My understanding (which could be wildly wrong, I only skimmed the thread) is that it's running in a standard 2-dimensional Game of Life grid, it just happens to start out as a 1x3.7B cell line.

The best way to run it is in the software Golly. It has the HashLife algorithm needed to make it run fast enough to see it finish.

After the first step it isn't 1D any more, so I don't think that visualization is possible

It's possible. It'd just be a 3D visualization and more importantly, stupendously huge. If each cell was a cubic millimeter, the shape would be 3700km wide, and stretch 1/3rd of the way to the moon.

And if each cell was a cubic micrometer (which is a side length 200-300 times smaller than a pixel on a typical screen and 50-100 times thinner than a human hair), it'd still stretch 3.7 kilometers, which is about the length of a commercial airport runway.

I'm having trouble finding a common image file format that has > 32bit resolution fields, fit it in.