Torus | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-10

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Torus


00:01:14 1 Geometry
00:05:17 2 Topology
00:07:27 3 Two-sheeted cover
00:10:40 4 in/i-dimensional torus
00:11:11 4.1 Configuration space
00:13:58 5 Flat torus
00:14:31 6 Genus ig/i surface
00:15:26 7 Toroidal polyhedra
00:18:18 8 Automorphisms
00:21:15 9 Coloring a torus
00:22:27 10 Cutting a torus
00:22:44 11 See also
00:23:11 12 Notes
00:26:34 13 References
00:27:28 14 External links
00:28:48 See also
00:28:57 Notes



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SUMMARY
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In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution.
Real-world examples of toroidal objects include inner tubes. A torus should not be confused with a solid torus, which is formed by rotating a disc, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world approximations include doughnuts, non-inflatable lifebuoys, and O-rings.
In topology, a ring torus is homeomorphic to the Cartesian product of two circles: S1 × S1, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane with itself. This produces a geometric object called the Clifford torus, a surface in 4-space.
In the field of topology, a torus is any topological space that is topologically equivalent to a torus. Famously, a coffee cup and a doughnut are both topological tori.