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Chair of the Council on Publications and Communications

Santa Clara University

ffarris@scu.edu

Have you enough mathematics in your home? To help me in that inevitable moment when guests ask me why I spend my life on mathematics, I commissioned artist Hans Schepker to realize in glass an image from my work. The story of the resulting window involves a trip to the fifth dimension in search of a color tool for visualizing complex functions. This talk can work as a light lunch talk or a more mathematical 50-minute lecture.

The rich are indeed getting richer, as measured by the Gini index, a number economists use to express the degree to which a distribution of resources is equitable. If everyone had a perfectly equal share of the pie, the Gini index would be 0. One person having it all would lead to a Gini of 1. In the United States, the Gini index for family income has been rising steadily since the 1960s, from a low of about 0.34 to a current value of about 0.41. The Gini index for wealth has also been increasing and is now about 0.80. In this talk, I'll develop the Gini index and discuss the problem of computing it from data. The main mathematical tool is a simple one: the definite integral that gives the area between two curves.

If you look at enough swatches of wallpaper, you will see centers of 2-fold, 3-fold, 4-fold, and 6-fold rotation. Why not 5-fold centers? They cannot occur, according to the Crystallographic Restriction, a fundamental result about wallpaper patterns, which are defined to be invariant under two linearly independent translations. Even so, we offer convincing pictures of wallpapers with 5-fold symmetry and ask "Who is lying?". The talk is intended to be accessible to students who know something about level curves in the plane and linear algebra.

What if the universe had an edge? Since "universe" is construed to indicate "all that is," such an edge would have to be inaccessible, "infinitely far away."

In this talk, we travel to a hypothetical universe, whose inhabitants, along with all the matter they use to measure their space, shrink as they approach the edge. In this shrinking-ruler universe, that boundary is indeed inaccessible.

The picture of what we call "hyperbolic wallpaper" helps us imagine this cosmos: In the world of the shrinking ruler, all of the peacock fans are exactly the same distance across. All of them. And there are infinitely many copies hidden down there near the edge, unseen by our outsider eyes.

Frank Farris completed a five-year term as editor of Mathematics Magazine in 2005 and then served again in 2009, aspiring to continue its tradition of challenging and inspiring teachers and students of mathematics at the undergraduate level. A native Californian, Frank did his undergraduate work at Pomona College and received his Ph.D. from M.I.T. in 1981. Awards include a Trevor Evans Award for his article "The Edge of the Universe" in Math Horizons and the David E. Logothetti Teaching Award at Santa Clara University, where he has taught since 1984.

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