Date: October 4, 2000
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:
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!