Hans Walser, [20170711]
Sphere and cylinder
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
The mentioned statement holds similar in other dimensions.
We use the following notations:
= Volume of the sphere in the ndimensional space
= Surface of the sphere in the ndimensional space
= Volume of the cylinder in the ndimensional space
= Surface of the cylinder in the ndimensional 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.
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.
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): ATweeting We Will Go. Building a Professional Network with Twitter. MAA FOCUS  JUNE/JULY 2017  maa.org/focus. 2225.
Websites
[1]
Wikipedia: nsphere
https://en.wikipedia.org/wiki/Nsphere#Volume_and_surface_area