Mathematical visualization

Mathematical visualization is an aspect of Mathematics which allows one to understand and explore mathematical phenomena 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.


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.

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.
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