Date: October 4, 2000
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:
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:
And here's the surprise: the sequence of real numbers
increases and is not bounded above.
Conclusion: for , the ``little'' ball is not contained in the box!