Right, a seemingly stronger statement but actually equivalent, because of
rescaling, is: Given a fixed E (regardless of how small) and a fixed R
(regardless of how big) there is a dimension n such that an n-cube with
edge size E cannot be covered by an n-ball of radius R.
One, well-known, way to approximate the area enclosed by a circle is to use
a tiling of sufficiently small squares, count how many are covered by the
circle and its interior (the 2-ball) and multiply that number by the area
of the small square archetype. Likewise, for approximating the volume of a
ball one can use a honeycomb of tiny cubes; and for the n-volume of a ball,
presumably, one could use an n-honeycomb of tiny n-cubes. Yet, the claim
above suggests that even covering a single seemingly tiny n-cube by an
apparent huge n-ball becomes increasingly difficult as n becomes big (not
to mention that the n-volume of a tiny n-cube decreases geometrically).
A few more little claims involving regular n-simplexes and n-balls follow
(the first one seems straightforward):
Claim 1. The n-volume of a regular n-simplex tends to zero as n increases.
The volume of a regular n-simplex with edges of length E (the left
argument) and dimension n (the right argument) is (see [0]),
v=. ((^ % !@:]) * %:@:((1 + ]) % 2 ^ ]))"0
1 v 0 1 2 3 4 5 6
1 1 0.433012702 0.11785113 0.0232923748 0.00360843918 0.000459331825
Claim 2. Given a fixed E (regardless of how big) and a fixed R (regardless
of how small) there is a dimension n such that an n-ball with radius R
cannot be covered by a regular n-simplex with edges of size E.
Claim 3. A regular n-simplex with edges of size 1 can be covered by an
n-ball of radius %%:2 in any dimension.
I could be wrong because I am guessing. ;)
Anyway, it is time for me to start getting ready for a coming attraction
due to the alignment of 3 big "balls" (aka, 'The Great American Total
Eclipse').
:)
[0] Volume
https://en.wikipedia.org/wiki/Simplex#Volume
On Thu, Aug 17, 2017 at 5:31 AM, R.E. Boss <[email protected]> wrote:
> That's easy: the diagonal point in the unit hyper-cube of dimension d,
> with coordinates (1 #~ d) has distance (%: d) from the origin, which grows
> above any R with increasing d.
>
>
> R.E. Boss
>
>
> > -----Original Message-----
> > From: Programming [mailto:[email protected]]
> > On Behalf Of Jose Mario Quintana
> > Sent: donderdag 17 augustus 2017 00:27
> > To: Programming forum <[email protected]>
> > Subject: Re: [Jprogramming] "n-volume" of an "n-sphere"
> >
> > The statement, "Suppose that R is fixed. Then the volume of an n-ball of
> > radius R approaches zero as n tends to infinity." is in the link that
> Raul
> > mentioned in his second post in this thread together with a couple of
> proofs.
> >
> > A related statement (assuming I am not mistaken) is: Given a fixed R
> > (regardless of how large) there is a dimension n such that the unit
> n-cube
> > centered at the origin is not contained by the n-ball of radius R
> centered at
> > the origin.
> >
> > How come? The following J sentence gives a clue,
> >
> > ( (+/&.:*:)@:(0.5$~ ])("0) i.222 ) i. (0 1 2 3 4 5 6 7)
> > 0 4 16 36 64 100 144 196
> >
> > OEIS A016742
> >
>
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