How can a shape be a torus and be flat at the same time?

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A torus can be flat in a mathematical sense, but it has a distinct three-dimensional shape. In geometry, a torus is defined as a surface of revolution generated by rotating a circle in three-dimensional space about an axis that does not intersect the circle.

In this sense, the torus has a well-defined shape and is not flat like a two-dimensional plane. However, the geometry of the surface of the torus can be described as flat, meaning that it has zero curvature. This means that the surface of the torus is locally Euclidean, which is to say that it satisfies the same rules of geometry as a flat plane.

In other words, while the torus has a distinct three-dimensional shape, the geometry of its surface is flat in a local sense, even though the overall shape of the torus is not flat. This is an example of how the concept of flatness in geometry can have different meanings depending on the context.

YouTube: The Tortuous Geometry of a Flat Torus - http://www.science4all.org/article/flat-torus/

 

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