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Mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
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). In contrast, 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.


Applications

Mathematical visualization is used throughout mathematics, particularly in the fields of
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
. Notable examples include
plane curve In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...
s,
space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s,
polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
,
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s,
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s (particularly numerical solutions, as in
fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
or
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
s such as
soap film Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Plat ...
s),
conformal map In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-prese ...
s,
fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
s, and
chaos Chaos or CHAOS may refer to: Science, technology, and astronomy * '' Chaos: Making a New Science'', a 1987 book by James Gleick * Chaos (company), a Bulgarian rendering and simulation software company * ''Chaos'' (genus), a genus of amoebae * ...
.


Geometry

Geometry can be defined as the study of shapes their size, angles, dimensions and proportions


Linear algebra


Complex analysis

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, 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 In complex analysis, domain coloring or a color wheel graph is a technique for visualizing complex functions by assigning a color to each point of the complex plane. By assigning points on the complex plane to different colors and brightness, do ...
.


Chaos theory


Differential geometry


Topology

Many people have a vivid “mind’s eye,” but a team of British scientists has found that tens of millions of people cannot conjure images. The lack of a mental camera is known as aphantasia, and millions more experience extraordinarily strong mental imagery, called hyperphantasia. Researchers are studying how these two conditions arise through changes in the wiring of the brain. Visualization played an important role at the beginning of topological knot theory, when polyhedral decompositions were used to compute the homology of covering spaces of knots. Extending to 3 dimensions the physically impossible
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s 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 fact that these imaginary objects have no "real" existence does not stand in the way of their usefulness for proving knots distinct. It was the key to Perko's 1973 discovery of the duplicate knot type in Little's 1899 table of 10-crossing knots.


Graph theory

Permutation groups have nice visualizations of their elements that assist in explaining their structure—e.g., the rotated and flipped regular p-gons that comprise the dihedral group of order 2p. They may be used to "see" the relationships among linking numbers between branch curves of dihedral covering spaces of knots and links.


Combinatorics


Cellular automata

Stephen Wolfram Stephen Wolfram ( ; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer algebra and theoretical physics. In 2012, he was named a fellow of the American Mathematical So ...
's book on
cellular automata A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
, ''
A New Kind of Science ''A New Kind of Science'' is a book by Stephen Wolfram, published by his company Wolfram Research under the imprint Wolfram Media in 2002. It contains an empirical and systematic study of computational systems such as cellular automata. Wolfram ...
'' (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.


Computation


Other examples

* 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 In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral sur ...
demonstrates the 27 lines on a cubic surface. *
Sphere eversion In differential topology, sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (the word ''wikt:eversion#English, eversion'' means "turning inside out"). It is possible to smoothly and continuou ...
– 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 ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume was published in 1953. Each issue of the magazine ...
'' regularly features a mathematical visualization.


See also

* Geometry Center *
Mathematical diagram Mathematical diagrams, such as charts and graphs, are mainly designed to convey mathematical relationships—for example, comparisons over time. Specific types of mathematical diagrams Argand diagram A complex number can be visually repres ...
*
Parametric surface A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that oc ...
*
List of mathematical art software Gallery See also * ASCII art * Computer-based mathematics education * Computer representation of surfaces * For loop * Fractal-generating software * Julia set * Lambert W function * Lens space * List of interactive geometry software * List of ...


References

*


External links


Virtual Math Museum
{{Computer science Geometry Visualization (graphics)