Hello. Yesterday we saw how a circle can become
a framework for discovering continuous triangles,
sine waves, and many other beautiful ideas. Today,
let's take the opposite approach. What happens
if we remove properties that seem fundamental
to shapes? What if distance no longer matters?
What if angles do not matter? What if straight
and curved lines are treated the same way? What
kind of geometry would that create? Well, this
leads to one of the strangest and most creative
areas of mathematics we call topology. Topology
studies shapes in a very unusual way. Rather
than focusing on measurements, topology focuses
on deeper structure. Shapes can stretch, and
bend like rubber without losing their essential
identity. And this leads to the surprising idea
that a coffee mug and donut are considered the
same shape. That does sound ridiculous at first,
but topologists are not focused on exact appearances.
They focus on something deeper. Literally, the
number of holes in the object. A donut has one
hole. A coffee mug has one hole, the handle.
So from a topological perspective, they belong
to the same family. Now what changed was not
the object. What changed was what we chose to
prioritize. See, for a long time, topology seemed
like one of those strange mathematical playgrounds,
but with little practical value. Just curious
people exploring abstract ideas. But over time,
These strange ideas began helping us understand
the universe itself. For example, at the quantum
level, the world of the incredibly small, topology
now helps physicists describe how particles and
fields behave. And at the larger scales, topology
helps scientists think about the possible shape
of the universe itself. Does space curve back
on itself? What is the overall shape of reality?
Questions that once sounded purely philosophical
became deeply mathematical. A new perspective
changed what we could see. And I think this is
part of what Eddie Wu means when he says that
mathematics gives us a new sense for seeing the
world. And as I let my imagination flow, Topology
challenges me to think differently about life.
To me, topology is an exercise in choosing to
focus on different things than I did before.
So, here's a question that I ask myself. What
if I stop evaluating people mainly by outward
measurements, such as their appearance, their
status, wealth, or even success? What if I focus
more on things like character, kindness, wisdom,
and love? That shift changes the focus of what
matters to me. And as I move into my second half
of life, I find myself caring more deeply about
relationships and presence, compassion, and listening
well. This way of seeing does not happen automatically.
It requires intentional focus. But the more I
practice, the more beauty I discover in life,
much like the surprising beauty found in topology.
So this week, as we think about transformation,
perhaps one of the first steps is learning to
look beyond the measurements that we thought
mattered most. Because some of the most important
things in life may not be something that we can
measure. they are deeply worth pursuing.