G313 Arnold’s Cat Map

There is a uniformity and beauty to the universe that we live in. Indeed, mathematicians are often able to perceive this beauty in their calculations.
Now, this being said, sometimes the simplicity and the beautify that lies inherent within a mathematical construct can be profound and stunning.

It can make the most skeptical of people into a believer of the divine.

Here is one such exercise.

In the 1960s, Soviet mathematician Vladimir Arnold mapped the square image of a cat to a torus, “stretched” (sheared) it as shown on that surface, then sliced the resulting image into pieces and recomposed them into a square.

As the process is repeated, any two points in the image quickly become separated, but, surprisingly, after sufficient repetitions the original image reappears.

A discrete analogue is below…

As the transformation is repeated, the image appears increasingly random or disordered, but the underlying cat can be glimpsed making occasional appearances, sometimes as a ghostly suggestion, sometimes in multiple smaller images, and occasionally (yowling, one imagines) even upside down.

It reappears again, unhurt, at the 300th iteration.

It’s called Arnold’s cat map. You can try it yourself here.

It implies a uniformity within our universe, and a glimpse into how we can perceive ourselves, our alternative world-lines, our past and our futures through the lenses of the momentary iterations of our own consciousness.