WEBVTT

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Hello. Yesterday we saw how a circle can become

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a framework for discovering continuous triangles,

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sine waves, and many other beautiful ideas. Today,

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let's take the opposite approach. What happens

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if we remove properties that seem fundamental

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to shapes? What if distance no longer matters?

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What if angles do not matter? What if straight

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and curved lines are treated the same way? What

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kind of geometry would that create? Well, this

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leads to one of the strangest and most creative

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areas of mathematics we call topology. Topology

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studies shapes in a very unusual way. Rather

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than focusing on measurements, topology focuses

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on deeper structure. Shapes can stretch, and

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bend like rubber without losing their essential

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identity. And this leads to the surprising idea

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that a coffee mug and donut are considered the

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same shape. That does sound ridiculous at first,

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but topologists are not focused on exact appearances.

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They focus on something deeper. Literally, the

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number of holes in the object. A donut has one

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hole. A coffee mug has one hole, the handle.

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So from a topological perspective, they belong

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to the same family. Now what changed was not

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the object. What changed was what we chose to

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prioritize. See, for a long time, topology seemed

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like one of those strange mathematical playgrounds,

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but with little practical value. Just curious

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people exploring abstract ideas. But over time,

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These strange ideas began helping us understand

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the universe itself. For example, at the quantum

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level, the world of the incredibly small, topology

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now helps physicists describe how particles and

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fields behave. And at the larger scales, topology

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helps scientists think about the possible shape

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of the universe itself. Does space curve back

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on itself? What is the overall shape of reality?

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Questions that once sounded purely philosophical

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became deeply mathematical. A new perspective

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changed what we could see. And I think this is

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part of what Eddie Wu means when he says that

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mathematics gives us a new sense for seeing the

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world. And as I let my imagination flow, Topology

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challenges me to think differently about life.

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To me, topology is an exercise in choosing to

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focus on different things than I did before.

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So, here's a question that I ask myself. What

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if I stop evaluating people mainly by outward

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measurements, such as their appearance, their

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status, wealth, or even success? What if I focus

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more on things like character, kindness, wisdom,

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and love? That shift changes the focus of what

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matters to me. And as I move into my second half

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of life, I find myself caring more deeply about

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relationships and presence, compassion, and listening

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well. This way of seeing does not happen automatically.

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It requires intentional focus. But the more I

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practice, the more beauty I discover in life,

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much like the surprising beauty found in topology.

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So this week, as we think about transformation,

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perhaps one of the first steps is learning to

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look beyond the measurements that we thought

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mattered most. Because some of the most important

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things in life may not be something that we can

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measure. they are deeply worth pursuing.