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There are algorithms where I think "Sure, with enough time and attention given to the problem, I might have thought of that." And then there are algorithms where I think "Oh, wow. That came from another planet. I would never have come up with that myself."

This one is definitely in the latter category.

Agreed. Further, for most algorithms, I can read a high-level description of it and go "ohhh, I get it now", and go away and implement it myself without further information. HashLife is not one of those algorithms! While I grok the concept of it, I'm pretty lost on how I'd turn that into functional code. I'm sure I could figure it out with enough further reading, though.
The Wikipedia article skips an important trick on how to arrange the quadtree nodes to make this work.

This link is better: http://www.drdobbs.com/jvm/an-algorithm-for-compressing-spac...

Yeah, I implemented this in 2010 and the Dr. Dobbs article was essential. I even had the original C code and couldn’t figure it out without this article.
It starts very good, but does this article have multiple pages? Clicking 2 or Next at the bottom shows "something went wrong. Please contact an administrator". Tried in firefox and chrome on android
You can click the tiny printer icon above the title to display the entire article as a single page.
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An interesting consequence of the HashLife algorithm is that when people design patterns in Conway's Game of Life they sometimes try to tailor them to the algorithm. The rule of thumb is that any repeated components should be laid out with spacings that are powers of two, and the period of any oscillations should be a power of two.
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Current AI is like memoized probability, then? Optimizations across a gigantic word cloud.

Suppose someone puts on a canvas one instance or snapshot of an intermediate representation of the board.

If it is contained in the "path" of a future HashLife, what would it mean?

I had a lot of fun a few years ago playing with this algorithm. I ended up with a little site that has life in life and some fun zooming and panning gestures on mobile.

https://github.com/natessilva/gol