# 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

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 n>9

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