Mathematical diagram
Mathematical diagrams, such as charts and graphs, are mainly designed to convey mathematical relationships—for example, comparisons over time.[1]
Specific types of mathematical diagrams
Argand diagram
A
The concept of the complex plane allows a
Butterfly diagram
In the context of fast Fourier transform algorithms, a butterfly is a portion of the computation that combines the results of smaller discrete Fourier transforms (DFTs) into a larger DFT, or vice versa (breaking a larger DFT up into subtransforms). The name "butterfly" comes from the shape of the data-flow diagram in the radix-2 case, as described below. The same structure can also be found in the Viterbi algorithm, used for finding the most likely sequence of hidden states.
The butterfly diagram show a data-flow diagram connecting the inputs x (left) to the outputs y that depend on them (right) for a "butterfly" step of a radix-2 Cooley–Tukey FFT algorithm. This diagram resembles a butterfly as in the Morpho butterfly shown for comparison, hence the name.
Commutative diagram
In mathematics, and especially in
Commutative diagrams play the role in category theory that equations play in algebra.
Hasse diagrams
A Hasse diagram is a simple picture of a finite partially ordered set, forming a drawing of the partial order's transitive reduction. Concretely, one represents each element of the set as a vertex on the page and draws a line segment or curve that goes upward from x to y precisely when x < y and there is no z such that x < z < y. In this case, we say y covers x, or y is an immediate successor of x. In a Hasse diagram, it is required that the curves be drawn so that each meets exactly two vertices: its two endpoints. Any such diagram (given that the vertices are labeled) uniquely determines a partial order, and any partial order has a unique transitive reduction, but there are many possible placements of elements in the plane, resulting in different Hasse diagrams for a given order that may have widely varying appearances.
Knot diagrams
In Knot theory a useful way to visualise and manipulate knots is to project the knot onto a plane—;think of the knot casting a shadow on the wall. A small perturbation in the choice of projection will ensure that it is one-to-one except at the double points, called crossings, where the "shadow" of the knot crosses itself once transversely[3]
At each crossing we must indicate which section is "over" and which is "under", so as to be able to recreate the original knot. This is often done by creating a break in the strand going underneath. If by following the diagram the knot alternately crosses itself "over" and "under", then the diagram represents a particularly well-studied class of knot, alternating knots.
Venn diagram
A Venn diagram is a representation of mathematical sets: a mathematical diagram representing sets as circles, with their relationships to each other expressed through their overlapping positions, so that all possible relationships between the sets are shown.[4]
The Venn diagram is constructed with a collection of simple closed curves drawn in the plane. The principle of these diagrams is that classes be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram. That is, the diagram initially leaves room for any possible relation of the classes, and the actual or given relation, can then be specified by indicating that some particular region is null or is not null.[5]
Voronoi diagram
A
In the simplest case, we are given a set of points S in the plane, which are the Voronoi sites. Each site s has a Voronoi cell V(s) consisting of all points closer to s than to any other site. The segments of the Voronoi diagram are all the points in the plane that are equidistant to two sites. The Voronoi nodes are the points equidistant to three (or more) sites
Wallpaper group diagrams
A wallpaper group or plane symmetry group or plane crystallographic group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinct groups.
Wallpaper groups are two-dimensional
Young diagram
A Young diagram or
Listing the number of boxes in each row gives a
Young tableaux were introduced by
Other mathematical diagrams
- Cremona diagram
- De Finetti diagram
- Dynkin diagram
- Elementary diagram
- Euler diagram
- Stellation diagram
- Ulam spiral
- Van Kampen diagram
- Taylor diagram
See also
- Category theory
- Logic diagram
- Mathematical jargon
- Mathematical model
- Mathematics as a language
- Mathematical visualization
- Statistical model
References
- ^ Working with diagrams at LearningSpace.
- ISBN 978-0-521-58807-2.)
- ISBN 978-0-914098-16-4.
- ^ "Venn diagram" Archived 2009-11-07 at the Wayback Machine, Encarta World English Dictionary, North American Edition 2007. Archived 2009-11-01.
- Clarence Irving Lewis (1918). A Survey of Symbolic Logic. Republished in part by Dover in 1960. p. 157.
Further reading
- Barker-Plummer, Dave; Bailin, Sidney C. (1997). "The Role of Diagrams in Mathematical Proofs". Machine Graphics and Vision. 6 (1): 25–56. CiteSeerX 10.1.1.49.4712. (Special Issue on Diagrammatic Representation and Reasoning).
- Barker-Plummer, Dave; Bailin, Sidney C. (2001). "On the practical semantics of mathematical diagrams". In Anderson, M. (ed.). Reasoning with Diagrammatic Representations. ISBN 978-1-85233-242-6.
- Kidman, G. (2002). "The Accuracy of mathematical diagrams in curriculum materials". In Cockburn, A.; Nardi, E. (eds.). Proceedings of the PME 26. Vol. 3. University of East Anglia. pp. 201–8.
- ISBN 978-3-540-23029-8.
- Puphaiboon, K.; Woodcock, A.; Scrivener, S. (25 March 2005). "Design method for developing mathematical diagrams". In Bust, Philip D.; McCabe, P.T. (eds.). Contemporary ergonomics 2005 Proceedings of the International Conference on Contemporary Ergonomics (CE2005). Taylor & Francis. ISBN 978-0-415-37448-4.
External links
- "Diagrams". The Stanford Encyclopedia of Philosophy. Fall 2008.
- Kulpa, Zenon. "Diagrammatics: The art of thinking with diagrams". Archived from the original on April 25, 2013.
- One of the oldest extant diagrams from Euclid by Otto Neugebauer
- Lomas, Dennis (1998). "Diagrams in Mathematical Education: A Philosophical Appraisal". Philosophy of Education Society. Archived from the original on 2011-07-21.