Hi, have you ever noticed how some of the most
beautiful things are surprisingly simple? A single
melody played on piano, a child's drawing, a
quiet snowfall, a circle drawn in the sand. Beauty
does not always come from complexity. Sometimes
beauty comes from simplicity that reaches deeper
than we expect. And I think mathematics often
works that way. One of the most famous mathematical
ideas is the Pythagorean theorem. A squared plus
B squared equals C squared. Now, at first glance,
it may not seem beautiful. Just letters and symbols.
Yet this simple relationship works for every
right triangle. And how do we know that? Well,
over thousands of years, mathematicians have
discovered hundreds of proofs. Some use geometry,
some use algebra, some use motion. Recently,
two high school students even found a proof using
trigonometry. So for fun, I recently reviewed
three different proofs myself, trying to see
it from different perspectives. And what surprised
me was that despite the different perspectives,
all three proofs depended on the same key point
inside the triangle. In two of the proves, the
point was obvious, but in the third, I had to
dig for it. But once I saw it, it felt almost
magical. Three completely different approaches,
yet all centered on the same hidden point. It
felt like discovering that three different paths
through a forest all lead to the same clearing.
So why would so many people spend so much time
proving the same thing? Because beautiful truths
invite us to see them from different angles.
And sometimes the simplest ideas open the door
to enormous worlds. One of the proofs came from
Euclid's elements from over 2 ,000 years ago.
Euclid not only proved the Pythagorean theorem,
but he built an entire geometry from a small
collection of assumptions about points, lines,
angles, and shapes. And one assumption concerned
parallel lines. So for centuries, mathematicians
wondered, what if that one assumption changed?
Surely the whole system would collapse. But rather
than collapsing, entirely new geometries emerge.
Beautiful geometries. Coherent geometries. Geometries
where space curves instead of remaining flat.
In those worlds, triangles no longer have angles
that sum to 180 degrees. And today, those geometries
help us describe the shape of the universe itself.
I'm drawn to this unique combination of simplicity
and vastness. First, a hidden point unites many
different proofs, and then a single changed assumption
creates an entirely new mathematical world. This
is why mathematics feels beautiful to me. Sometimes
beauty comes from discovering that many things
are secretly connected. And sometimes beauty
comes from discovering how much we can grow from
one simple idea.