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Happy New Year! Just like every year, there were tons of fantastic discoveries and theorems proved in Conway's Game of Life in 2024. This is a thread for my three favourites (and the context behind them to try to convince you that they're interesting). 🧵 #MathSky

#1 (1/4): Phoenices. A phoenix is an oscillator in which every cell dies in every generation (and thus every alive cell was dead in the previous generation, hence the name). The first known phoenix has period 2: conwaylife.com/wiki/Phoenix_1

#1 (2/4) That phoenix was found in the early days of Life (1970 or 1971), and since then people have been wondering whether or not there are phoenices of any other periods. And previously, only a few scattered periods had been ruled out.

#1 (3/4) In 2000, Stephen Silver showed that period 3 phoenices don't exist. In 2019, Alex Greason showed that period 5 phoenices don't exist. And in 2023, Keith Amling showed that period 7, 9, and 11 phoenices don't exist.

#1 (4/4) In 2024, Adam P. Goucher completely resolved this problem by showing that there does not exist a phoenix oscillator of any period other than 2. The proof is computer-assisted, but the idea behind it is very readable: cp4space.hatsya.com/2024/01/20/e...

#2 (1/4): Still life glider synthesis. A glider synthesis is a way of crashing together 2 or more gliders so as to create another object. For example, in the image below three gliders collide so as to create a lightweight spaceship.

#2 (2/4) Glider synthesis is the key ingredient of Life that makes it possible to build most of the complex mega-patterns that you hear about. If you zoom in on a pattern like Gemini, for example, you'll see that it's almost entirely made of gliders: conwaylife.com/wiki/Gemini

#2 (3/4) For this reason, people have been cataloging glider syntheses of patterns in Life ever since the early 1970s. We've known how to synthesize all small (say 10 cell or smaller) still lifes and oscillators since those early days. Recently this was pushed to all <= 21-cell still lifes in 2022.

#2 (4/4) In 2024, a large collaborative effort extended this to all 22-cell still lifes. To get a sense of scale, there are 672172 still lifes with 22 cells. That's 672172 different patterns to construct by colliding gliders together: conwaylife.com/forums/viewt...

#3 (1/4): Oscillator density. In the early days of Life, it was conjectured that the maximum density (i.e., maximum ratio of alive cells to dead cells) of an infinitely large still life is 0.5. This density is easily attained by alternating rows of dead and alive cells.

#3 (3/4) However, it's natural to ask whether or not the same is true of infinitely large oscillators. There are plenty of infinitely large oscillators with average density equal to 0.5, but until 2023 the best known *upper* bound on the average density of an oscillator was 8/13 ≈ 0.615.

#3 (4/4) This bound has now been improved, culminating in 2024 with Keith Amling proving an upper bound of 1176/2087 ≈ 0.563. We still don't know the exact answer, but this is the first progress that was made in over 3 decades: conwaylife.com/forums/viewt...