Amazing behaviour in high dimensions

[Michel Bierlaire]

Here is an interesting geometrical problem that J. Mersch told me last month.

1) In dimension 2

You take a square of side 2. You divide it into 4 squares of side 1.
You inscribe a circle (of diameter 1, of course) in each of these 4 squares.
Take care on the small circle that is tangent to the 4 circles.
Its diameter is
(main_diagonal - 2*diameter) / 2 = sqrt(2) - 1

2) In dimension 3

You take a cube of side 2.
You divide it in8 cube of side 1.
You insribe a sphere (of diameter 1) in each cube.
You yake care of the small sphre tangent to the 8 spheres.
Its diameter is sqrt(3) - 1

3) In higher dimension, you always have that the diameter of the "small" hypersphere
is sqrt(n) - 1

So in dimension 4, the inner hypersphere has exactly the same dimension than the
inscribed ones

In dimension 9, the diameter of the hypersphere is the same as the side of the
square.

When the dimension is higher than 9, the "small" sphere is higher than the
first square !!!

Amazing, no ?

Michel Bierlaire
m...@math.fundp.ac.be

[Dr. C.D. Wright]

This is a fairly well known phenomenon that deserves more
wonder than it usually attracts. It shows that, in some
sense, spheres in high dimensions are "pointy", in so far
as slices tend to have small volume, and they "poke through"
poles.

-------------- Original follows ------------------

Michel Bierlaire (m...@math.fundp.ac.be) wrote:

: Here is an interesting geometrical problem that J. Mersch told me last month.

[Gerald Edgar]

In <CAMJn...@liverpool.ac.uk> cd...@liverpool.ac.uk (Dr. C.D. Wright) wrote:
>This is a fairly well known phenomenon that deserves more
>wonder than it usually attracts. It shows that, in some
>sense, spheres in high dimensions are "pointy", in so far
>as slices tend to have small volume, and they "poke through"
>poles.
>

"In high-dimensional worlds, most people live in the tropics."

Take the surface measure within distance epsilon on each side
of the equator. As a fraction of the total surface of the n-dimensional
sphere, it approaches 1 (at an exponential rate) as n increases.

--
Gerald A. Edgar Internet: ed...@math.ohio-state.edu
Department of Mathematics Bitnet: EDGAR@OHSTPY
The Ohio State University telephone: 614-292-0395 (Office)
Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)