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:

• The distance from a point to itself is zero.
  d(p,p)=0.
• The distance from one point p to another point q is the same as the distance from second point q to first point p.
  d(p,q)=d(q,p).,
• The triangle inequality: For any three points, p, q, and z, we can form a triangle so that the distances between each pair of points satisfies
  d(p,q) ≤ d(p,z)+d(z,q) .

• The simplest metric space is Euclidean space which is the geometry of a plane. We've all learned formulas for figuring out the distances between points and know how to estimate the lengths of sides of triangles. In fact the distance between points in Euclidean space is the length of the line segment joining two points.
• The space X could be the Bronx and the distance between points or locations could be defined by taking the length of the shortest path walking through the streets of the Bronx to get from one point to the other. Paths aren't allowed to go though blocks or buildings but most go along the streets and around the corners. This is similar to a geometry called Taxicab geometry, where the space is the plane and the distance between two points is the length of the shortest curve between the points that travels only in horizontal and vertical directions but not slanted. So one curve between p=(a,b) and q=(c,d) is to go up the line segment from (a,b) to (a,d) then across to (c,d). The length is then |c-a| from the upwards part of the path + |d-b| from the across part. So the length of this curve is |b-a|+|c-d|. Now the distance from p to q has to be the shortest of all possible lengths. Can you guess what it is? Try some other curves and check their lengths. I get d(p,q)=|b-a|+|c-d|! This is a good project for students who are studying the cartesian plane and the absolute value.
• The space X could be the surface of the earth and the distance between the points could be the length of the shortest path between the points where the path is on the surface of the earth. We can let these paths cross the oceans but we won't allow them to be tunnels to China. So to get from NYC to China, the distance would be measured by going around the earth. If you get out a globe and taking a string, you can measure the distance between points by stretching the string between the points, marking the ends and then measuring the length with a ruler. This is a fun project for all ages. It is called spherical geometry.
• More spaces that can be studied are cylinders which are cardboard rolls of paper towels that go on forever to either side. Distances on them can be measured using strings just like on a globe.

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

• BALLS: A ball of radius r about a point p is the set of all points, q, such that the distance from q back to p is less than r. A ball looks like the inside of a circle in Euclidean geometry. What do balls look like in the spaces we described above? In one of them the balls look like squares!
• LINES: A line is a path so that if two points p and q are on the path then the length of the path from p to q is the distance from p to q. A line looks like a straight line in Euclidean geometry. What do they look like in taxicab geometry? What about the cylinder? Did you know there are no lines in spherical geometry?! Can you prove it?
• GEODESICS: A geodesic is a path so that if two points p and q are on the path and are near each other then the length of the path from p to q is the distance from p to q.We say points are near each other if the distance between them is small The lines of longitude on a globe are geodesics. This can be checked by stretching a string between nearby points on a line of longitude and noticing that the string runs along the line of longitude. This is not true for lines of lattitude except for one special line of lattitude. Which one? Airlines use geodesics to plan the routes between cities because they are the most efficient paths. An interesting project is to make triangles out of geodesics on a globe and study the angles at the corners. For example, what is the length of the hypotenuse of a right triangle on a globe? Pythagorus formula doesn't work! What about the law of cosines?
• DIAMETER: If S is a set of points in a metric space X, the diameter of S is the longest distance between two points in S. In a cylinder the diameter of a ball is twice the radius. And the same is true for a small ball in spherical geometry. But what about a large ball in spherical geometry?

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.