### Breaking a Degenerate Hyper-Dimensional Game of Life

tldr: Demonstrated with interactive visualizations and simulations — over the course of a month, we were able to discover successive new mathematical properties of a “degenerate” hyper-dimensional game of life" to take a “7 dimensions may just barely be possible on a commercial PC, could we ever reach 10 dimensions?” to “10 dimensions is easy enough to be run on any modern browser (jump to spoilers here), and 60 dimensions can be reached with a compiled language”.

This is a story about breaking a degenerate hyper-dimensional game of life via interactive exploratory visualizations and math!

T’was the night before Thursday, December 17, 2020, the release of “Conway Cubes”. It was Day 17 of Advent of Code 2020, a series of fun little themed coding puzzles building up to Christmas; I always enjoyed these puzzles because they are so self-contained and tidy that they are often *open-ended* in the interesting ways you can solve them or expand on them (which I’ve written many blog posts on).

On the surface, Day 17 seemed to be a straightforward extension of Conway’s Game Of Life (“GoL”). GoL is a simulation played out on a 2D grid, where cells are “on” and “off”, and at each step of the simulation the states spread and propagate in interesting ways based on the state of their neighbors (a 2D cellular automaton). The twist of the Advent of Code puzzle is it asks what would happen if we played out the rules of GoL in 3D instead, and then 4D.

I submitted my solution on my assigned puzzle input with a naive implementation (placing 66 and 66 on the leaderboards for that day), concluding the “competitive” part. Of course, the real fun always starts after. When discussing with some friends (on the subreddit and freenode’s `##adventofcode`

channel), we started talking about the trade-offs of different implementations and realized that the extra dimensionality was no joke: as you upped the number of dimensions, the number of points you have to consider grow exponentially, and so does the number of neighbors at each point to check. 4D can be solved naively, but anything higher is going to be strained. My naive solution on 6D took three minutes, and 7D in a reasonable amount of time (requiring as much as 612,220,032 points with 2,186 neighbors each) seemed *impossible* on commercial consumer hardware because of the sheer number of points in 7D space. But I thought…what if a breakthrough in optimization was possible? I set an (arbitrary) personal goal of reaching 10D (3,570,467,226,624 points with 59,048 neighbors each), not knowing if it would ever be possible.

And soon…a breakthrough did come! Someone brought up that if we look at the 3d version, we see there’s actually a *mirror symmetry*! Because everything starts off on the xy plane, with z=0, the resulting progression must be symmetrical on both sides (positive and negative z).