Graphs and diagrams (mathematics)

Summary: Graphs and diagrams are one way to represent mathematical information and may convey it more clearly than other methods or reveal interesting patterns and relationships.

Graphical representations have been found since antiquity in such places as cave drawings and maps. Modern graphs are fundamental to the organization and presentation of information. The concept of a graph developed along with advances in printing, mathematical theory, and empirical observations, especially in such fields as astronomy, cartography, chemistry, crystallography, calculus, geometry, probability, and statistics. Quantitative information, such as data points or functions, is often exhibited and analyzed in graphs. In twenty-first-century classrooms, students of all ages explore various types of graphs. Graph theory is a branch of mathematics that studies mathematical graphs in which vertices or nodes, representing objects, are connected by edges that represent relationships between the objects.

Early Graphs

Attempts to depict familial relationships led to a variety of family trees and graphs, some of which survived from the Middle Ages. Family trees have long been of historical and personal interest in tracing ancestry and nobility relationships. The rise of genealogical social networks at the beginning of the twenty-first century led to huge family trees. Researchers and software developers have created new ways to visually represent ever-changing family relationships, including divorce and remarriage.

Some graphical representations arose in the context of puzzles or games. For instance, variants of a game known as Men’s Morris have long appeared in carvings on Roman buildings and in cathedrals in medieval England. In the thirteenth century publication of Alfonso X of Castile, the Libro de los Juegos (Book of Games), an illustration, below, shows a Morris game board with nodes that represent the positions of game counters and connections between them that represent the moves. The beginnings of graph theory are often attributed to eighteenth-century mathematician Leonhard Euler. In 1736, he presented a solution showing that it was impossible to continuously traverse the seven bridges of Konigsberg, Russia, without retracing the same path or lifting the writing utensil. However, his paper does not contain any graphs, although it does contain maps of Konigsberg. Continuous figure tracing also appeared in Danish folk puzzles, as well as in the Angola, Zaire, and Zambia region in Africa, and in the New Ireland and Vanuatu regions in Oceana. Euler also did not use graphs in his 1759 work on a Knight’s Tour, where a knight must traverse each square on a chessboard without repetition. In 1771, Alexandre-Theophile Vandermonde used a graph drawing in this context.

One common notion of a graph is a pictorial representation of a function. The graph of a function passes the vertical line test, so that each input has one assigned output. Egyptologists Somers Clarke and Reginald Engelbach noted that an ancient Egyptian architect’s diagram showed a curve with vertical lines and coordinate measurements expressed in units of cubits, palms, and digits. The graphical depiction of changing quantities where one quantity depends on another can be found in the fourteenth-century publications of Nicole d’Oresme and in De latitudinibus formarum (the Latitudes of Forms), which may also have been written by d’Oresme. The development of coordinate geometry, coordinate axis systems, and the notion of a function in the seventeenth and later centuries, through the work of René Descartes, Pierre de Fermat, Gottfried Leibniz, Peter Dirichlet, and others, allowed for the graphical representations of algebraic formulas, curves, and other mathematical objects. Thomas Hankins noted that graphs started appearing in 1770 in the context of

…the statistical atlases of William Playfair, the indicator diagrams of James Watt and the writings of Johann Heinrich Lambert.… That leaves us with the question of what is to count as a graph. If we include maps and geometrical and astronomical diagrams, graphs are very old indeed. What was new in the late eighteenth century was a diagram with rectangular coordinates that showed the relationship between two measured quantities. Lambert called them Figuren, Watt called them “diagrams,” and William Playfair called them “lineal arithmetic.” William Whewell, who seemed to rename everything that he came into contact with, called them the “method of curves.”

Gaspard Monge’s eighteenth-century work also influenced the development of graphs as well as fields like architecture and engineering. He is known as the “father of descriptive geometry,” which studies three-dimensional geometry through two-dimensional images.

The earliest known uses of the terms “graph,” “graph paper,” and “graph theory” originated in the nineteenth and twentieth centuries. Mathematician James Sylvester is noted as the first to use the term “graph” in the publication Nature in 1878 when he described a chemical graph. Graphs in chemistry originated earlier, such as in the eighteenth century when chemist William Cullen referred to an “affinity diagram” to model molecular forces. Alexander Brown depicted molecules as graphs in 1864. Mathematician Arthur Cayley developed graph theory in the 1870s in the context of chemistry. Some have cited Julius Peterson’s late-nineteenth-century work as the start of the field of graph theory. The Peterson graph that is named for him is explored in graph theory classes. George Chrystal referred to the “graph of a function” in his 1886 algebra text: “This curve we may call the graph of the function.” Graph paper was originally known as “squared paper” or “coordinate paper” and was patented by Dr. Buxton in the late eighteenth century. The use of “graph” as a verb may date to an 1898 work on applied mechanics, in which John Perry advised: “Students will do well to graph on squared paper some curves like the following… in each case calculate y. Plot the values of x and y as co-ordinates of points on squared paper, and draw the curve passing through the points.…”

Types of Graphs

The study of logical statements, their implications, and their relationships resulted in a variety of different types of diagrams. Young children use Venn diagrams, which represent set containments and intersections using overlapping circles. These were named for philosopher and mathematician John Venn because of his nineteenth century work to formalize and generalize them. The concept of a Eulerian Circle, named for Euler, is related. Aristotle’s square of opposition is named for the ancient Greek philosopher. Aristotle analyzed deductive logic among various statements. Fourth-century mathematician and philosopher Anicius Manlius Severinus Boethius also explored the logical relations.

In some versions, the square of opposition was presented as a square diagram that contained propositions that were represented inside circles. Lines that connected the circles represented the relationships between the propositions. College students and researchers in fields like logic, topology, algebra, and geometry use commutative diagrams with arrows or other symbols to represent mappings or logical relationships.

Educational Graphs

Students in the twenty-first century investigate a wide variety of graphical and diagram representations. In primary schools in the United States, students represent and analyze problems using graphs, charts, data, and functions; the graphs also serve as a subject of study themselves. William Playfair’s 1786 publication The Commercial and Political Atlas is noted as the beginning of charts, such as bar charts and line charts, and perhaps the first appearance of statistical time series graphs. He also invented the pie chart in 1801. In the middle grades, students also generalize patterns with graphs and identify and contrast linear and nonlinear graphs.

In addition, they convert between symbolic algebraic formulas and graphical representations and learn about graphical features, such as the slope or intercept of a line and the changing quantities in a graph. In high school, students continue to create graphical representations and they approximate the rate of change of a function from its graph. In calculus, students use graphs to further understand the properties of functions, such as their derivatives, integrals, and the notion of concavity. The integral is defined as the area under a curve, and students use Riemann sums, named for nineteenth-century mathematician Bernhard Riemann, to approximate the area using rectangles.

The widespread use of graphing calculators and computer software in the late twentieth century changed the way that students explored graphs. They were able to quickly graph complex equations and large amounts of data to look for patterns. Students and teachers explore candidates for categories like the most beautiful graph, the funniest graph, or the worst graph, which some define as the most misleading and others as the most confusing.

Debate continues regarding what is the desired balance between by-hand graphing skills versus a reliance on graphical methods on the computer or calculator. Some teachers argue that if students do not understand how to create graphs, they will not be able to fully understand misrepresentations or analyses. Another area that has taken on new prominence in twenty-first century schools and colleges is discrete mathematics and graph theory.

Bibliography

Ascher, Marcia. “Graphs in Cultures: A Study in Ethnomathematics.” Historia Mathematica 15 (1988).

Biggs, N., E. Keith Lloyd, and R. Wilson. Graph Theory 1736–1936. New York: Oxford University Press, 1999.

Friendly, Michael. “Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization.” http://www.math.yorku.ca/SCS/Gallery/milestone/milestone.pdf.

Hankins, Thomas. “History of Science Society Distinguished Lecture: Blood, Dirt, and Nomograms, A Particular History of Graphs.” Isis 90 (1999).

Kruja, Eriola, Joe Marks, Ann Blair, and Richard Waters. “A Short Note on the History of Graph Drawing.” 9th International Symposium on Graph Drawing. Berlin: Springer-Verlag, 2002.