Description
This is my most frequent (and possibly my favorite)
talk. It evolved from a lecture I gave in graduate school motivated by
the
pedagogical complaint that we never tell our students the history
of the math we teach them. With a mix of transparencies, posters, and
models, this talk meanders through twenty-three centuries
of mathematical history. The path is marked by a list of equivalents
of Euclid's Parallel Postulate, and the talk concludes with a
demonstration that each of these equivalents fails in the non-Euclidean
plane.
Online Flier for Radical Pi Talk
If you click on "Non-Euclidean Geometry", you will find
Joel Castellanos's package for making compass and straightedge
constructions in the Poincare disk. If you click on the blue tiling,
you will see an Escher print that I use to illustrate the Poincare disk
in this talk.
Printed Flier for Ross Program Talk
Description
This talk is closely related to the non-Euclidean geometry talk
described above.
I developed this presentation for
Ohio State University's
Undergraduate Recognition Ceremony.
The department invited me to give a talk pertaining to the Mathematical Association of America
theme for the year, Mathematics and Art. I could not resist
returning to my old geometric haunts, and the result is a less
comprehensive but more accessible glimpse of the non-Euclidean
universe.
After a brief survey of Escher's mathematical and artistic themes, we discuss the polygonal tilings underlying Escher's tesselations, focusing on the triangular reflection tilings. We see that in the Euclidean plane and on the surface of the sphere, there are strong restrictions on the types of triangles that allow reflection tilings, and only finitely many triangular shapes can tile each surface. However, in the non-Euclidean plane, there are infinitely many such tilings. As we go, we see Escher prints and sculptures constructed in all three geometries.
I am indebted to Daniel Shapiro of the Ohio State University for suggesting this focus for the talk.
Description
The mathematics behind this talk comes from the discussion of
compass and straightedge constructions in
What is Mathematics by Courant and Robbins and from
the
discussion of origami constructions on
Thomas Hull's Origami Mathematics Page.
After sketching a proof of the impossibility of trisection
by compass and straightedge, we perform the Archimedes
compass-and-ruler
trisection and discuss the mathematics that makes trisection by
paper folding possible. At the end of the talk, each member of
the audience trisects an angle by origami.
Description
The central idea of this talk is the mathematical effect of
phyllotaxis, the growth process in plants whereby leaves (or seeds
or petals) of a plant emerge with a fixed angle between consecutive
leaves. We explore why the phyllotaxis angle in sunflower seed heads
and in pinecones and in many other plants is the golden secton of the
circle and why this angle produces Fibonacci patterns in the spirals
of such plants. Continued fractions play a key role in unraveling the
mystery.
For source materials, see R. Knott's excellent discussion of Fibonacci Numbers and Nature, page 1 and page 2, and the references at the bottom.
Printed Flier for Radical Pi Talk
Description
Hexaflexagons
are gadgets invented by the topologist
Arthur H. Stone and
popularized in
Martin Gardner's Scientific American Column. The article appears
in his book Mathematical Puzzles and Diversions.
This presentation is more of a workshop than a talk. First, we construct hexaflexagons with three faces and with six faces and learn how to flex them. In playing with the hexahexaflexagon, novices are quickly struck by the difficulty of locating all six faces. We then have a discussion of how to resolve this problem, and if a solution is reached, we discuss how to construct hexaflexagons with different numbers of faces and different internal structures.
Online Flier for Radical Pi Talk (with video clip)
Printed Flier for Radical Pi Talk
Description
The talk is presented as a wager. I take a sheet of paper,
set it on a desktop, mark its position, and have an audience member
crumple the sheet and place it within the marks. I then propose
the following wager: I will bet any member of the audience that
there is a point on the page that lies directly over its pre-crumple
position. For the remainder of the talk, I convince the audience
that I may collect on this bet.
This talk was inspired by a lecture given by Francis Su (creator of the Math Fun Facts web page) to the Harvard Graduate Student Seminar. His subject was applications of Sperner's Lemma, one of which is a straightforward proof of the Brouwer Fixed Point Theorem.
Transparencies for Brown SUMS Talk (HTML)
Description
Among my expository lectures, this talk is exceptional
in that it concerns an area of my own
mathematical research. As a consequence, it is much more variable
in title and content than the talks above.
Suppose f is a polynomial with rational coefficients. Repeated application of f to the rational numbers produces a discrete dynamical system. To study the properties of this dynamical system, we bring to bear both classical complex dynamics and the number-theoretic properties of rational numbers.
For a sufficiently sophisticated audience, I describe the use of p-adic dynamics in the study of dynamical systems over Q. I can also formulate this talk as a motivation for constructing the p-adic numbers and studying p-adic analysis.
Needless to say, I also give research talks on this subject for more advanced audiences.
