The Universe in Your Eyeball

How can you catch a glimpse of an infinite extent in one single glance? For large objects, you can step back and zoom out, but for…. infinitely large objects?

The (observable) Universe in its glorious beauty. This representation is a scientific artwork by Pablo Carlos Budassi

There is a surprisingly simple technique to represent infinite planes in a finite area.

Let’s first be clear and acknowledge that by representing an infinite object onto a limited area with a discrete resolution (i.e. your screen), we will have to cope with some compromise.
In the case we will see here, that compromise means parts of the represented infinite will feature less detail (or, if you prefer, a coarser resolution).

The idea has been experimented with by several mathematicians and artists in the last century or so; to visualize it, I sketched a quick illustration with Inkscape:

Imagemagick function to create an hyperbolically distorted image: convert birdline.png -channel RGBA -virtual-pixel Gray -fx ‘p{i+sinh((i-w/2)/50)), j}’ birdline_hyperbolic.jpg

In this sketch we use a simplified model, in that the objects we repeat (a tessellation borrowed from Escher) are placed on a two-dimensional hyperbole, like if we had unrolled an infinite ribbon. The Universe, as we know it, has instead at least three “geometrical” dimensions, but…