Mathematical visualization

Mathematical phenomena can be understood and explored via visualization. Classically this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century), while today it most frequently consists of using computers to make static two or three dimensional drawings, animations, or interactive programs. Writing programs to visualize mathematics is an aspect of computational geometry.

For mathematical visualization in the context of science, see scientific_visualization § In_mathematics.
The Mandelbrot set, one of the most famous examples of mathematical visualization.

Applications

Mathematical visualization is used throughout mathematics, particularly in the fields of geometry and analysis. 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.

Geometry
An illustration of Desargues' theorem, an important result in Euclidean and projective geometry

Linear algebra
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.

Complex analysis
Domain coloring of:
f(x) = (x2−1)(x−2−i)2/x2+2+2i

In complex analysis, functions of the complex plane are inherently 4-dimensional, but there is no natural geometric projection into lower dimensional visual representations. Instead, colour vision is exploited to capture dimensional information using techniques such as domain coloring.

Chaos theory
A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3

Differential geometry
Costa's Minimal Surface

Topology
A table of all prime knots with seven crossings or fewer (not including mirror images).

Graph theory
A force-based network visualization.[1]

Combinatorics
An example of change ringing (with six bells), one of the earliest nontrivial results in graph theory.

Cellular automata
Gosper's Glider Gun creating "gliders" in the cellular automaton Conway's Game of Life[2]

Stephen Wolfram's book on cellular automata, A New Kind of Science (2002), is one of the most intensely visual books published in the field of mathematics. It has been criticized for being too heavily visual, with much information conveyed by pictures that do not have formal meaning.[3]

Computation
"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.

Other examples
A proof without words of the Pythagorean theorem in Zhoubi Suanjing.
  • Proofs without words have existed since antiquity, as in the Pythagorean theorem proof found in the Zhoubi Suanjing Chinese text which dates from 1046 BC to 256 BC.
  • The Clebsch diagonal surface demonstrates the 27 lines on a cubic surface.
A Morin surface, the half-way stage in turning a sphere inside out.
  • Sphere eversion – that a sphere can be turned inside out in 3 dimension if allowed to pass through itself, but without kinks – was a startling and counter-intuitive result, originally proven via abstract means, later demonstrated graphically, first in drawings, later in computer animation.

The cover of the journal The Notices of the American Mathematical Society regularly features a mathematical visualization.

Three random walks

See also

References
  1. Published in Grandjean, Martin (2014). "La connaissance est un réseau". Les Cahiers du Numérique. 10 (3): 37–54. doi:10.3166/lcn.10.3.37-54. Retrieved 2014-10-15.
  2. Daniel Dennett (1995), Darwin's Dangerous Idea, Penguin Books, London, ISBN 978-0-14-016734-4, ISBN 0-14-016734-X
  3. Berry, Michael; Ellis, John; Deutch, David (15 May 2002). "A Revolution or self indulgent hype? How top scientists view Wolfram" (PDF). The Daily Telegraph. Retrieved 14 August 2012.
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