2D with 1D surface

• Bézier curve
  Bézier curve
  A Bézier curve is a parametric curve frequently used in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case....
  
  : (As + Bt)ⁿ | 0 ≤ s ≤ 1 ∧ 0 ≤ t ≤ 1 ∧ s + t = 1, Aⁿ, A^n − 1B, ..., Bⁿ are vectors
• circle
  Circle
  A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
  
  : x² + y² = r²
• ellipse
  Ellipse
  In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
  
  
• parabola
  Parabola
  In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
  
  
• hyperbola
  Hyperbola
  In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
  
  
• plane
  Plane (mathematics)
  In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
  
  
• polygon
  Polygon
  In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
  
  
  • chiliagon
  • decagon
    Decagon
    In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and each internal angle equal to 144°...
    
    
  • enneagon
    Enneagon
    In geometry, a nonagon is a nine-sided polygon.The name "nonagon" is a prefix hybrid formation, from Latin , used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century...
    
    
  • googolgon
  • hectagon
  • heptagon
  • hendecagon
    Hendecagon
    In geometry, a hendecagon is an 11-sided polygon....
    
    
  • hexagon
  • myriagon
  • octagon
  • pentagon
    Pentagon
    In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
    
    
  • quadrilateral
    Quadrilateral
    In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
    
    
  • triangle
    Triangle
    A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
    
    

3D with 2D surface

• Bézier triangle: (As + Bt + Cu)ⁿ | 0 ≤ s ≤ 1 ∧ 0 ≤ t ≤ 1 ∧ 0 ≤ u ≤ 1 ∧ s + t + u = 1, A^pB^qC^r vectors if p + q + r = n and p, q, r are nonnegative integers
• cylinder
  Cylinder (geometry)
  A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
  
  
• hyperplane
  Hyperplane
  A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
  
  
• möbius strip
  Möbius strip
  The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
  
  
• platonic solid
  Platonic solid
  In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
  
  
  • dodecahedron
  • hexahedron
    Hexahedron
    A hexahedron is any polyhedron with six faces, although usually implies the cube as a regular hexahedron with all its faces square, and three squares around each vertex....
    
     (cube)
  • icosahedron
    Icosahedron
    In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
    
    
  • octahedron
    Octahedron
    In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
    
    
  • tetrahedron
    Tetrahedron
    In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
    
    
• torus
  Torus
  In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
  
   (doughnut)
• quadric
  Quadric
  In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
  
  
  • cone
    Cone (geometry)
    A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...
    
    
  • cylinder
    Cylinder (geometry)
    A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
    
    
  • ellipsoid
    • spheroid
      Spheroid
      A spheroid, or ellipsoid of revolution is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters....
      
      
      • sphere
        Sphere
        A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
        
        
  • hyperboloid
  • paraboloid
    Paraboloid
    In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....
    
    
  • sphere
    Sphere
    A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
    
    

4D with 3D surface

• polychoron
  Polychoron
  In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...
  
  
  • hecatonicosachoron
  • hexacosichoron
  • hexadecachoron
  • icositetrachoron
  • pentachoron
    Pentachoron
    In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or hyperpyramid...
    
     (simplex
    Simplex
    In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
    
    )
  • tesseract
    Tesseract
    In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
    
    
• spherical cone
  Spherical cone
  In geometry, a spherical cone is the surface of revolution obtained by rotating a sector of a circle about one of its straight edges.- External links :* at MathWorld...
  
  

5D with 4D surfaces

• regular 5-polytopes:
  • 5-dimensional simplex
    Simplex
    In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
    
    
  • 5-dimensional cross-polytope
    Cross-polytope
    In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of . The cross-polytope is the convex hull of its vertices...
    
    
  • 5-dimensional hypercube
    Hypercube
    In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
    
     (5-measure polytope)

Fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

s

• Apollonian gasket
  Apollonian gasket
  In mathematics, an Apollonian gasket or Apollonian net is a fractal generated from triples of circles, where each circle is tangent to the other two. It is named after Greek mathematician Apollonius of Perga.-Construction:...
  
  
• Cantor set
  Cantor set
  In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
  
  
• Dragon curve
  Dragon curve
  A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems.-Heighway dragon:...
  
  
• Koch snowflake
  Koch snowflake
  The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described...
  
  
• Lévy C curve
  Lévy C curve
  In mathematics, the Lévy C curve is a self-similar fractal that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and G...
  
  
• Lyapunov fractal
  Lyapunov fractal
  In mathematics, Lyapunov fractals are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values A and B.A Lyapunov fractal is constructed by mapping the regions of stability and chaotic...
  
  
• Mandelbrot set
  Mandelbrot set
  The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
  
  
• Sierpinski carpet
  Sierpinski carpet
  The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions . Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional...
  
  
• Peano curve
• Sierpinski triangle
  Sierpinski triangle
  The Sierpinski triangle , also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral...
  
  

See also

• List of mathematical topics
• The Periodic table of mathematical shapes
  Periodic table of shapes
  The Periodic table of mathematical shapes is a concept thought up by Professor Alessio Corti, from the Department of Mathematics at Imperial College London. It aims to categorise all three-, four- and five-dimensional shapes into a single table, analogous to the periodic table of chemical elements...