Mathematical diagram

Euclid's Elements, ms. from Lüneburg, A.D. 1200

Mathematical diagrams are diagrams in the field of mathematics, and diagrams using mathematics such as charts and graphs, that are mainly designed to convey mathematical relationships, for example, comparisons over time.^[1]

[edit] Overview

Mathematicians use diagrams to convey information, demonstrate algorithms, and convey geometric relationships.^[2] Diagrams can also play a role in the construction of mathematical concepts. They provide an intuitive and mathematical explanation able to enhance the understanding of concepts that are difficult to grasp or that appear to be obscure and/or epistemologically unjustified. They can also help create new previously unknown concepts. Especially for users of mathematics, diagrams provide an efficient method of representation of complex information. To make the knowledge accessible to them, and managing complexity and intricacy of mathematical knowledge, one must use more efficient knowledge representation methods, both in the sense of efficiency of encoding and storing the knowledge and efficiency of reading and absorbing the knowledge by the user. Diagrams offer readable general comprehension of some part of knowledge "at a glance", allowing also for representation of precise structural relationships.^[3]

The use of diagrams in mathematics has a long history.^[3] Visual fields of mathematics such as geometry, rely heavily on diagrams, although algebraic subjects such as homological algebra, not traditionally considered a visual field, also do so. In number theory simple diagrams of a numbers axis played a role in the acceptance of negative numbers as mathematical entities. Also the invention of the complex plane diagram, the argand diagram stimulated the acceptance of imaginary numbers and the development of complex analysis. Despite this long history, mathematical diagrams are rarely treated as important enough to deserve a rigorous study.^[3]

[edit] History

Quantitative graphics have been central to the development of science from the earliest attempts to analyse data.^[4] In the early geometric diagrams and in the making of maps to aid in navigation and exploration data visualization arose.

Page with text and geometrical diagrams from Manuale Mathematicum, Straßburg 1619.

Galileo's demonstration of the law of the space traversed in case of uniformly varied motion. It is the same demonstration that Nicole Oresme had made centuries earlier.

In Greek mathematics, diagrams were used in both the writing and the reading of the texts. Nertz (1999)^[5]argues that in the presentation of a theorem, even orally, the diagram was assumed as complete and given from the outset, rather than constructed along the way. The text and the diagram are mutually interdependent in subtle and complex ways. In his study, Nertz demonstrates that Greek geometrical diagrams are frequently 'underspecified' or 'unspecified': that is, points or lines turn up in the diagrams which have been nowhere defined, or at best sketchily defined, in the body of the text. This leads to the conclusion that, first, many theorems cannot stand on their textual presentation alone but require their diagrams to make sense, and, second, that the diagram is not directly constructable from the text alone.^[5][6]

Planetary Movements diagrams from the 10th century A.D..^[4]

By the tenth century A.D. medieval astronomers depicted planetary movements as cyclic lines on a spatial-temporal grid, see image. This diagram from an unknown Astronomer in a translation of Commentary of Macrobius on Cicero's Somnium Scipinious, is the earliest known attempt^[7] to show changing values graphically positions of the sun, moon, and planets throughout the year. The diagram is strikingly similar to modern line graphs.^[4] In the fourteenth century Nicole Oresme conceived the idea of employing what we should now call rectangular coordinates, in modern terminology, a length proportionate to the longitudo was the abscissa at a given point, and a perpendicular at that point, proportional to the latitudo, was the ordinate. Oresme shows that a geometrical property of such a figure could be regarded as corresponding to a property of the form itself. The parameters longitudo and latitudo can vary or remain constant. Oresme defines latitudo uniformis as that which is represented by a line parallel to the longitude, and any other latitudo is difformis; the latitudo uniformiter difformis is represented by a right line inclined to the axis of the longitude. Oresme proved that this definition is equivalent to an algebraic relation in which the longitudes and latitudes of any three points would figure: i.e., he gives the equation of the right line, and thus long precedes Descartes in the invention of analytical geometry. In this doctrine, Oresme extends to figures of three dimensions.

By the 16th century, techniques and instruments for precise observation and measurement of physical quantities were well developed, and drew the beginnings of the husbandry of visualization. The 17th century saw great new growth in theory and the dawn of practice, with the rise of analytic geometry, theories of errors of measurement, the birth of probability theory, and the beginnings of demographic statistics and “political arithmetic”.^[7] Many familiar forms, including bivariate plots, statistical maps, bar charts and coordinate paper were used in the 18th century.^[4]

Over the 18th and 19th centuries, numbers pertaining to people—social, moral, medical, and economic statistics began to be gathered in large and periodic series; moreover, the usefulness of these bodies of data for planning, for governmental response, and as a subject worth of study in its own right, began to be recognized.^[7] This birth of statistical thinking was also accompanied by a rise in visual thinking: diagrams were used to illustrate mathematical proofs and functions; nomograms were developed to aid calculations; various graphic forms were invented to make the properties of empirical numbers—their trends, tendencies, and distributions—more easily communicated, or accessible to visual inspection. As well, the close relation of the numbers of the state (the origin of the word “statistics”) and its geography gave rise to the visual representation of such data on maps, now called “thematic cartography.^[7]

[edit] Mathematical diagram topics

[edit] Visualization in mathematics

Mathematicians have always used their “mind’s eye” to visualize the abstract objects and processes, that arise in all branches of mathematical research.^[8] The familiar types of visualization in mathematics and using mathematics are:

A pie chart.

Graph of the function

A simple, closed curve: a hypotrochoid.

A scatter plot.

Chart

A chart is a graphic depicting the relationship between two or more variables with a discrete or continuous value range, used, for instance, in visualising scientific data. Charts are often used to make it easier to understand large quantities of data and the relationship between different parts of the data. Charts can usually be read more quickly than the raw data that they come from. They are used in a wide variety of fields, and can be created by hand, often on graph paper, or by computer using a charting application.

Diagrams

A diagram is a 2D geometric symbolic representation of information according to some visualization technique. Sometimes, the technique uses a 3D visualization which is then projected onto the 2D surface. The word "diagram" is often coupled together with the word "chart" as in "charts and diagrams", within the larger conceptual framework of qualitative rather than quantitative displays. Charts can contain both quantitative and qualitative information. The term "charts and diagrams" is specially meant to address one class of genre: the kind that communicates qualitative information.^[9]

Geometric figures

The geometric shape of an object located in some space refers to the part of space occupied by the object as determined by its external boundary — abstracting from other aspects the object may have such as its colour, content, or the substance of which it is composed, as well as from the object's position and orientation in space, and its size.

Graph of a function

In mathematics a graph of a function f is the collection of all ordered pairs (x, f(x)). In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes.

The graph of a function on real numbers is identical to the graphic representation of the function. For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the closed graph theorem in functional analysis.

Mathematical curves

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle. In everyday use of the term "curve", a straight line is not curved, but in mathematical parlance curves include straight lines and line segments. A large number of other curves have been studied in geometry.

The distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading. Terminology here is not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more common in vector calculus and differential geometry.

Mathematical table

A table is both a mode of visual communication and a means of arranging data. The use of tables is pervasive throughout all communication, research and data analysis.

Creating tables is a common code optimization technique, and works as well for computers as humans. In computers, use of such tables is done in order to speed up calculations in those cases where a table lookup is faster than the corresponding calculations (particularly if the computer in question doesn't have a hardware implementation of the calculations). In essence, one trades computing speed for the computer memory space required to store the tables.

Plots

A plot is is a graphical technique for presenting a data set drawn by hand or produced by a mechanical or electronic plotter. It is a graph depicting the relationship between two or more variables used, for instance, in visualising scientific data.

[edit] Developing mathematical diagrams

Diagrams can act as cognitive "externalizations" enhancing cognition by mapping problem elements to a display so that solutions become apparent, according to Puphaiboon (2005)^[10]. The diagrams thus has become part of a larger cognitive process involving perceptual pattern finding and cognitive symbolic operations. Thus a diagram's effectiveness to some extent, depends on how well it is designed as an input to the perceptual system. It is important to apply perceptual psychology into the design guideline for designers.^[10]

[edit] Diagrammatic representations

Diagrams are a kind of analogical knowledge representation mechanism, that is characterized by a parallel, though not necessarily isomorphic, correspondence between the structure of the representation and the structure of the represented.^[11] For example, relative positions and distances of certain marks on a map are in direct correspondence to relative positions and distances of the cities they represent, whereas in a propositional representation, its parts or relationships between them need not correspond explicitly to any parts and relations within the thing denoted. The analogical representation, according to Zenon Kulpa (1997)^[12], can be said to model or depict the thing represented, whereas the propositional representation rather describes it. A similar distinction can be made regarding the method of retrieving information from the representation. The needed information can usually be simply observed or measured in the diagram, whereas it must be inferred from the descriptions of the facts and axioms comprising the propositional representation.^[12]

Advantages of diagrammatic representations are:^{[12][13] [14]}

• Possibilities of specifying and structuring in two dimensions: topological, metric, morphological: by connectivity, proximity, shape, etc., compared to those available in a one-dimensional linear strings of symbols in propositional formulations.^[12]
• Explicit representation and direct retrieval of information especially structural and spatial relations, that can be represented only implicitly in other types of representations and then has to be computed (or inferred), sometimes at great cost, to make it explicit for use.^[12]
• The ease of visual processing, because humans possess a well developed apparatus for making easy perceptual inferences on a diagram.^[12]

[edit] Mathematical visualization

Mathematical Visualization is a young new discipline. It offers efficient visualization tools to the classical subjects of mathematics, and applies mathematical techniques to problems in computer graphics and scientific visualization. Originally, it started in the interdisciplinary area of differential geometry, numerical mathematics, and computer graphics. In recent years, the methods developed have found important applications.^[15]

[edit] Specific types of mathematic diagrams

[edit] Argand diagram

Argand diagram.

A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745–1818).^[16] Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.

The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates – the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.

Butterfly diagram

[edit] 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. This diagram resembles a butterfly as in the Morpho butterfly shown for comparison), hence the name.

Commutative diagram.

[edit] Commutative diagram

In mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition.

Commutative diagrams play the role in category theory that equations play in algebra.

Hasse diagram.

[edit] Hasse diagrams

A Hasse diagram is a simple picture of a finite partially ordered set, forming a drawing of the transitive reduction of the partial order. Concretely, one represents each element of S as a vertex on the page and draws a line segment or curve that goes upward from x to y if 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. Furthermore it is required that the vertices are positioned in such a way that each curve 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 diagram.

[edit] 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 ^[17]

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.

[edit] 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.^[18]

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 notnull.^[19]

Voronoi centerlines.

[edit] Voronoi diagram

A Voronoi diagram is a special kind of decomposition of a metric space determined by distances to a specified discrete set of objects in the space, e.g., by a discrete set of points. This diagram is named after Georgy Voronoi, also called a Voronoi tessellation, a Voronoi decomposition, or a Dirichlet tessellation after Lejeune Dirichlet.

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 diagram.

[edit] 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 symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional crystallographic groups, also called space groups. Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns which are very different in style, color, scale or orientation may belong to the same group.

[edit] Young diagram

A Young diagram or Young tableau, also called Ferrers diagram, is a finite collection of boxes, or cells, arranged in left-justified rows, with the row sizes weakly decreasing (each row has the same or shorter length than its predecessor).

Young diagram.

Listing the number of boxes in each row gives a partition λ of a positive integer n, the total number of boxes of the diagram. The Young diagram is said to be of shape λ, and it carries the same information as that partition. Listing the number of boxes in each column gives another partition, the conjugate or transpose partition of λ; one obtains a Young diagram of that shape by reflecting the original diagram along its main diagonal.

Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians.

[edit] Other mathematical diagrams

• Cremona diagram
• De Finetti diagram
• Dynkin diagram
• Stellation diagram
• Ulam spiral
• Van Kampen diagram

[edit] Applications

Some of the familiar applications of mathematical visualization techniques are:

• Educational tool to augment plaster models of mathematical surfaces that inhabit display cases in many mathematics centers and the line drawings of textbooks.^[8]
• Use of mathematical visualization software to obtain fresh insights concerning complex and poorly understood mathematical objects^[8]
• Visual embodiment for scientists, who need and use mathematics but are not completely at ease with abstract mathematical notations and formulas. They can often better understand the mathematical concepts they have to deal with if these concepts can be given a visual embodiment.^[8]
• Aesthetic appeal: mathematical visualization has a strong aesthetic appeal, even to the lay public.^[8]
• Use by mathematicians to understand and prove mathematical theorems.

[edit] See also

• Category theory
• Logic diagram
• Mathematical jargon
• Mathematical model
• Mathematics as a language
• Statistical model

[edit] References

[edit] Further reading

• D. Barker-Plummer & S. Bailin (1997). "The Role of Diagrams in Mathematical Proofs". In: Machine GRAPHICS and VISION. 6(1): 25–56. (Special Issue on Diagrammatic Representation and Reasoning).
• D. Barker-Plummer & S.C. Bailin (2001). "On the practical semantics of mathematical diagrams". In: M. Anderson (ed.). Reasoning with Diagrammatic Representations. Springer Verlag.
• Kidman, G. (2002). "The Accuracy of mathematical diagrams in curriculum materials". In: Cockburn A and Nardi E. (Eds) Proceeding of the PME 26 Vol.3 201–208. UK:University of East Anglia.
• Zenon Kulpa (2004). "On Diagrammatic Representation of Mathematical Knowledge". In: Mathematical Knowledge Management, Andrea Asperti, Grzegorz Bancerek & Andrzej Trybulec (eds.). pp. 190–204.
• K. Puphaiboon, A. Woodcock & S. Scrivener (2005). "Design method for developing mathematical diagrams". In: Contemporary ergonomics 2005 Proceedings of the International Conference on Contemporary Ergonomics (CE2005), 5–7 April 2005 Hatfield, UK. Philip D. Bust (ed.). Taylor & Francis.

[edit] External links

Wikimedia Commons has media related to:

Mathematical diagram

• Diagrams, The Stanford Encyclopedia of Philosophy, Fall 2008.
• Diagrammatics: The art of thinking with diagrams by Zenon Kulpa
• One of the oldest extant diagrams from Euclid by Otto Neugebauer
• Diagrams in Mathematical Education: A Philosophical Appraisal, Dennis Lomas 1998.