# The Best Little Horosphere in Kazan

## Document Type

Presentation Abstract

2-1-2007

## Abstract

Trigonometry is easy to develop in Euclidean geometry. Indeed, one can summarize the process in two sentences. In a right triangle, an acute angle determines the remaining angle (since the angles must sum to ∏), which then determines the triangle’s shape (by AAA-similarity), and thus, the ratios of its sides. Once we express these ratios as functions of the acute angle, a few simple manipulations yield the laws of cosines and sines, which suffice to solve all trigonometric problems.

In the hyperbolic plane, none of this works. Here, triangles’ angle sums vary widely, similar triangles do not exist, and side-ratios cease to be functions of one acute angle. In this lecture, I shall explain the beautiful and surprising means by which N. I. Lobachevski (our hero) overcame these obstacles to derive the trigonometric formulae of hyperbolic geometry. The tale includes an unexpected excursion into three-dimensional hyperbolic space, a lovely surface therein called the horosphere, and a new definition of parallelism.

The details are prickly, but in this lecture I shall confine myself to a descriptive overview that will be accessible to all. He who has ears to hear, let him hear.