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### Formulas for N-Dimensional Spheres

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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/
```
Associated Topics:
College Analysis
College Higher-Dimensional Geometry
High School Analysis
High School Functions
High School Higher-Dimensional Geometry

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