Re: HyperVolumes
David W. Cantrell
> As a non-mathematician one naively expects volumes to continually
> increase (be monotonic) with increasing dimension; i.e.
>
> 1) for Cubes of edge length a and dimension n, V(Cube) = a^n. This is
> obviously correct up to n = 3, and (I assume) for any value of n.
Yes, V = a^n. But I don't know why, as n increases, you'd expect volume to
increase. Rather, as n increases, volume increases or decreases depending,
respectively, on whether a is greater than or less than 1.
> 2) However for (geometric) spheres it seems that V(n) is not monotonic
> and is in fact determined by Gamma (n) with a maximum when n is more than
> 7. For n > 7, V(Sphere) reduces with increasing n.
I suspect that you mean 5, rather than 7. For example, see the first graph
at
<http://gregegan.customer.netspace.net.au/APPLETS/29/HypercubeNotes.html>,
an interesting page, mentioned in another sci.math thread recently.
Some people point out, as though it were of special interest, that, for an
n-ball of unit radius, volume reaches its maximum at n = 5. But I see no
reason why that should be of special interest. The value of n at which
maximum volume is reached is _radius-dependent_.
What is interesting is that, regardless of the radius of the n-ball, its
volume must _eventually_ decrease as n increases.
David Cantrell
Terry Padden
"David W. Cantrell" <DWCan...@sigmaxi.org> wrote in message
news:20050613235208.158$d...@newsreader.com...
>
>> 2) However for (geometric) spheres it seems that V(n) is not monotonic
>> and is in fact determined by Gamma (n) with a maximum when n is more than
>> 7. For n > 7, V(Sphere) reduces with increasing n.
>
> I suspect that you mean 5, rather than 7. For example, see the first graph
> at
>
> <http://gregegan.customer.netspace.net.au/APPLETS/29/HypercubeNotes.html>,
>
> an interesting page, mentioned in another sci.math thread recently.
>
> Some people point out, as though it were of special interest, that, for an
> n-ball of unit radius, volume reaches its maximum at n = 5. But I see no
> reason why that should be of special interest. The value of n at which
> maximum volume is reached is _radius-dependent_.
>
> What is interesting is that, regardless of the radius of the n-ball, its
> volume must _eventually_ decrease as n increases.
>
Thanks for the reference and other insights. Is there a standard text on or
that covers basic hyper geometry ?
Dave L. Renfro
> What is interesting is that, regardless of the
> radius of the n-ball, its volume must _eventually_
> decrease as n increases.
In fact, for each fixed radius the volume decreases
to zero as n --> oo
The n-volume of an n-ball of radius r is
Pi^(n/2) * r^n / gamma(n/2 + 1).
Using Stirling's Formula we find that gamma(n/2 + 1) is
asymptotic to sqrt(n*Pi) * (n / 2e)^(n/2), and hence an
n-ball's n-volume is asymptotic to P*Q, where
P = 1 / sqrt(n*Pi)
Q = (2*Pi*e*r^2 / n) ^ (n/2).
Since both P and Q approach zero as n --> oo (for a fixed
value of r), it follows that the n-volume of an n-ball
approaches zero as n --> oo.
This has an interesting consequence that might not be
very well-known. Given any r > 0, there are parallel
hyperplanes arbitrarily close to each other such
that for sufficiently large n, essentially 100%
of the n-volume of some n-ball with radius r lies
between these two hyperplanes.
More precisely, given r > 0, epsilon > 0, and delta > 0,
there exists n such that given any n-ball B of radius r
there exists two parallel hyperplanes separated by
a distance of delta with the center of B midway between
them such that the volume of the part of B that doesn't
lie between these two hyperplanes is less than epsilon.
To see this, consider the percentage of the n-volume
of an n-cube with edge lengths 2r that two hyperplanes
passing through the n-cube and parallel to one of the
n-cube's faces determine as compared to the percentage
of the n-volume of this same cube that an inscribed
n-ball of radius r takes up. The former is proportional
to the distance between the hyperplanes (and the constant
of proportionality is independent of n), while the latter
approaches zero as n --> oo.
Dave L. Renfro
double d
Terry Padden
"Dave L. Renfro" <renf...@cmich.edu> wrote in message
news:1119035063.5...@o13g2000cwo.googlegroups.com...
>
> The n-volume of an n-ball of radius r is
>
> Pi^(n/2) * r^n / gamma(n/2 + 1).
>
Can you provide a basic readily available reference text, or on-line
resource, showing details of how this is developed. The usual advanced
calculus texts stop at 3D and don't show the Gamma dependency.
Interesting.
G. A. Edgar
Padden <TPa...@bigpond.net.au> wrote:
> "Dave L. Renfro" <renf...@cmich.edu> wrote in message
> news:1119035063.5...@o13g2000cwo.googlegroups.com...
> >
> > The n-volume of an n-ball of radius r is
> >
> > Pi^(n/2) * r^n / gamma(n/2 + 1).
> >
>
> Can you provide a basic readily available reference text, or on-line
> resource, showing details of how this is developed. The usual advanced
> calculus texts stop at 3D and don't show the Gamma dependency.
K.R. Stromberg, AN INTRODUCTION TO CLASSICAL REAL ANALYSIS, p. 394
Smith & Vamanamurty, Mathematics Magazine 62 (1989) 101-107
L. Badger, Amer. Math. Monthly 107 (2000) 256
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Dave L. Renfro
>> The n-volume of an n-ball of radius r is
>>
>> Pi^(n/2) * r^n / gamma(n/2 + 1).
>
> Can you provide a basic readily available reference text,
> or on-line resource, showing details of how this is
> developed. The usual advanced calculus texts stop
> at 3D and don't show the Gamma dependency.
Sorry, I didn't see this reply until now. Edgar has
given three references, but in my usual spirit of
overkill (the main reason being for everyone's later
reference via google's sci.math archive), here are
some places you can look. Not all of these give actual
derivations of the volume and/or surface area of an
n-ball, but they all relate to these issues.
No, I didn't look all this stuff up just now! It's in
a manuscript of miscellaneous neat results that I've been
working on in my spare time over the past few years.
************************ TEXTBOOKS ************************
[1] Tom M. Apostol, "Calculus (Volume II): Multi-Variable
Calculus and Linear Algebra, with Applications to
Differential Equations and Probability", 2'nd edition,
Blaisdell Publishing Company, 1969, xxiv + 673 pages.
[MR 40 #1542; Zbl 316.26006]
http://www.emis.de/cgi-bin/Zarchive?an=0185.11402
See Section 11.32: Example 4 (pp. 411-413) and
Exercise 33 (p. 416)
[2] Robert G. Bartle, "The Elements of Real Analysis",
2'nd edition, John Wiley & Sons, 1976, xvi + 480 pages.
[MR 52 #14179; Zbl 309.26003]
http://www.emis.de/cgi-bin/MATH-item?0309.26003
See Exercises 45.S & 45.T (pp. 454-455)
[3] Richard Courant and Fritz John, "Introduction to Calculus
and Analysis", Volume II, Wiley-Interscience, 1974,
xxvi + 954 pages. [MR 90j:00002b; Zbl 294.26003]
http://www.emis.de/cgi-bin/MATH-item?0294.26003
See Section 4.11b (pp. 455-459)
[4] Charles Henry Edwards, "Advanced Calculus of Several
Variables", Academic Press, 1973, xii + 457 pages.
[MR 50 #4828; Zbl 308.26002]
http://www.emis.de/cgi-bin/MATH-item?0308.26002
See Examples IV.5.1, V.4.7, V.4.8 and
Exercises IV.5.17-19, IV.6.3, IV.6.13, V.4.10, V.4.11
[5] Harold M. Edwards, "Advanced Calculus: A Differential
"Forms Approach", Birkhäuser, 1994, xvi + 508 pages.
[MR 94k:26001; Zbl 796.26003]
http://www.emis.de/cgi-bin/MATH-item?0796.26003
See Section 9.6, Exercise 12 (p. 425)
[6] Wendell Fleming, "Functions of Several Variables",
2'nd edition, Undergraduate Texts in Mathematics,
Springer-Verlag, 1977, xii + 411 pages.
[MR 54 #10514; Zbl 348.26002]
http://www.emis.de/cgi-bin/MATH-item?0348.26002
See Section 5.9 (p. 220)
[7] Jerrold E. Marsden, "Elementary Classical Analysis",
W. H. Freeman and Company, 1974, xvi + 549 pages.
[MR 50 #10161; Zbl 285.26005]
http://www.emis.de/cgi-bin/MATH-item?0285.26005
See Chapter 9, Example 3 (pp. 324-325)
********************** PAPERS, ETC. **********************
[8] Lee Badger, "Generating the measures of n-balls",
American Mathematical Monthly 107 (2000), 256-258.
[Zbl 999.28003]
http://www.emis.de/cgi-bin/MATH-item?0999.28003
[9] David Ross Barr, "A packing problem revisited",
Operations Research 18 (1970), 746-747.
[ZbL 197.16903]
http://www.emis.de/cgi-bin/Zarchive?an=0197.16903
[10] Harold Scott MacDonald Coxeter, "Regular Polytopes",
3'rd edition, Dover Publications, 1973, xiv + 321 pages.
[MR 51 #6554; Zbl 118.35902]
http://www.emis.de/cgi-bin/Zarchive?an=0118.35902
See Section 7.3 (pp. 125-126)
[11] Gerald B. Folland, "How to integrate a polynomial
over a sphere", American Mathematical Monthly 108
(2001), 446-448. [MR1837866; Zbl 1046.26503]
http://www.emis.de/cgi-bin/MATH-item?1046.26503
[12] N. Gauthier and J. R. Gosselin, "Volume of a sphere
in N dimensions", International Journal of Mathematical
Education in Science and Technology 23 (1992), 902-903.
[Zbl 768.26004]
http://www.emis.de/cgi-bin/MATH-item?0768.26004
[13] M. G. Goluzina, A. A. Lodkin, B. M. Makarov, and
A. N. Podkorytov, "Selected Problems in Real Analysis",
Translations of Mathematical Monographs #107, American
Mathematical Society, 1992. [MR 93i:26001; Zbl 765.26001]
http://www.emis.de/cgi-bin/MATH-item?0765.26001
See Section VIII.6 (pp. 104-108, 290-300)
[14] Larry C. Grove and Olga Yiparaki, "Contumacious spheres",
College Mathematics Journal 31 (2000), 35-41.
[MR 2001e:52035; Zbl 995.52505]
http://www.emis.de/cgi-bin/MATH-item?0995.52505
[15] Omar Hijab, "The volume of the unit ball in C^n",
American Mathematical Monthly 107 (2000), 259.
[MR1742126; Zbl 982.32001]
http://www.emis.de/cgi-bin/MATH-item?0982.32001
[16] Greg Huber, "Gamma function derivation of
n-sphere volumes", American Mathematical Monthly
89 (1982), 301-302.
[17] Andrew Keith Jobbings, "The volume of the n-ball (II)",
Mathematical Gazette 82 #493 (March 1998), 105-106.
[18] Maurice George Kendall, "A Course in the Geometry
of n Dimensions", Griffin's Statistical Monographs
& Courses #8, Charles Griffin & Company, 1961,
viii + 63 pages. [Reprinted by Dover in 2004.]
[MR 25 #476; Zbl 103.12002]
http://www.emis.de/cgi-bin/Zarchive?an=0103.12002
See Sections 41-42 (pp. 35-36)
[19] R. Kuruganti and Prashant S. Sansgiry, "B(m,n)
= Gamma(m)*Gamma(n) / Gamma(m+n) and volume of
an n-dimensional sphere", International Journal
of Mathematical Education in Science and Technology
22 (1991), 475-476. [Zbl 742.33002]
http://www.emis.de/cgi-bin/MATH-item?0742.33002
[20] Jean B. Lasserre, "A quick proof for the volume of
n-balls", American Mathematical Monthly 108 (2001),
768-769. [Zbl 1054.26503]
http://www.emis.de/cgi-bin/MATH-item?1054.26503
[21] Jeffrey Lynn Nunemacher, "The largest unit ball in any
Euclidean space", Mathematics Magazine 59 (1986), 170-171.
[22] Walter Rudin, "A generalization of a theorem of
Archimedes", American Mathematical Monthly 80 (1973),
794-796. [MR 48 #7100; Zbl 278.50003]
http://www.emis.de/cgi-bin/MATH-item?0278.50003
[23] J. A. Scott, "The volume of the n-ball (I)", Mathematical
Gazette 82 #493 (March 1998), 104-105.
[24] David Breyer Singmaster, "On round pegs in square holes
and square pegs in round holes", Mathematics Magazine
37 (1964), 335-337. [Zbl 123.15704]
http://www.emis.de/cgi-bin/Zarchive?an=0123.15704
[25] David J. Smith and Mavina K. Vamanamurthy, "How small
is a unit ball?", Mathematics Magazine 62 (1989),
101-107. [MR 90e:51029; Zbl 702.52006]
http://www.emis.de/cgi-bin/MATH-item?0702.52006
For more even precise estimates of the n-volume of the
unit n-ball, see Alzer (MR 2001m:26036; Zbl 972.26015)
and Anderson/Qiu (MR 98h:33001; Zbl 881.33001).
[26] Duncan M'Laren Young Sommerville, "An Introduction
to the Geometry of N Dimensions", Dover Publications,
1929/1958, xx + 196 pages.
[MR 20 #6672; Zbl 86.35804; JFM 55.0953.01]
http://www.emis.de/cgi-bin/Zarchive?an=0086.35804
http://www.emis.de/cgi-bin/JFM-item?55.0953.01
See Section VIII.13 (pp. 135-137)
Dave L. Renfro
Dave L. Renfro
>[25] David J. Smith and Mavina K. Vamanamurthy, "How small
> is a unit ball?", Mathematics Magazine 62 (1989),
> 101-107. [MR 90e:51029; Zbl 702.52006]
> http://www.emis.de/cgi-bin/MATH-item?0702.52006
>
> For more even precise estimates of the n-volume of the
> unit n-ball, see Alzer (MR 2001m:26036; Zbl 972.26015)
> and Anderson/Qiu (MR 98h:33001; Zbl 881.33001).
I meant to say "even more precise asymptotic estimates
as n --> oo of the n-volume of the unit n-ball".
Dave L. Renfro
Terry Padden
"G. A. Edgar" <ed...@math.ohio-state.edu.invalid> wrote in message
news:210620051004076562%ed...@math.ohio-state.edu.invalid...
Many thanks - but - the book is not readily available. Don't know about the
magazine or periodical; I'll see if the nearest (20 miles away) university
takes them.
Many of us are non-affiliated and do not have ready access to facilities
that are normal in academia.
Terry Padden
> in my usual spirit of
> overkill (the main reason being for everyone's later
> reference via google's sci.math archive), here are
> some places you can look. Not all of these give actual
> derivations of the volume and/or surface area of an
> n-ball, but they all relate to these issues.
> No, I didn't look all this stuff up just now! It's in
> a manuscript of miscellaneous neat results that I've been
> working on in my spare time over the past few years.
Muchos gratias. Overkill very welcome - and very astute.
G. A. Edgar
<TPa...@bigpond.net.au> wrote:
> Many of us are non-affiliated and do not have ready access to facilities
> that are normal in academia.
The internet is not yet as good as a real hands-on library.
Maybe in 10 years it will be?
Herman Rubin
>>> The n-volume of an n-ball of radius r is
>>> Pi^(n/2) * r^n / gamma(n/2 + 1).
>> Can you provide a basic readily available reference text,
>> or on-line resource, showing details of how this is
>> developed. The usual advanced calculus texts stop
>> at 3D and don't show the Gamma dependency.
A very easy way to do this is by induction on n.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Greg Egan
(Herman Rubin) wrote:
> In article <1119480126....@g44g2000cwa.googlegroups.com>,
> Dave L. Renfro <renf...@cmich.edu> wrote:
> >Terry Padden wrote:
>
> >>> The n-volume of an n-ball of radius r is
>
> >>> Pi^(n/2) * r^n / gamma(n/2 + 1).
>
> >> Can you provide a basic readily available reference text,
> >> or on-line resource, showing details of how this is
> >> developed. The usual advanced calculus texts stop
> >> at 3D and don't show the Gamma dependency.
>
> A very easy way to do this is by induction on n.
> --
Take it as given that:
I_n = integral_0^Pi sin^n(t) dt
= sqrt(Pi) Gamma(n/2 + 1/2) / Gamma(n/2 + 1)
which itself can be proved by induction. Then suppose that the volume of
an n-ball of radius r is given by:
V(r,n) = A(n) r^n
which is obviously true for n=1 with A(1)=2.
The volume of an (n+1)-ball can then be found by integrating the volume of
n-balls in successive n-dimensional slices:
V(r,n+1) = integral_{-r}^r V(sqrt(r^2-u^2), n) du
= A(n) integral_{-r}^r (r^2-u^2)^{n/2} du
Change variables, putting u = r cos t, du = -r sin t dt. Then:
V(r,n+1) = A(n) r^{n+1} I_{n+1}
so A(n+1) = A(n) I_{n+1}
Assume that A(n) = Pi^(n/2) / Gamma(n/2 + 1).
Since Gamma(1/2)=sqrt(Pi), and Gamma(x+1) = x Gamma(x), Gamma(3/2) = (1/2)
sqrt(Pi), and A(1)=2, which agrees with the obvious result for n=1.
To get the induction step,
A(n) I_{n+1}
= (Pi^(n/2) / Gamma(n/2+1)) (sqrt(Pi) Gamma(n/2+1) / Gamma((n+1)/2 + 1))
= Pi^((n+1)/2) / Gamma((n+1)/2 + 1))
= A(n+1)
--
Greg Egan
Email address (remove name of animal and add standard punctuation):
gregegan netspace zebra net au