Hans Walser, [20170711]

Sphere and cylinder

1     The original problem

In (Richeson 2017, p 23) I found:

 

ŇJust noticed that the ratio of the volume of a sphere to the volume of the cylinder containing it is 2:3.

Likewise, the surface area. Eureka!Ó

 

Fir. 1: Sphere and cylinder

2     In other dimensions

The mentioned statement holds similar in other dimensions.

We use the following notations:

 

 = Volume of the sphere in the n-dimensional space

 

 = Surface of the sphere in the n-dimensional space

 

 = Volume of the cylinder in the n-dimensional space

 

 = Surface of the cylinder in the n-dimensional space

 

 

Examples:

 

Ratio

Ratio

2

3

4

5

6

7

Tab. 1: Examples

In every dimension there is the same ratio. In even dimensions the ratio is irrational.

3     General case

Notation:

 

                                                                                                                 (1)

 

Hence we get:

 

                                                                                        (2)

 

And for the volume of the cylinder:

 

                                                               (3)

 

For the surface of the cylinder we get:

 

       (4)

 

Remark:

 

                                                                                                       (5)

 

The volumes of the sphere and the cylinder have the ratio:

 

                                                               (6)

 

For the surfaces we get the ratio:

 

                                                     (7)

 

From (6) and (7) we see that the ratios are equal in any dimension.

4     Explicit formulas

According to [1] we have:

 

                                                                                           (8)

 

 

From (6) and (8) we get:

In even dimensions we have:

 

                                                  ratio =                                              (9)

 

!! denotes the double faktorial, defined for odd integers  by:

 

                                                      (10)

 

 

In odd dimensions  we get:

 

                                                     ratio =                                               (11)

 

References

Richeson, David (2017): A-Tweeting We Will Go. Building a Professional Network with Twitter. MAA FOCUS | JUNE/JULY 2017 | maa.org/focus. 22-25.

 

Websites

[1] Wikipedia: n-sphere

https://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_area