A little ball in the middle.

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 $\mathcal{C}_2$ the circle whose center is at the origin and which is tangent to the four previous circles.

Figure 1: The little circle in the middle.
\mbox{\epsfig{file=CirclesInSquare.eps,width=5cm} }\end{figure}

Denote by E the center of the ``big'' circle in the first quadrant; its coordinate are $\left( \frac 12 , \frac 12 \right)$, thus we have: $OE= \sqrt{ \left( \frac 12 \right)^2 + \left( \frac 12 \right)^2} = \frac {\sqrt{2}}{2}$. Therefore the radius of the ``little'' circle $\mathcal{C}_2$ in the middle is equal to $\frac {\sqrt{2}}{2} - \frac 12 = \frac {\sqrt{2}-1}{2}$. 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 $( \pm 1, \pm 1, \pm 1)$. 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.

Figure 2: The little sphere in the middle.
\subfigure[view from any side]{\epsfig{file=Sph...
...bfigure[global view]{\epsfig{file=BallInMiddle.eps,width=5cm} }

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

Let us now generalize this construction: in an n-dimensional Euclidean space, draw a ``hypercube'' whose vertices have coordinates $( \pm 1, \pm 1, \dots , \pm 1)$. We will call this a box. Inscribe in this box balls whose radius is equal to 1, and whose centers have respective coordinates $( \pm \frac 12, \pm \frac 12, \dots , \pm \frac 12)$. Then we can inscribe in the middle a ``little'' ball verifying the following properties:

And here's the surprise: the sequence of real numbers $x_n=\frac {\sqrt{n} -1}{2}$ increases and is not bounded above. Actually:

\begin{displaymath}x_n > 1 \Longleftrightarrow \frac {\sqrt{n} -1}{2} >1 \Longleftrightarrow \sqrt{n} -1 > 2


\begin{displaymath}\sqrt{n} > 3

and this means that n>9

Conclusion: for $n \geq 10$, the ``little'' ball is not contained in the box!