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:

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) .
So what are some metric spaces? Very ordinary things we deal with every
day:
 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 ca from the upwards part of the path +
db from the across
part. So the length of this curve is ba+cd.
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)=ba+cd!
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.
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.
 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 NonEuclidean 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:
DMS0102279. She is a tenured faculty member at
Lehman College
and the CUNY Graduate Center.
The page was last updated in May 2002.