Explore Geometry!
Metric Spaces

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.

One of the simplest ideas in geometry today is that of a metric space. 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.

It begins with a set of points, which we can call a space, X. Between every two points, say p and q, sitting in X we define a distance, d(p,q). The distance has a few rules based on common sense:

So what are some metric spaces? Very ordinary things we deal with every day: Now all these spaces are called length spaces because they are defined using lengths of paths. Length spaces are an important study in the field of noneuclidean geometry. They are studied by faculty at CUNY and NYU and many other universities.

Here are some fun concepts in metric space theory that anyone can explore.

There are many more projects and concepts at all levels! Check out MAT346 Geometry(undergraduate) and MAT636 Non-Euclidean Geometries (masters) taught by Professor Sormani at Lehman College this coming Fall 2002. Calculus I and II are required to take the courses but you'll learn ideas that can be taught to high school students of various levels.

Check out http://comet.lehman.cuny.edu/sormani/explore/projects.html for possible class projects in the future.

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. The page was last updated in May 2002.