Exploring the infinite

-“Deep, deep infinity! Rest, dreaming removed from the nervous tensions of everyday life; sailing over a calm sea, on the bow of a ship, towards a horizon that always recedes; staring at the waves that go by and listening to their monotonous, soft murmuring; dreaming away toward unconsciousness…”

1. Although we are accustomed to the idea that the infinite is something endless, infinity can be found within a closed shape with definite boundaries. Even if there will be something lying outside this closed shape, infinity could be found inside this shape in different forms. For example an object of definite shape may consist of an infinite number of smaller parts (such as an atom).

2. If instead of an object we have a process or a function, then a process may repeat itself over and over again, so that even if the process itself is definite, it can last forever. Our own thought is an everlasting process, with functions which can be well-defined, but with no boundaries. Thought can be regarded as an object with definite shape (a circle or sphere for example), composed of an infinite number of functions, or interests. Even if the interest eventually dies out, the void takes its place for eternity.

3. There are many different ways in which we may divide an object or a process into smaller parts. Instead of using atoms or particles, we may use a small number of fundamental structures or shapes. For example we may tessellate a flat surface using tiles. This method has the advantage that there are no gaps between the constituents.

4. Such a tessellation process, or regular division of the plane as the artist called it, is depicted in the picture. A couple of geometric shapes in the form of some creature, either black or white, are alternated in a repetitive pattern, until they fill a circle, becoming smaller and smaller as we move from the center to the perimeter. Although the circle has limits, the number of little ‘angels’ and ‘devils’ goes to infinity. Thus even though the picture is finite, its constituent parts are infinite.

-“When one dives into endlessness, in both time and space, farther and farther without stopping, one needs fixed points or milestones past which one speeds. Without these, one’s movement does not differ from standing still. There must be stars along which one shoots, beacons from which one can measure the road covered.”

5. Can we define space without the stars which comprise it? Can we define time without following the path of a comet moving in the sky? As with any other object or process of thought, ultimately spacetime is composed of the planetary objects and their motions. As the stars are countless, so the universe must be infinite, so that its motion is eternal.

6. But even though the universe seems to be so vast, we can find a small piece of it within ourselves. And although comparatively such a piece is just a small part, strangely enough it seems that it contains the picture of the whole universe. Is it space and time which bring together all the separate pieces in this eternal process of division and recombination, or is it the symmetry underlying the process which produces space and time as we know it?

-“Long before there were people on earth, crystals grew in the earth’s crust. One day a human being saw for the first time such a glimmering little piece of regularity lying about, or he hit it with his stone pickax, and it broke off and fell in front of his feet, and he picked it up and looked at it in his open hand, and he was amazed.”

8/7/2018

Picture: Circle Limit I, M.C. Escher

Abstracts (in italic) from the book: Escher on Escher: Exploring the Infinite, M.C. Escher

[https://www.amazon.com/Escher-Exploring-Infinite-Maurits-Cornelis/dp/0810924145]