Mathematical visualization

In contrast, today it most frequently consists of using computers to make static two- or three-dimensional drawings, animations, or interactive programs.

Notable examples include plane curves, space curves, polyhedra, ordinary differential equations, partial differential equations (particularly numerical solutions, as in fluid dynamics or minimal surfaces such as soap films), conformal maps, fractals, and chaos.

Extending to 3 dimensions the physically impossible Riemann surfaces used to classify all closed orientable 2-manifolds, Heegaard's 1898 thesis "looked at" similar structures for functions of two complex variables, taking an imaginary 4-dimensional surface in Euclidean 6-space (corresponding to the function f=x^2-y^3) and projecting it stereographically (with multiplicities) onto the 3-sphere.

In the 1920s Alexander and Briggs used this technique to compute the homology of cyclic branched covers of knots with 8 or fewer crossings, successfully distinguishing them all from each other (and the unknot).

By 1932 Reidemeister extended this to 9 crossings, relying on linking numbers between branch curves of non-cyclic knot covers.

The Mandelbrot set , one of the most famous examples of mathematical visualization.
An illustration of Desargues' theorem , an important result in Euclidean and projective geometry
In three-dimensional Euclidean space , these three planes represent solutions of linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.
Domain coloring of:
f ( x ) = ⁠ ( x 2 −1)( x −2− i ) 2 / x 2 +2+2 i ⁠
A plot of the Lorenz attractor for values r = 28 , σ = 10 , b = 8/3
Costa's Minimal Surface
A table of all prime knots with seven crossings or fewer (not including mirror images).
A force-based network visualization. [ 2 ]
An example of change ringing (with six bells), one of the earliest nontrivial results in graph theory .
Gosper's Glider Gun creating " gliders " in the cellular automaton Conway's Game of Life [ 4 ]
"Inelegant" is a translation of Knuth's version of the algorithm with a subtraction-based remainder-loop replacing his use of division (or a "modulus" instruction). Derived from Knuth 1973:2–4.
A proof without words of the Pythagorean theorem in Zhoubi Suanjing .
A Morin surface , the half-way stage in turning a sphere inside out .
Three random walks