## Generalizing volumes of revolution to higher dimensions

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### Generalizing volumes of revolution to higher dimensions

How can we generalize volumes of revolution (3D) to higher dimensions?

And can we evaluate bulk or surcell volume of that shape?

1. In 3D, a sphere can be constructed by rotating a half circle by coordinate axis.
Then in 4D, can 4D hypersphere be constructed by rotating half sphere by stationary plane?

2. volume of a sphere can be calculated via disk method ( Integrating cross-sectional areas , http://tutorial.math.lamar.edu/Classes/ ... Rings.aspx )
Can disk method be generalized to higher dimensions?
hy.dodec
Dionian

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### Re: Generalizing volumes of revolution to higher dimensions

Volumese of revolution can certainly be generalized to higher dimensions.

The cubinder, for example, can be generated by revolving a 3D cube around a plane that involves the 4th dimension.

Revolving a sphere around its center produces a 4D sphere.

Revolving a sphere around a point outside its surface produces a spherindrical torus, one of the analogues of a 3D torus (there are multiple possibilities in 4D).

Revolving a cylinder around the plane formed by its axis and the 4th axis produces a spherinder. OTOH, revolving it around the plane parallel to its lids produces instead a duocylinder.

These are just some of the simplest shapes you can make. There are many, many more.
quickfur
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