolation theoryMultiplicative cascadeThis table shows a summary of regular polytope counts by dimension.
There are no nonconvex Euclidean regular tessellations in any number of dimensions.
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 elementFace, a 2-dimensional elementCell, a 3-dimensional elementHypercell or Teron, a 4-dimensional elementFacet, an (n-1)-dimensional elementRidge, an (n-2)-dimensional elementPeak, an (n-3)-dimensional elementFor 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.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.
PointThere is only one polytope in 1 dimension, whose boundaries are the two endpoints of a line segment, represented by the empty Schläfli symbol {}.
PolygonEquilateralCyclic polygonConvex polygonStar polygonPentagramRegular polygonEquilateral triangleSimplexSquareCross-polytopeHypercubePentagonHexagonHeptagonOctagonEnneagonDecagonHendecagonDodecagonTridecagonTetradecagonPentadecagonHexadecagonHeptadecagonOctadecagonEnneadecagonIcosagonHectogonChiliagonRegular polygonDegenerate (spherical)
HenagonDigonstar polygonPentagramHeptagramOctagramEnneagramDecagramApeirogonpolyhedronPlatonic solidTetrahedron, the 3-space SimplexCube, the 3-space hypercubeOctahedron, the 3-space Cross-polytopeDodecahedronIcosahedronhosohedrondihedronHenagon#In spherical geometryKepler–Poinsot polyhedraSmall stellated dodecahedronGreat dodecahedronGreat stellated dodecahedronGreat icosahedronEuclidean tilings
Square tilingTriangular tilingHexagonal tilingApeirogonDihedronHyperbolic tilings
Lobachevski planeHyperbolic tilingHyperbolic star-tilings
Order-7 heptagrammic tilingHeptagrammic-order heptagonal tilingOrder-9 enneagrammic tilingEnneagrammic-order enneagonal tilingconvex regular 4-polytope5-cell, the 4-space Simplex8-cell, the 4-space Hypercube16-cell, the 4-space Cross-polytope24-cell120-cell600-cellDitopeHosotope3-sphereStar or (Schläfli–Hess) regular 4-polytopeIcosahedral 120-cellSmall stellated 120-cellGreat 120-cellGrand 120-cellGreat stellated 120-cellGrand stellated 120-cellGreat grand 120-cellGreat icosahedral 120-cellGrand 600-cellGreat grand stellated 120-cellHoneycombCubic honeycombHosohedronDihedronOrder-2 apeirogonal tilingApeirogonal hosohedronOrder-4 square hosohedral honeycombOrder-6 triangular hosohedral honeycombHexagonal hosohedral honeycombOrder-2 square tilingOrder-2 triangular tilingOrder-2 hexagonal tilingOrder-4 dodecahedral honeycombOrder-5 dodecahedral honeycombOrder-5 cubic honeycombIcosahedral honeycombOrder-3 icosahedral honeycombOrder-4 octahedral honeycombTriangular tiling honeycombSquare tiling honeycombOrder-4 square tiling honeycombOrder-6 tetrahedral honeycombOrder-6 cubic honeycombOrder-6 dodecahedral honeycombHexagonal tiling honeycombOrder-4 hexagonal tiling honeycombOrder-5 hexagonal tiling honeycombOrder-6 hexagonal tiling honeycombFive-dimensional regular polytopes and higher
5-polytopeHoneycombTetracombhoneycombsTesseractic honeycomb16-cell honeycomb24-cell honeycombTessellations of Euclidean 5-space and higher
Hypercubic honeycombHypercubeSquare tilingCubic honeycombTesseractic honeycomb5-cube honeycomb6-cube honeycomb7-cube honeycomb8-cube honeycombHypercubic honeycombhoneycombsOrder-5 5-cell honeycomb120-cell honeycombOrder-5 tesseractic honeycombOrder-4 120-cell honeycombOrder-5 120-cell honeycombOrder-4 24-cell honeycombCubic honeycomb honeycombSmall stellated 120-cell honeycombPentagrammic-order 600-cell honeycombOrder-5 icosahedral 120-cell honeycombGreat 120-cell honeycomb5-orthoplex honeycomb24-cell honeycomb honeycomb16-cell honeycomb honeycombOrder-4 24-cell honeycomb honeycombTesseractic honeycomb honeycombApeirotopeApeirogonApeirohedronRegular skew polyhedronAbstract polytope11-cell57-cellConvex polygonConcave polygonConstructible polygonCyclic polygonEquiangular polygonEquilateral polygonRegular polygonPenrose tilePolyformBalbisGnomonGolygonStar without crossing linesStar polygonHexagramStar of DavidHeptagramOctagramStar of LakshmidecagramPentagramPolygons named for their number of sides
List of uniform tilingsUniform tilings in hyperbolic planeArchimedean tilingSquare tilingTriangular tilingHexagonal tilingTruncated square tilingSnub square tilingTrihexagonal tilingTruncated hexagonal tilingRhombitrihexagonal tilingTruncated trihexagonal tilingSnub hexagonal tilingElongated triangular tilingRegular polyhedronPlatonic solidTetrahedronCubeOctahedronDodecahedronIcosahedronKepler–Poinsot polyhedron (regular star polyhedra)Great icosahedronSmall stellated dodecahedronGreat dodecahedronGreat stellated dodecahedronAbstract regular polyhedra (Projective polyhedron)HemicubeHemi-octahedronHemi-dodecahedronHemi-icosahedronArchimedean solidTruncated tetrahedronCuboctahedronTruncated cubeTruncated octahedronRhombicuboctahedronTruncated cuboctahedronSnub cubeIcosidodecahedronTruncated dodecahedronTruncated icosahedronRhombicosidodecahedronTruncated icosidodecahedronSnub dodecahedronPrismatic uniform polyhedronPrismAntiprismUniform star polyhedronCatalan solidTriakis tetrahedronRhombic dodecahedronTriakis octahedronTetrakis hexahedronDeltoidal icositetrahedronDisdyakis dodecahedronPentagonal icositetrahedronRhombic triacontahedronTriakis icosahedronPentakis dodecahedronDeltoidal hexecontahedronDisdyakis triacontahedronPentagonal hexecontahedronnon-convexPyramidBipyramidDisphenoidParallelepipedCuboidRhombohedronTrapezohedronFrustumTrapezo-rhombic dodecahedronRhombo-hexagonal dodecahedronTruncated trapezohedronDeltahedronZonohedronPrismatoidCupolaBicupolaDihedronHosohedronConvex uniform honeycombDual uniform honeycombDisphenoid tetrahedral honeycombRhombic dodecahedral honeycombOthersTrapezo-rhombic dodecahedral honeycombWeaire–Phelan structureConvex uniform honeycombs in hyperbolic spaceOrder-4 dodecahedral honeycombOrder-5 cubic honeycombOrder-5 dodecahedral honeycombIcosahedral honeycombPolyhedral compound and Uniform polyhedron compound4-polytopeHecatonicosachoronHexacosichoronHexadecachoronIcositetrachoronPentachoronTesseractSpherical coneConvex regular 4-polytope5-cell, Tesseract, 16-cell, 24-cell, 120-cell, 600-cellAbstract regular polytope11-cell, 57-cellSchläfli–Hess 4-polytope (Regular star 4-polytope)Icosahedral 120-cell, Small stellated 120-cell, Great 120-cell, Grand 120-cell, Great stellated 120-cell, Grand stellated 120-cell, Great grand 120-cell, Great icosahedral 120-cell, Grand 600-cell, Great grand stellated 120-cellUniform 4-polytopeRectified 5-cell, Truncated 5-cell, Cantellated 5-cell, Runcinated 5-cellRectified tesseract, Truncated tesseract, Cantellated tesseract, Runcinated tesseractRectified 16-cell, Truncated 16-cellRectified 24-cell, Truncated 24-cell, Cantellated 24-cell, Runcinated 24-cell, Snub 24-cellRectified 120-cell, Truncated 120-cell, Cantellated 120-cell, Runcinated 120-cellRectified 600-cell, Truncated 600-cell, Cantellated 600-cellPrismatic uniform polychoronGrand antiprismDuoprismTetrahedral prism, Truncated tetrahedral prismTruncated cubic prism, Truncated octahedral prism, Cuboctahedral prism, Rhombicuboctahedral prism, Truncated cuboctahedral prism, Snub cubic prismTruncated dodecahedral prism, Truncated icosahedral prism, Icosidodecahedral prism, Rhombicosidodecahedral prism, Truncated icosidodecahedral prism, Snub dodecahedral prismUniform antiprismatic prismTesseractic honeycomb24-cell honeycombSnub 24-cell honeycombRectified 24-cell honeycombTruncated 24-cell honeycomb16-cell honeycomb5-cell honeycombOmnitruncated 5-cell honeycombTruncated 5-cell honeycombOmnitruncated 5-simplex honeycombregular 5-polytope5-dimensional cross-polytope5-dimensional hypercube5-dimensional simplexFive-dimensional space, 5-polytope and uniform 5-polytope5-simplex, Rectified 5-simplex, Truncated 5-simplex, Cantellated 5-simplex, Runcinated 5-simplex, Stericated 5-simplex5-demicube, Truncated 5-demicube, Cantellated 5-demicube, Runcinated 5-demicube5-cube, Rectified 5-cube, 5-cube, Truncated 5-cube, Cantellated 5-cube, Runcinated 5-cube, Stericated 5-cube5-orthoplex, Rectified 5-orthoplex, Truncated 5-orthoplex, Cantellated 5-orthoplex, Runcinated 5-orthoplexPrismatic uniform 5-polytopeFor each polytope of dimension
n, there is a prism of dimension
n+1.
5-cubic honeycomb5-simplex honeycombTruncated 5-simplex honeycomb5-demicubic honeycombSix-dimensional space, 6-polytope and uniform 6-polytope6-simplex, Rectified 6-simplex, Truncated 6-simplex, Cantellated 6-simplex, Runcinated 6-simplex, Stericated 6-simplex, Pentellated 6-simplex6-demicube, Truncated 6-demicube, Cantellated 6-demicube, Runcinated 6-demicube, Stericated 6-demicube6-cube, Rectified 6-cube, 6-cube, Truncated 6-cube, Cantellated 6-cube, Runcinated 6-cube, Stericated 6-cube, Pentellated 6-cube6-orthoplex, Rectified 6-orthoplex, Truncated 6-orthoplex, Cantellated 6-orthoplex, Runcinated 6-orthoplex, Stericated 6-orthoplex122 polytope, 221 polytope6-cubic honeycomb6-simplex honeycomb6-demicubic honeycomb222 honeycombSeven-dimensional space, uniform 7-polytope7-simplex, Rectified 7-simplex, Truncated 7-simplex, Cantellated 7-simplex, Runcinated 7-simplex, Stericated 7-simplex, Pentellated 7-simplex, Hexicated 7-simplex7-demicube, Truncated 7-demicube, Cantellated 7-demicube, Runcinated 7-demicube, Stericated 7-demicube, Pentellated 7-demicube7-cube, Rectified 7-cube, 7-cube, Truncated 7-cube, Cantellated 7-cube, Runcinated 7-cube, Stericated 7-cube, Pentellated 7-cube, Hexicated 7-cube7-orthoplex, Rectified 7-orthoplex, Truncated 7-orthoplex, Cantellated 7-orthoplex, Runcinated 7-orthoplex, Stericated 7-orthoplex, Pentellated 7-orthoplex132 polytope, 231 polytope, 321 polytope7-cubic honeycomb7-demicubic honeycomb331 honeycomb, 133 honeycombEight-dimensional space, uniform 8-polytope8-simplex, Rectified 8-simplex, Truncated 8-simplex, Cantellated 8-simplex, Runcinated 8-simplex, Stericated 8-simplex, Pentellated 8-simplex, Hexicated 8-simplex, Heptellated 8-simplex8-orthoplex, Rectified 8-orthoplex, Truncated 8-orthoplex, Cantellated 8-orthoplex, Runcinated 8-orthoplex, Stericated 8-orthoplex, Pentellated 8-orthoplex, Hexicated 8-orthoplex,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-cube8-demicube, Truncated 8-demicube, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube142 polytope, 241 polytope, 421 polytope, Truncated 421 polytope, Truncated 241 polytope, Truncated 142 polytope, Cantellated 421 polytope, Cantellated 241 polytope, Runcinated 421 polytope8-cubic honeycomb8-demicubic honeycomb521 honeycomb, 251 honeycomb, 152 honeycomb9-polytope9-cube9-demicube9-orthoplex9-simplexE9 honeycomb10-polytope10-cube10-demicube10-orthoplex10-simplexRegular polytope and List of regular polytopesSimplexHypercubeCross-polytopeUniform polytopeDemihypercubeUniform 1k2 polytopeUniform 2k1 polytopeUniform k21 polytopeHoneycombsHypercubic honeycombAlternated hypercubic honeycomb 
