**Thierry Dana-Picard**

*Date:* October 4, 2000

Department of Applied Mathematics, Jerusalem College of Technology

Havaad Haleumi Str. 21, POB 16031, Jerusalem 91160,Israel

email: dana@mail.jct.ac.il

The little mathematical story that we will tell you here is not new. Gardiner told it about 45 years ago already, but it is still very nice.

Choose a coordinate system in the Euclidean plane and draw a square whose vertices are
*A*(1,1), *B*(-1,1), *C*(-1,-1) and *D*(1,-1) (the side length of the square is equal to 2).
In the square *ABCD* we draw four circles whose radius is equal to 1; each circle is tangent to two of the sides of
the square and to two other cicles, as can be seen in Figure 1. The four circles determine
a star-shaped region; we denote by
the circle whose center is at the origin and which is tangent
to the four previous circles.

Denote by *E* the center of the ``big'' circle in the first quadrant; its coordinate are
,
thus we have:
.
Therefore the radius of the ``little'' circle
in the middle is equal to
.
By symmetry, we could have made our computations with any
other ``big'' circle.

Consider a similar situation, now in 3-dimensional Euclidean space: we choose a coordinate system and draw a cube
whose vertices are at the points with coordinates
.
Here we can draw eight spheres inscribed
in the cube, every one having radius 1: in fact there are four spheres beneath the *xy*-plane, and four spheres
above it, as shown in Figure 2.

By a computaton similar to the one above, we have:

- The radius of each ``big'' sphere is equal to ;
- The distance from the origin to the center of one ``big'' sphere is equal to .
- The radius of the ``little'' sphere in the middle is equal to .

Let us now generalize this construction: in an *n*-dimensional Euclidean space, draw a ``hypercube'' whose vertices
have coordinates
.
We will call this a box. Inscribe in this box balls whose radius
is equal to 1, and whose centers have respective coordinates
.
Then we can inscribe in the middle a ``little'' ball verifying the following properties:

- The radius of each ``big'' ball is equal to ;
- The distance from the origin to the center of one ``big'' sphere is equal to .
- Thus, the radius of the ``little'' sphere in the middle is equal to .

And here's the surprise: the sequence of real numbers
increases and is not bounded above.
Actually:

i.e.

and this means that

Conclusion: for , the ``little'' ball is not contained in the box!