So let us then try to climb the mountain, not by stepping on what is below us, but to pull us up at what is above us, for my part at the stars; amen

Escher

How can we visualize the infinity?

Maurits Cornlis Escher contemplated this problem. His tessellations of the plane repeat forever in every direction. Yet although this implies the infinite it does not give us the feeling of infinity.

To solve this problem Escher drew circular patterns where the repeated shapes grew smaller and smaller as they approached a central point. If the first shape has a width of 1cm, then the second has a width of 1/2cm, the third 1/4cm etc. The pattern has infinitely many shapes and yet all together they fill a finite space.

If you add up the widths of the shapes you get the infinite sum:

1 = 1/2 = 1/4 = 1/8 …. cm, which must be finite, otherwise it won't fit on the page!

In fact 1 + 1/2 + 1/4 + 1/8 + … = 2.

We have bridged the gap between the infinite (sum) on the left hand side and the finite on the right.