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One of the most fundamental properties of geometry is connectedness: when a shape is a single continuous entity. But how do you define this idea precisely so that you can apply it even to extremely bizarre shapes in very strange spaces?

=Chapters=
0:00 The Topologist's Sine Curve
1:54 The goal of this video
3:06 PathConnectedness
6:56 A new definition?
7:37 Topology basics
10:27 A snag
12:04 Connectedness 2.0 and the Topologist's Sine Curve
15:48 Who's right?
17:38 The Ordered Square
19:32 The connection between the definitions
21:18 The underappreciated art of crafting definitions
22:16 Brilliant ad


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CREDITS

The songs used in this video are (in order of first appearance) "Dream Escape", "Checkmate", "Rubix Cube", and "Twinkle in the Night".

The song "Rubix Cube" comes courtesy of Audionautix.com


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The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here:
https://github.com/morphomatters/mor...