Formulas for N-Dimensional Spheres

Date: 10/26/2000 at 00:49:38
From: Shalanna Collins Weeks
Subject: Geometry in 4 dimensions (4D circle?)

Hi! I have a puzzle book that I have used in my math classes (I tutor 
math for gifted and talented 7th-8th grade kids) for the students who 
have finished their work. There is a "finish-the-sequence" question 
that one of the students called to my attention, and I could not 
answer it for him, and there was no answer in the book, so I came 
here. This may not even be answerable, but... (It's not essential for 
schoolwork, but we were both curious!)

The first term in the sequence was the formula for the area of a 
circle (pi r^2); the second term in the sequence was the formula for 
the volume of a sphere (4/3 pi r^3); what would be the third term?

We agreed that the next term in this sequence should logically be the 
formula to find the area contained within a four-dimensional sphere. 
You know, like a tesseract is supposed to be the four-dimensional 
extension of a cube. A beach ball for somebody with a fourth spatial 
dimension.

Well, we didn't know what this is called, or if there is a book that 
would talk about these kinds of things. Does this appeal to you as 
something you would like to take a guess at answering? Anything you 
come up with would probably fascinate us. Thank you in advance for any 
help with this one. (I've already lent the student my copy of Madeline 
L'Engle's _A Wrinkle in Time_, which talks about a tesseract as a way 
into the fourth dimension... in a fantasy, fanciful way, of course.)

Shalanna Collins Weeks
Teacher, pedant, dabbler, charlatan

Date: 10/26/2000 at 15:27:32
From: Doctor Rob
Subject: Re: Geometry in 4 dimensions (4D circle?)

Thanks for writing to Ask Dr. Math, Shalanna.

The sequence is

     r^n * pi^(n/2) / Gamma(n/2)

Here pi is the familiar ratio of circumference to diameter of a 
circle. Gamma(x) is a function with the property that if k is a 
positive integer, then

     Gamma(k) = (k-1)!

Gamma(x) is the smooth continuous version of the discrete factorial 
function.

When n is even, say n = 2*k, this becomes

     r^n * pi^(n/2) / (n/2-1)!,   n even.

You can see that this works for n = 2, when you get

     r^2 * pi / 0! = pi * r^2

When n is odd, say n = 2*k - 1, you need the additional fact that

     Gamma(k-1/2) = sqrt(pi)*(2*k)!/(k!*4^k)

In this case, the expression you get is

     r^n * pi^([n-1]/2) * ([n+1]/2)! * 2^(n+1)/(n+1)!, n odd.

You can see that this works for n = 3, when you get

       r^3 * pi * 2!*2^4 / 4!
     = (4/3) * Pi * r^3

For n = 4, the expression is

       r^4 * pi^2 / 1!
     = Pi^2 * r^4

For n = 5, the expression is

       r^5 * pi^2 * 6!/(3!*4^3)
     = (15/8)* Pi^2 * r^5

Observe that the exponent of r increases by 1 each time, but the 
exponent of pi increases by 1 every second time, and the coefficients 
obey the peculiar sequence given above. A larger part of the sequence 
is:

     n       F(n)
     0        0
     1       2*r
     2       Pi*r^2
     3       (4/3)*Pi*r^3
     4       Pi^2*r^4
     5       (15/8)*Pi^2*r^5
     6       (1/2)*Pi^3*r^6
     7       (16/105)*Pi^3*r^7
     :           :

For more, see "Formula for the Surface Area of a Sphere in Euclidean 
N-Space" from the Sci.Math FAQ (editor Alex Lopez-Ortiz) at:
  
  http://db.uwaterloo.ca/~alopez-o/math-faq/node75.html   

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/