Visualization in mathematics

Summary: Visualization is a useful practice when doing or learning mathematics and computers can help create visualizations of difficult concepts.

The ability to form a mental image is a fundamental process and has been incorporated in many theories about knowledge acquisition. The advent of the printing press and perspective drawings allowed for an unprecedented sharing of realistic pictures, graphs, and inventions, and this led in part to the Industrial Revolution.

The development of coordinate geometry gave rise to graphical representations of data and algebraic concepts. With the popularity of computers and computer graphics, mathematicians, artists, and programmers have created visualizations of mathematical objects and huge amounts of data. Mathematicians also found new ways to visualize and share abstract ideas such as the fourth dimension. Dynamic image manipulation features, such as rotation or zooming, further increased the accessibility of visualized objects by facilitating new perspectives and comprehension of hard-to-see surfaces. Mathematical visuals have been fundamental in both research and entertainment contexts like for computer-generated imagery (CGI) used in modeling, computational geometry, or movies. Various types of visualization, including spatial visualization and visuals of data and graphs, are important components of all levels of mathematics and statistics classrooms in the twenty-first century. Visualization is an interdisciplinary topic and researchers from a diverse range of fields contribute, including mathematicians, computer scientists, psychologists, engineers, and neuroscientists. Educators and researchers create visualizations, study visualization ability, and design new ways to help students visualize.

Early History

Visualization has been as important in mathematics and statistics research as in education and mathematicians in many fields throughout history created visual representations. Representations of maps are as ancient as the earliest societies from which there exists evidence of stone tablets and animal skins. Another important historical research area related to visualization and mathematics was the field of optics. For example, ancient people created lenses. Euclid of Alexandria investigated geometry and perspective in his book on optics. Many mathematicians and scientists worked to understand vision, including mathematician Abu Ali al-Hasan ibn al-Haytham, who wrote a seven volume work on optics and visual perception, which is noted by some as the first work to correctly demonstrate understanding that light is reflected from an object to the eye. Self-taught mathematician and scientist Tobias Mayer was one of many to formulate a theory for color perception and he also modeled the limits of vision, noting, “there is a certain visual angle below which an object presented to the eye appears either not distinct enough or not even distinct at all, but only confused and as though it had vanished from sight.… We shall call this angle the limit of vision, and we shall investigate its angle by experiment.”

In the seventeenth century, René Descartes made significant progress in coordinate geometry. The Cartesian plane that is named for him allowed for new representations of data and algebraic equations. Mathematicians, statisticians, social scientists, and others began to investigate ways to visually present graphs and data to facilitate analysis, interpretation, and understanding. Social issues motivated many researchers in the nineteenth century. For example, William Playfair created color-coded graphical representations of the English national debt and the trade balances between England and other countries. Adolphe Quetelet graphed the distributions of anthropometric data to show both the center and variability, leading in part to the measure now known as Body Mass Index. Florence Nightingale developed the polar area chart as part of her campaign for improved sanitation in medical facilities. John Snow used graphical mapping techniques to trace the source of a London cholera outbreak. Graphs of mortality statistics and many other naturally occurring phenomena also proliferated. Philosopher and logician John Venn developed Venn diagrams in 1881, which are also used in many mathematics classrooms.

Recent Developments

The rise of computers in the twentieth century led to mind-bending visualizations and new fields of research in mathematics as well as beautiful artistic forms. Mathematician Benoit Mandelbrot popularized the field of fractals. The computer visualization of some objects helped clarify their mathematical properties. One example is Enneper’s surface, which had been introduced by Alfred Enneper in the nineteenth century. In the mid-twentieth century, Steven Smale proved that it was possible to turn a sphere inside out in three dimensions without creating any creases. This idea stretched the imagination and mathematicians tried to visualize it. For instance, mathematicians at the Geometry Center for the Computation and Visualization of Geometric Structures produced a video called Outside In, which visualized William Thurston’s sphere eversion method. Geometer Thomas Banchoff pioneered visualizations of four-dimensional objects. Mathematicians in the twenty-first century attempted to visually model the Internet using hyperbolic geometry in order to reduce the load on routers. Researchers from interdisciplinary fields have participated in conferences on topics like visualization algorithms or data visualization. Mathematicians have designed visualization software and techniques for many areas in mathematics, including linear algebra, group theory, and complex analysis. Some of these visualizations are used in classrooms, while others are the focus of research investigations or artistic exhibitions.

Visualization Ability

The connections between visualization ability and mathematical success also have a long and varied history. In the nineteenth century, scientist and mathematician Sir Francis Galton conducted studies to examine the relationship between visual imagery and abstract thought. Some have noted that nineteenth-century mathematician Henri Poincaré had poor eyesight as a student and scored a zero on an entrance exam for the École Polytechnique; however, he had a great memory because he was able to mentally translate concepts he heard aurally into visual representations of the same concepts. Poincaré later wrote about the ability to form retina images and what he referred to as “pure visual space.” The Poincaré disc model of hyperbolic geometry is named for him, and twenty-first-century students explore this in interactive computer models that are designed to help visualize and explore mathematical topics, including the variation in the sum of the angles for differently sized triangles.

Other visual challenges, like “stereoblindness” and “subitizing” difficulties, have also been tied to mathematics. Stereoblindness, the inability to properly combine images in the mind to see in three dimensions, was once thought of as impossible to cure. Subitizing is the ability to rapidly perceive and differentiate the number of distinct items in a small group of objects, like dots on a cube. Some researchers in the first part of the twentieth century investigated the importance of subitizing to the understanding of numbers, counting, and abstract thinking and educational psychologists in the second half of the twentieth century continued this work and developed a variety of theories. While the specific mechanisms are still the topic of debate, in the twenty-first century, vision and subitizing therapies have been successfully implemented in the optometry profession and are thought to help mathematics students. Some proponents of left-brain versus right-brain dominance theories assert that visualization is focused in the right brain, while other mathematical skills, like logic and analysis, are focused in the left side of the brain. Psychobiologist Roger Sperry was awarded the Nobel Prize in 1981 in part for his split-brain experiments. However, medical imaging scans of people performing mathematical tasks has shown regions from both sides of the brain highlighted and researchers continue to investigate this issue.

Gender

In the latter half of the twentieth century, researchers investigated gender differences in spatial visualization ability. In 1978, geneticists Steven Vandenburg and Allan Kuse developed a mental rotation test that has been used in part to quantify spatial visualization ability. In 1980, Camilla Benbow and Julian Stanley, referred to as psychologists and educators, asserted that gender differences in mathematics might result from “greater male ability in spatial tasks.” Their statements were widely publicized in the media. Later researchers found that visual training by video games or certain changes in testing conditions, like removing “I don’t know” as an answer or eliminating time constraints, could reduce these observed gender differences. Research on stereotype threat, where the effort to counter societal perceptions about a whisper of inferiority can negatively impact performance, has further complicated visualization research efforts.

Education

Various educational learning models and theories stress the importance of visualization. In Piaget’s theory, named for epistemologist Jean Piaget, spatial skills develop at various age levels or stages and according to experience. For instance, he proposed that young children could understand two-dimensional space, while the mental manipulation of three-dimensional objects in space comes later on. Mathematician Walter Whiteley has proposed research questions related to visualization and suggested a variety of ways in which teachers might intentionally train students to “see like a mathematician.” He noted:

Curriculum suggests that 2-D is easier than 3-D, although it is cognitively less natural for many modes of reasoning, and 3-D skills are the needed goal for later work. The domination of analytic over synthetic reasoning encourages the pattern that 2-D is the starting point, and the disconnection between early childhood reasoning, and latter problem solving both of which engage 3-D reasoning.

The van Hiele model of geometric thought, developed by educators Dina van Hiele-Geldof and Pierre van Hiele, listed visualization as its first level. Additional learning models presented by mathematicians and educators have also stressed the importance of interweaving visualization training with other skills.

Bibliography

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