Just like in the case of SmoothLife (which Lenia generalizes), the square grid used for this implies a lack of isotropy(we could also say that it "introduces anisotropy"; but that makes it sounds like a good thing), EVEN IF you turn the entire thing continuous like done here and use . This last fact becomes relatively obvious if you mash the "random" button in the demo, you'll see 'square' noise.
This really ought to use a triangular grid (or the dual of it, a hexagonal one) to ensure isotropy; Frisch, Hasslacher & Pomeau showed this in 1986 while working on two dimensional Navier-Stokes equations:
https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.56.1505 (Ctrl+F isotropy)
Section 4.4.4 of the OP paper DOES acknowledges that one could generalize this further (which includes generalizing to a hexagonal grid, doesn't mention a triangular grid however), but unfortunately it doesn't mention isotropy even once, anywhere, not even in section 3.1.1 on Spatial invariance (albeit, I suppose, the section does semi-address it indirectly, but what's described there to address it seems like a hack, and the 'empirical' evidence from the random button would seem to agree with it.).
Having said all that:
I still consider Lenia DAMN impressive, and I think other people have pointed out this lack of isotropy to the author before. I suspect he'll follow up with more generalizations, but, based on the Github activity (See here: https://github.com/Chakazul/Lenia - last commit in July, long open & relatively trivial pull requests, etc.) and when the OP paper got published to arXiv, I suspect he's been busy writing the paper & with other academic duties and that he could almost certainly use a lot of help with this project. I lack the time (and admittedly, skills) to do so, but I figure if I point this out here, maybe some people might want to pick up on it, especially since, as the author points out, Lenia & further generalizations of it could serve as a great benchmark for various machine learning related matters, which Hacker News users seem to have a great interest in.
Edit: I've decided to us this opportunity to do some renewed literature research:
1.
A:
If you want to read more about Euclidean automatONS beyond the citation #27 from the OP paper, you might want to check here (I suspect the author of the OP paper either overlooked the existence of this book chapter, or hasn't had time to study it yet):
https://link.springer.com/chapter/10.1007/978-1-84996-217-9_... Pivato, Marcus - RealLife (Chapter from "Game of Life Cellular Automata" edited by Andrew Adamatzky)
Which seems like a follow up work to: https://arxiv.org/abs/math/0503504 Pivato, Marcus - RealLife: the continuum limit of Larger Than Life cellular automata
B: There also seems to exist a separate idea ALSO called Euclidean AutomatA, first described here:
https://www.aaai.org/ocs/index.php/SSS/SSS14/paper/viewFile/... Kornai, András - Euclidean Automata
Citations here: https://scholar.google.com/scholar?cites=2906127536931309801
The author of this seems unaware of the works by Pivato, the same applies to this follow up work:
https://ruj.uj.edu.pl/xmlui/bitstream/handle/item/51661/EA01... Gyenis, Zálan - Skeleton in the Euclidean closet
2. More on Triangular & Hexagonal automata:
A:
Research by Carter Bays:
https://link.springer.com/referenceworkentry/10.1007%2F978-1... Cellular Automata in Triangular, Pentagonal, and Hexagonal Tessellations
"First Online: 28 November 2018" - but this is basically a republication of a 2009 version of an expanded & updated version of http://wpmedia.wolfram.com/uploads/sites/13/2018/02/15-3-4.p..., which stems from 2005!
Citations of the 2005 version (doesn't seem directly listed in Google Scholar):
https://scholar.google.com/scholar?q=%22A+Note+on+the+Game+o...
Citations of the 2009 version:
https://scholar.google.com/scholar?cites=1062581570709413178...
And there's also an even older version from 1994, only addressing triangular tesselations:
http://wpmedia.wolfram.com/uploads/sites/13/2018/02/08-2-4.p...
Citations here: https://scholar.google.com/scholar?cites=9274761406110052964
B:
Research by Andrew Wuensche:
http://users.sussex.ac.uk/~andywu/downloads/papers/self_rep.... Self-reproduction by glider collisions: the beehive rule
Citations here:
https://scholar.google.com/scholar?cites=1226081815033511695...
On an off-topic side note: Navigating citation space is an absolute mess, as the above shows. Researchers constantly have to cut corners when it comes to studying the literature because it's too damn hard to navigate, which leads to constant overlooking of convergently evolved research. A Peter Landin quote comes to mind:
"Most papers in computer science describe how their author learned what someone else already knew."
If this bothers you even nearly as much as it bothers me, I suggest you see how you can help the Initiative for Open Citations:
So tools like http://citationgecko.com/ & http://cluster.cis.drexel.edu/~cchen/citespace/ can actually help put an end to this mess.