Geometry is one of the oldest forms of mathematics, used in every ancient culture from Egyptians and Greeks to Mayas and Azteks. Today it is an active field applied to the study of the universe, crystals, and many other objects of interest. Although in the time of Euclid, geometry was modelled on a flat plane, in the past century mathematicians have turned to the study of curved spaces like the surface of the earth and more exotic spaces like the grid of streets navigated by a taxi driver. Here we introduce these concepts at the level of a high school or junior high school student.
We have links to lesson plans for teachers that introduce spherical, hyperbolic and taxicab geometry. These plans were written by my masters students, many of whom teach at nearby high schools.
We also have a page about metric spaces. Although such a space can be explained to a high school student, the study of metric spaces is one of the most advanced fields of mathematics.
We would like to recommend Go Geometry which has a a variety of beautiful proofs of classical theorems.
We would like to recommend Jeff Week's Geometry Games Webpage.
We would also like to recommend the cut-the-knot website which has alot of cool math puzzles.
Here is a webpage for undergraduate math majors describing Perelman's work and the Poincare Conjecture.
We recommend Jos Ley's website which has lots of beautiful images based on advanced geometry.
For a nice description about problems in minimal surface theory see Michael Hutchings' website.
For a neat introduction to planimeters see Foote's planimeter webpage.
For an animated proof of the Pythagorean Theorem Foote's Animated Pythagorean Theorem
Some neat animated limit sets can be found on Jeffrey Brock's Webpage
Some interesting billiards can be found on Richard Schwartz's Webpage
This page was written by Christina Sormani whose work on metric spaces has been supported by NSF Grant: DMS-0102279. She is a tenured faculty member at Lehman College and the CUNY Graduate Center.