Resources

Books

Here are some recommended books on mathematical topics. A number of them are available through the SWTJC Libraries; they can also be checked out to students from the SRSU Wildenthal Library. If you don’t find what you need at either of those, you can always obtain any book through the convenient interlibrary loan program. Library staff will be happy to walk you through the process.

How to Read a Book by Mortimer Adler. This should be at the top of every reading list. Adler, an American scholar of Aristotle, instructs the reader in the art of reading. As he puts it, there are two kinds of teachers: the living and the dead. The living teach in the classroom. The dead teach through their written works. If you ignore the latter, you're cutting yourself off from the greatest treasures of our civilization. But knowing the mechanics of reading isn't enough; you have to know how to think, how to analyze what you're reading. At the same time, reading the great books intelligently will teach you how to think. "[I]t is only by struggling with difficult books, books over one’s head, that anyone learns to read" (Adler).

Use Adler's book list to look up the great works of mathematics and related fields. Read Aristotle's Prior and Posterior Analytics; Euclid's Elements; the Conics of Apollonius of Perga; the works of Archimedes; Ptolemy's Almagest; the works of Copernicus and Kepler and Galileo; the Geometry of René Descartes; and so on. David Eugene Smith's Source Book in Mathematics (Dover Books) contains excerpts from the writings of the great mathematicians published since the invention of printing. These books are all available at our college libraries, but they just sit there, gathering dust. Go get them off the shelves! Or download them, since most are available for free online.

Textbooks package information into a palatable, easily digested form. Reading them is a bit like inserting a DVD into your brain and hitting "play." They have their place, but don't let them stand between you and the true great teachers of our culture. Don't let them (or me, or your other professors) tell you what to think. Learn to think, and then think for yourself; make up your own mind.

Flatland: A Romance of Many Dimensions by Edwin Abbott Abbott. An eccentric Victorian fantasy novella set (mostly) in Flatland, a two-dimensional world inhabited by polygonal creatures. The narrator, A. Square, describes the customs and history of his land (a satire on Victorian social mores) before recounting the revelation by which he became aware of the existence of other dimensions. He visits Pointland, inhabited by a single self-satisfied point; Lineland, whose denizens can't move past one another; and, finally, Spaceland, which he can only approach by way of analogy. This analogy is the means by which we "Spacelanders" can conceive of a fourth spacial dimension. Originally published in 1884, Flatland really is a rather profound book. It received widespread attention only after Einstein proposed his theory of relativity.

The unabridged text (pictures included) can be found here; it is also cheaply available as a Dover Thrift Edition.

Introduction to Geometry by H. S. M. Coxeter. An advanced undergraduate textbook that picks up where Euclid’s Elements leaves off, written by one of the twentieth century's foremost geometers. It covers a rich variety of topics, such as symmetry groups in the Euclidean plane, the golden ratio, the Platonic solids, complex numbers, non-Euclidean geometry, and the differential geometry of curves and surfaces. It's a textbook to be worked through slowly and carefully, but should be accessible to anyone with a basic background in algebra, geometry, and calculus.

Also recommended is Coxeter's Regular Polytopes (Dover Books), which provides in-depth coverage of the regular polyhedra (the Platonic solids) and their analogues in higher dimensions.

Euclid’s Elements. "At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world" (Bertrand Russell). Written c.300 BC, Euclid's Elements is one of the great texts of world literature, and stands as both the pioneer and the epitome of the mathematical project. Every teacher of high school mathematics should spend at least some time in Euclid.

Thomas Heath's excellent translation and commentary is in the public domain and available in full online; it can be found at the Internet Archive. Try searching for "euclid’s elements heath". It's also available in a three-volume Dover edition.

Click here to explore the complete work through interactive Java applets.

Click here to view Byrne's eccentric yet beautiful 1847 edition.

If you do decide to delve into Euclid, I recommend Benno Artmann's Euclid – The Creation of Mathematics as an excellent expository companion and cultural-historical guide.

The Character of Physical Law by Richard Feynman. A Modern Library book based on the Nobel laureate's series of public lectures explaining the importance of mathematical reasoning in our understanding of the laws of nature. As he puts it: "To those who do not know mathematics it is difficult to get across a real fealing as to the beauty, the deepest beauty, of nature… It is reputed—I do not know if it is true—that when one of the kings was trying to learn geometry from Euclid he complained that it was difficult. And Euclid said, "There is no royal road to geometry." And there is no royal road… If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form; we are not so unhumble as to demand that she change before we pay attention."

Also highly recommended are The Feynman Lectures on Physics, a thorough, three-volume account of modern physics based on Feynman's undergraduate physics courses at Cal Tech. Mathematically substantial, yet readable.

Artforms of Nature by Ernst Haeckel. Originally published in German as Kunstformen der Natur, this is a book of pictoral plates arranged by a nineteenth-century evolutionary biologist to identify instances of symmetry and aesthetic design in organic form. Perhaps it will serve to get you thinking about the role of group theory in the study of natural growth and form.

Click here to visit a site with fully zoomable images.

Also recommended in this vein are D'Arcy Wentworth Thompson's On Growth and Form and Adolf Portmann's Animal Forms and Patterns.

Geometry and the Imagination by David Hilbert and S. Cohn-Vossen. Hilbert is recognized as one of the great mathematicians of the late nineteenth and early twentieth centuries. The book, based on lectures delivered by Hilbert to the citizens of Göttingen, makes high-level topics in modern geometry and topology accessible to the average reader. It discusses conic sections, quadric surfaces, tilings, crystal groups, the Platonic solids, projective geometry, the differential geometry of curves and surfaces, and more.

Shapes, Space, and Symmetry by Alan Holden. Written by a non-mathematician, this beautiful little book provides an introduction to the Platonic and Archimedean solids and their relatives. What makes it especially delightful is that the author illustrated it with black-and-white photographs of hundreds of models that he constructed from card, wire, and glass. In the Preface he gives this advice: "The best way to learn about these objects is to make them, next best to handle them." To this end, he closes with a description of how to reproduce his work.

Science and Music by Sir James Jeans. A popular exploration of the science of sounds, sound perception, and music, written by a great physicist of the twentieth century. The exposition touches on harmonic analysis, the decomposition of complicated waveforms into series of sines and cosines, often studied in courses on Linear Algebra and Differential Equations; the subject has applications in accoustics, electronics, and quantum mechanics.

The Grammar of Ornament by Owen Jones. This is a book about aesthetics, not mathematics. It's a Victorian handbook of ornamental symmetry culled from cultures around the world. Anyone learning about group theory (as in our Modern Abstract Algebra course) should peruse it. It contains beautiful examples of many cyclic and dihedral groups, all seven frieze groups, and sixteen out of the seventeen lattice groups. Although available in full online at this link, it has also recently been reissued in an inexpensive edition by DK Books.

The Structure of Scientific Revolutions by Thomas Kuhn. Kuhn, a philosopher of science and scientific history, outlines the process by which established worldviews are overturned by new ideas. Unlike most historians, he sees the advance of science, not as a march of progress perenially hindered by obstructionists, but as a cycle in which old paradigms come to be viewed as inadequate and are replaced by new paradigms after a period of crisis.

Also recommended is Kuhn's The Copernican Revolution. The rise of new scientific theories in the Renaissance figures largely in the history of mathematics.

The Fractal Geometry of Nature by Benoit Mandelbrot. The seminal text on fractal geometry by the pioneer of the field. Although rather scattered and inaccessible to the beginner, it contains a wealth of beautiful ideas and illustrations. A fractal, broadly described, is a point set whose Hausdorff dimension differs from its topological dimension. The Cantor set, the Koch snowflake, the Sierpinski triangle, and the Mandelbrot set are well known examples.

The Child's Conception of Number by Jean Piaget. The parallels between the set-theoretic underpinnings of the modern science of number and the development of the idea of counting in the child's mind are explored through a sequence of experiments by the great child psychologist. Also recommended is The Child's Conception of Space. Many of the ideas touched on in these volumes make an appearence in our Survey of Basic Mathematical Theory sequence for future elementary educators.

How to Solve It by George Pólya. The title is self-explanatory. Written by a prominent twentieth-century mathematician, How to Solve It attempts to break the problem-solving process down into four basic steps. It also provides a handbook of strategies that are frequently helpful. Indispensible to the future teacher of mathematics.

Symmetry by Hermann Weyl. Weyl was a prominent twentieth-century mathematician who help formulate the theoretical underpinnings of quantum mechanics. The key to understanding quantum theory, it turns out, is group theory, which can be seen as the study of symmetry operations on spacial arrangements. Weyl's Symmetry is based on a lecture series delivered by the author. It touches on the role of symmetry in art and nature, in snowflakes, honeycombs, jellyfish, crystals, cathedrals, mosques, atoms, molecules, the universe, and the human body. It's addressed to the layman, but makes excellent reading for our Modern Abstract Algebra course, which culminates in a study of the ornamental symmetry groups.

Links

Byrne's Euclid A digitized copy of Byrne's beautiful, eccentric 1847 edition of Euclid, "in which coloured diagrams and symbols are used instead of letters for the greater ease of learners." Each page is a work of art reminiscent of twentieth-century Bauhaus designs.

Cut the Knot A veritable bonanza of geometrical topics, with numerous Java applets and diagrams.

Euclid's Elements in Java The complete text of the Elements, with Java applets in place of static diagrams.

GeoGebra Free dynamic geometry software supported by a worldwide network of users. Very highly recommended.

George Hart The website of a research professor and geometrical sculptor, with pictures of beautiful polyhedral sculptures, links to instructional materials, and much more.

MacTutor History of Mathematics Archive Well-researched biographies for every famous mathematician in history, and articles on a variety of mathematical-historical topics.

Michael Ortiz, Ph.D.
Assistant Professor of Mathematics
Department of Natural and Behavioral Sciences

Sul Ross State University Rio Grande College

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