Hi. So far this week, we've explored infinity
through numbers and infinite processes. In a
sense, we chose infinity as our input. We chose
the prime numbers. We chose infinite sums. But
today, infinity appears in a different way. This
time, it was already there waiting to be discovered.
Today, we witness infinity appearing inside geometry
itself. In a weird sort of way, I think of it
as infinity will impose itself on us and we must
decide how to deal with it. Let's begin with
a simple square. Imagine a square with side length
one. Now draw a diagonal from one corner to the
opposite corner. What is the length of that diagonal?
Using the Pythagorean theorem, The answer is
the square root of the sum of 1 squared plus
1 squared. So the diagonal has length square
root of 2. Now at first, that may not seem unusual.
We have a calculator button for this problem
for Pete's sake. But the intriguing part is we
cannot write square root of 2 as a fraction of
integers. Forever. Not only do the digits continue
without end, they never repeat. An infinite decimal
hidden inside a simple square. Cool. Okay, now
let's consider another simple shape, a circle.
If you take the circumference of any circle and
divide it by its diameter, you always get the
same number. And that number begins 3 .14159
dot dot dot. And it continues forever. And we
call this number pi. And just like square root
of 2, the decimal expansion of pi never ends
and never repeats. So today, infinity is not
showing up because we invented an infinite process.
It emerges naturally. from shapes themselves,
from space itself. And historically, this was
shocking. The ancient Greeks believed that all
lengths could be written as fractions. And then
square root of two appeared and shattered that
assumption. Infinity revealed that reality was
deeper than they imagined. And that's what makes
today... Such an important step in our journey.
Infinity is no longer just an abstract idea that
is connected to counting or adding forever. Now
it appears woven into geometry itself, into circles,
into squares, into the structure of the world
around us. And once again, humanity did not invent
this. We discovered it. People across centuries
and cultures kept encountering these same strange
truths hidden inside mathematics. And every time
they looked deeper, more structure appeared beneath
the surface. Infinity keeps doing that. It takes
something simple, a square, a circle, a diagonal,
and reveals... There was something far deeper
hidden all along. So when I think about infinity
appearing inside basic shapes, always existing
and never created, I just cannot help but wonder,
where did this come from?