Hi, have you ever seen a flower you thought was
beautiful? Or a sunset that took your breath
away? Or listened to a piece of music that moved
something deep inside you? The answer to at least
one of these questions is likely yes. But have
you ever seen math that you thought was beautiful?
So how do we go from math to beauty? Let's start
with one of those funny fishing pictures where
someone holds a fish towards the camera and suddenly
the fish looks enormous. Now, the fish isn't
actually bigger. The mathematics has not changed,
but perspective changes what we see. And artists
have understood this idea for centuries. Perspective
shapes how we experience beauty. And hidden underneath
perspective, is mathematics. And this week, our
focus is on beautiful math. Now, most people
do not think of math that way. I know I did not.
For many years, I considered math as formulas,
homework, calculations, or just getting the correct
answer. Something useful, perhaps, but not beautiful.
Yet, many mathematicians describe math as the
purest form of art. So my obvious challenge this
week is that I cannot use pictures or diagrams
and only words. So I will have to be creative
while talking about how mathematics itself can
be creative. Math is not just about patterns,
but about perspectives. One of my favorite math
authors, Paul Lockhart, compares doing mathematics
to climbing a mountain. Yes, climbing takes effort,
you get tired, and you struggle. But the goal
is not merely exercise. The goal is the view.
To stand somewhere higher and suddenly see something
you could not see before. The vastness of the
landscape. The surprising details of a flower.
The beauty that was always there, waiting to
be noticed. And I think mathematics is like that.
One of my favorite examples comes from over 2
,000 years ago. In an example we've discussed
before, the Greek mathematician Euclid wanted
to understand prime numbers, you know, 2, 3,
5, 7, 11, so on. And he asked a simple question,
are there infinitely many prime numbers? Now,
what amazes me is that he did not answer the
question by listing all the primes. That would
be impossible. Instead, Euclid had a unique perspective.
He had a beautiful idea. He showed that no matter
how many prime numbers you collect, there must
always be another prime beyond the previous.
In other words, prime numbers can never be contained.
And I find that perspective breathtaking. Not
because it helps me calculate something, but
because it reveals something unexpected and true.
Euclid transformed a question about infinity
into a question about containment. And with that
simple shift in perspective, he revealed something
beautiful. So this week, we will explore creative
and beautiful ideas for mathematics. And I want
to share some of that beauty that keeps drawing
me back. to behold.