Topology - Part 2 | Summary and Q&A

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December 22, 2016
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MyWhyU
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Topology - Part 2

TL;DR

The video discusses different ways to visualize and understand space, including 2-dimensional cylindrical and toroidal spaces, as well as 3-dimensional curved spaces.

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Questions & Answers

Q: What is space and how is it different on smaller scales?

Space is a way to connect points, but its properties may vary on different scales that are not measurable with our current technology. On smaller scales, quantum effects may come into play, causing space to have different properties.

Q: How are cylindrical and toroidal spaces created?

Cylindrical space is created by connecting points on the top and bottom boundaries, visualized as a flat sheet rolled up into a cylinder. Toroidal space is created by connecting points on both the top and bottom and left and right boundaries, visualized as a torus or donut shape.

Q: Can toroidal space be visualized differently?

Yes, toroidal space can also be visualized by stretching the cylinder and connecting the top and bottom boundaries like a garden hose. However, in both visualizations, the resulting shape is still a torus.

Q: How does curved space fit into the discussion?

Curved space is a concept in Einstein's theory of general relativity. It allows for different topologies of space, including unbounded, curved spaces. In the video, a 3-dimensional spherically curved space is mentioned as an example.

Summary & Key Takeaways

  • Space can be thought of as a way of connecting points, and on smaller scales, it may have different properties than what we perceive.

  • By connecting points on the top and bottom boundaries, a 2-dimensional cylindrical space is created, which can also be visualized as a flat sheet rolled up into a cylinder.

  • By connecting points on both the top and bottom and left and right boundaries, a 2-dimensional torus (donut shape) is created.

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