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Geometry (MTH 3301)

Some time around the year 300 BC, Euclid, the patriarch of mathematics as we know it, founded a school of geometry in Alexandria; his Elements remained the standard textbook on mathematics for two millennia, and is still the basis for most modern textbooks on geometry. In this course we will explore a modernized version of Euclid's system, with an emphasis on learning how to write proofs involving basic geometrical objects such as points, lines, triangles, and circles. We will proceed slowly, step by step, for, as Euclid said, "There is no royal road to geometry." At the same time we will learn how to make constructions with a compass and straightedge and how to build paper models of polyhedra. We will take excursions into humanistic mathematics, visiting upon the golden ratio, the golden spiral, the Fibonacci numbers, phyllotaxis in sunflowers, and the Platonic solids, coming into contact with Pythagoras, Plato, Archimedes, Pacioli, Leonardo, Kepler, and Gauss.

Prerequisite: MATH 2413 or permission of instructor.

Modern Abstract Algebra (MTH 4301)

"Symmetry, as wide or narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection" (Weyl, Symmetry). Look where you will in the worlds of art and of nature, and you will see examples of symmetry: band ornaments, tiles, lattices, tessellations, diatoms, flowers, seashells, honeycombs, snowflakes, crystals, atoms, neutrinos, quarks. Think of a cube fixed in space. Try to imagine all of the "symmetry moves" or rigid rotations of space that would take the cube into itself. It turns out that there are 24, if we count the "move" that does nothing at all. If we compose two of these moves, performing first one and then another, then the combination is itself a third move, one of the 24. The set of such moves forms what we call a group. The study of groups is one of the chief ends of modern abstract algebra.

The goal of this course is to study some of the symmetry groups that have applications in art and nature, and to learn something of their historical and cultural significance. We will begin with familiar notions of sets and natural numbers and build up group theory axiomatically. Students will also read Symmetry, the classic exposition by Hermann Weyl, one of the great mathematicians and physicists of the twentieth century.

Prerequisite: MTH 3304. Students who have taken MTH 3310 and are interested in the course should contact me, even if they don't have the prerequisite.

Readings and Research (MTH 4327)

All senior math majors close to graduation are encouraged to participate in our capstone course this spring. It will be run as an informal readings course; students will work through and discuss Coxeter's classic Introduction to Geometry. The course will culminate in the deliverance of an oral presentation and a research paper in a topic of the student's choice; Coxeter's book should suggest many possibilities. This would be a good opportunity to obtain a letter of reference, especially if you are considering going into teaching.

Contact me if interested.

Additional Course Information

The two-year course rotation of the RGC mathematics program may be viewed here. Course descriptions may be viewed here. If you are a new math major at RGC, I recommend making sure you've taken Calculus I and II right away. Begin your studies here with Geometry and Discrete Mathematics, and any other courses you have the prerequisites for.

If you have any questions about advising, or would simply like to talk to me about your degree or mathematics in general, please do not hesitate to contact me.

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

Sul Ross State University Rio Grande College

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