6 ms·

Lenia – Biology of Artificial Life

89 points by p1esk 8y ago

hirundo 8y ago
I honestly don't know if this a simulation of life or an example of it. Or if those are mutually exclusive.
- taneq 8y ago
  From the video, they seem to be more like stable moving patterns than life. They don't appear to self-reproduce or have behaviours.
  - AareyBaba 8y ago
    That is in the future works section in the pdf. 8. Do self-replicating or pattern-emitting lifeforms exist in Lenia?
    - taneq 8y ago
      It seems kind of a stretch to call it "biology" then. "I designed a new life form! points to lego brick It exists and you can move it! Future work will involve making it self-replicate and have artificial intelligence."
whatshisface 8y ago
>We describe the methods of constructing and studying Lenia, including its mathematical definition, in silico simulation [...] It's funny how different fields of science have all invented their own names for "computer program." Physicists and many engineers call them "codes," and biologists love fitting in the buzzword "in silico."
- tokai 8y ago
  I would guess it's a play on in situ.
  - taneq 8y ago
    Or 'in vitro' ('in glass' ie. 'in a test tube'), as opposed to 'in vivo' ('in life').
    - veddox 8y ago
      Yes, in biology we use in silico to extend this previous distinction between in vivo and in vitro.
  - p1esk 8y ago
    In my field (analog hardware for deep learning) "in situ" (e.g. in situ training) means operations are executed in hardware, while "ex-situ" means a simulation of the hardware running the model is done in software (on CPU/GPU).
whatshisface 8y ago
Here is an implementation: https://chakazul.github.io/Lenia/JavaScript/Lenia.html https://chakazul.github.io/Lenia/JavaScript/Lenia.html Here is the github: https://github.com/Chakazul/Lenia https://github.com/Chakazul/Lenia
oblosys 8y ago
Youtube video showing some of the persistent life forms: https://www.youtube.com/watch?v=iE46jKYcI4Y https://www.youtube.com/watch?v=iE46jKYcI4Y
kvark 8y ago
Huh, they extend the Game of Life with a complex neighbor kernel, and get more complex patterns emerging. This is a cool experiment, but hardly "biology of artificial life".
- veddox 8y ago
  A short, sweet summary of my opinion :D
dkural 8y ago
Figure 20 from the paper is particularly interesting. Life solves multiple problems, and for unicellular life, one of them is the problem of locomotion. At least here Lenia and real world biology seems to have some amount of overlap.
- ianai 8y ago
  If there’s any overlap at such a simplified model then the simplified model just might be a good place to start figuring out the actual system.
  - veddox 8y ago
    All due respect - I think that for pretty much any simple shape you'll find an actual organism that "looks just like it".
    - dkural 8y ago
      I do tend to agree that unicellular life has a mind boggling diversity of forms and you can find similarities to most things, simple or not-simple :) That said, humans have a very specific notion of "simple shape", intuitively incorporating symmetry into it. Symmetry is far from guaranteed in mathematics (although mathematicians, being human, also tend to search and study symmetric things). Even in the most symmetric objets - let's say groups, there are many more non-abelian (i.e. not commutative ab != ba) groups than there are commutative groups.
fao_ 8y ago
The title reminds me of the work done by Karl Simms and published in SIGGRAPH94 http://www.karlsims.com/papers/alife94.pdf http://www.karlsims.com/papers/alife94.pdf http://www.karlsims.com/papers/siggraph94.pdf http://www.karlsims.com/papers/siggraph94.pdf (video here: https://archive.org/details/sims_evolved_virtual_creatures_1994 https://archive.org/details/sims_evolved_virtual_creatures_1...)
sytelus 8y ago
TLDR; "Artificial life" is a misnomer for efforts like this which are simply play on cellular autometa type variations. This phrase is extensively used by biologist to constructing actual living physical cells using non-living material. This paper is not about that and I hope folks stop calling this "artificial life". Important bit: Life can be defined as the capabilities of self-organizing (morphogenesis), selfregulating (homeostasis), self-directing (motility), self-replicating (reproduction), entropy reduction (metabolism), growth (development), response to stimuli (sensitivity), response to environment (adaptability), and evolving through mutation and selection (evolvability). Lenia, the subject of this paper, is able to achieve many of them, except self-replication that is yet to be discovered. Above definition wouldn't be agreed upon by many biologist because the entire concept of environment that is as complex as universe is missing. It's easy to create something that can "thrive" in 2D grids with bunch of simple rules but quite another thing that can thrive in a computer called universe which is simulating 10^80 atoms at once where all quantum and space-time rules aren't even known to us.
- dj-wonk 8y ago
  However you would prefer the term be defined and used, for decades, people have used the term "artificial life" to include cellular automata. Sources: https://en.wikipedia.org/wiki/Artificial_life https://en.wikipedia.org/wiki/Artificial_life https://en.wikipedia.org/wiki/History_of_artificial_life https://en.wikipedia.org/wiki/History_of_artificial_life
  - ianai 8y ago
    Further, the study of how life arises has to start somewhere very simple, like drastically simple/simplified 2D rule systems.
- AareyBaba 8y ago
  Too late. The term has been co-opted. There are Artifical Life conferences that have been running for more than 15 years now. http://2018.alife.org/accepted-papers-oral/ http://2018.alife.org/accepted-papers-oral/
- 8y ago
lioeters 8y ago
Strange and fascinating. It raises interesting questions about we consider "life". The implementation [0] contains a whole taxonomy tree of variations, of which Conway's Game of Life is but a branch. It's certainly an ecosystem of mathematical "life" forms. Reminds me of one of my favorite books, Vehicles: Experiments in Synthetic Psychology [0]. [0] https://chakazul.github.io/Lenia/JavaScript/Lenia.html https://chakazul.github.io/Lenia/JavaScript/Lenia.html [1] https://mitpress.mit.edu/books/vehicles https://mitpress.mit.edu/books/vehicles
ambrop7 8y ago
I wonder if cellullar automata could exhibit at least some approximation of classical mechanics, i.e. mass, inertia, forces etc.? From my understanding, motion as seen here is purely due to the structure of the objects.
- no_identd 8y ago
  Yes. See here: https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.56.1505 https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.56.1505 Frisch, Hasslacher, Pomeau - Lattice Gas Automata for the Navier-Stokes Equation (1986) Citations (5k+) here: https://scholar.google.com/scholar?cites=5993777633272177716 https://scholar.google.com/scholar?cites=5993777633272177716 See also my other comment here: https://news.ycombinator.com/item?id=18757946 https://news.ycombinator.com/item?id=18757946
veddox 8y ago
Hm, sounds like mathematician giving fancy biological-sounding names to some procedurally generated abstract art forms. Frankly, I had hoped for more, given the title. There's some really cool work that has been done with digital artificial life that has real biological significance. For example, Richard Lenski's group at Michigan State University developed a platform called AVIDA (https://en.wikipedia.org/wiki/Avida https://en.wikipedia.org/wiki/Avida) that has been used for evolutionary biology research. Slightly different, 2012 saw the publication of the first whole-cell computational model (https://www.sciencedirect.com/science/article/pii/S0092867412007763 https://www.sciencedirect.com/science/article/pii/S009286741...). And in the past 30 years, computer simulations of ecosystems have slowly been gaining momentum and acceptance as a tool for addressing ecological questions that are very hard to study in real life (this happens to be the field I work in). Lenia, on the other hand, is nothing more than an abstraction of Conway's Game of Life. A pretty mathematical pastime that is fun to play around with and gives you some interesting patterns - but nothing that merits the description "biology". The whole paper just reeks with misapplied terms: from "taxonomy" to "ecology" and even "physiology". (They don't have anything to do with their real counterparts.) In short, I fail to find any significance for our knowledge of life on earth, or any other life, for that matter. Pretty graphs, though.
- trevyn 8y ago
  Molecular biology: https://youtu.be/_SGoDIkscXg?t=39 https://youtu.be/_SGoDIkscXg?t=39 Game of Life: https://youtu.be/xP5-iIeKXE8 https://youtu.be/xP5-iIeKXE8 Sure, one is 2D and discrete, the other is 3D and continuous... oh wait, Lenia is 2D and continuous. That’s interesting.
  - nivertech 8y ago
    2D and discrete -> Conway's Game of Life 2D and continuous -> Lenia 3D and discrete -> Minecraft ? 3D and continuous -> Molecular biology
  - tree_of_item 8y ago
    Sure, being continuous is cool, but your parent is absolutely right that calling anything about this "biology" is way out of line.
no_identd 8y ago
Pet Peeve Rant: 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 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 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_12 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 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/7696/7742 https://www.aaai.org/ocs/index.php/SSS/SSS14/paper/viewFile/... Kornai, András - Euclidean Automata Citations here: https://scholar.google.com/scholar?cites=2906127536931309801 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.pdf?sequence=1&isAllowed=y 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%2F978-1-4939-8700-9_58 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.pdf 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+of+Life+in+Hexagonal+and+Pentagonal+Tessellations%22 https://scholar.google.com/scholar?q=%22A+Note+on+the+Game+o... Citations of the 2009 version: https://scholar.google.com/scholar?cites=1062581570709413178&as_sdt=2005&sciodt=0,5&hl=en 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.pdf http://wpmedia.wolfram.com/uploads/sites/13/2018/02/08-2-4.p... Citations here: https://scholar.google.com/scholar?cites=9274761406110052964 https://scholar.google.com/scholar?cites=9274761406110052964 B: Research by Andrew Wuensche: http://users.sussex.ac.uk/~andywu/downloads/papers/self_rep.pdf 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=12260818150335116959 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: https://i4oc.org/ https://i4oc.org/ So tools like http://citationgecko.com/ http://citationgecko.com/ & http://cluster.cis.drexel.edu/~cchen/citespace/ http://cluster.cis.drexel.edu/~cchen/citespace/ can actually help put an end to this mess.

89 points by p1esk 8y ago

hirundo 8y ago

I honestly don't know if this a simulation of life or an example of it. Or if those are mutually exclusive.

taneq 8y ago
From the video, they seem to be more like stable moving patterns than life. They don't appear to self-reproduce or have behaviours.
- AareyBaba 8y ago
  That is in the future works section in the pdf. 8. Do self-replicating or pattern-emitting lifeforms exist in Lenia?
  - taneq 8y ago
    It seems kind of a stretch to call it "biology" then. "I designed a new life form! points to lego brick It exists and you can move it! Future work will involve making it self-replicate and have artificial intelligence."

whatshisface 8y ago

>We describe the methods of constructing and studying Lenia, including its mathematical definition, in silico simulation [...] It's funny how different fields of science have all invented their own names for "computer program." Physicists and many engineers call them "codes," and biologists love fitting in the buzzword "in silico."

tokai 8y ago
I would guess it's a play on in situ.
- taneq 8y ago
  Or 'in vitro' ('in glass' ie. 'in a test tube'), as opposed to 'in vivo' ('in life').
  - veddox 8y ago
    Yes, in biology we use in silico to extend this previous distinction between in vivo and in vitro.
- p1esk 8y ago
  In my field (analog hardware for deep learning) "in situ" (e.g. in situ training) means operations are executed in hardware, while "ex-situ" means a simulation of the hardware running the model is done in software (on CPU/GPU).

whatshisface 8y ago

Here is an implementation: https://chakazul.github.io/Lenia/JavaScript/Lenia.html https://chakazul.github.io/Lenia/JavaScript/Lenia.html Here is the github: https://github.com/Chakazul/Lenia https://github.com/Chakazul/Lenia

oblosys 8y ago

Youtube video showing some of the persistent life forms: https://www.youtube.com/watch?v=iE46jKYcI4Y https://www.youtube.com/watch?v=iE46jKYcI4Y

kvark 8y ago

Huh, they extend the Game of Life with a complex neighbor kernel, and get more complex patterns emerging. This is a cool experiment, but hardly "biology of artificial life".

veddox 8y ago
A short, sweet summary of my opinion :D

dkural 8y ago

Figure 20 from the paper is particularly interesting. Life solves multiple problems, and for unicellular life, one of them is the problem of locomotion. At least here Lenia and real world biology seems to have some amount of overlap.

ianai 8y ago
If there’s any overlap at such a simplified model then the simplified model just might be a good place to start figuring out the actual system.
- veddox 8y ago
  All due respect - I think that for pretty much any simple shape you'll find an actual organism that "looks just like it".
  - dkural 8y ago
    I do tend to agree that unicellular life has a mind boggling diversity of forms and you can find similarities to most things, simple or not-simple :) That said, humans have a very specific notion of "simple shape", intuitively incorporating symmetry into it. Symmetry is far from guaranteed in mathematics (although mathematicians, being human, also tend to search and study symmetric things). Even in the most symmetric objets - let's say groups, there are many more non-abelian (i.e. not commutative ab != ba) groups than there are commutative groups.

fao_ 8y ago

The title reminds me of the work done by Karl Simms and published in SIGGRAPH94 http://www.karlsims.com/papers/alife94.pdf http://www.karlsims.com/papers/alife94.pdf http://www.karlsims.com/papers/siggraph94.pdf http://www.karlsims.com/papers/siggraph94.pdf (video here: https://archive.org/details/sims_evolved_virtual_creatures_1994 https://archive.org/details/sims_evolved_virtual_creatures_1...)

sytelus 8y ago

TLDR; "Artificial life" is a misnomer for efforts like this which are simply play on cellular autometa type variations. This phrase is extensively used by biologist to constructing actual living physical cells using non-living material. This paper is not about that and I hope folks stop calling this "artificial life". Important bit: Life can be defined as the capabilities of self-organizing (morphogenesis), selfregulating (homeostasis), self-directing (motility), self-replicating (reproduction), entropy reduction (metabolism), growth (development), response to stimuli (sensitivity), response to environment (adaptability), and evolving through mutation and selection (evolvability). Lenia, the subject of this paper, is able to achieve many of them, except self-replication that is yet to be discovered. Above definition wouldn't be agreed upon by many biologist because the entire concept of environment that is as complex as universe is missing. It's easy to create something that can "thrive" in 2D grids with bunch of simple rules but quite another thing that can thrive in a computer called universe which is simulating 10^80 atoms at once where all quantum and space-time rules aren't even known to us.

dj-wonk 8y ago
However you would prefer the term be defined and used, for decades, people have used the term "artificial life" to include cellular automata. Sources: https://en.wikipedia.org/wiki/Artificial_life https://en.wikipedia.org/wiki/Artificial_life https://en.wikipedia.org/wiki/History_of_artificial_life https://en.wikipedia.org/wiki/History_of_artificial_life
- ianai 8y ago
  Further, the study of how life arises has to start somewhere very simple, like drastically simple/simplified 2D rule systems.
AareyBaba 8y ago
Too late. The term has been co-opted. There are Artifical Life conferences that have been running for more than 15 years now. http://2018.alife.org/accepted-papers-oral/ http://2018.alife.org/accepted-papers-oral/
8y ago

lioeters 8y ago

Strange and fascinating. It raises interesting questions about we consider "life". The implementation [0] contains a whole taxonomy tree of variations, of which Conway's Game of Life is but a branch. It's certainly an ecosystem of mathematical "life" forms. Reminds me of one of my favorite books, Vehicles: Experiments in Synthetic Psychology [0]. [0] https://chakazul.github.io/Lenia/JavaScript/Lenia.html https://chakazul.github.io/Lenia/JavaScript/Lenia.html [1] https://mitpress.mit.edu/books/vehicles https://mitpress.mit.edu/books/vehicles

ambrop7 8y ago

I wonder if cellullar automata could exhibit at least some approximation of classical mechanics, i.e. mass, inertia, forces etc.? From my understanding, motion as seen here is purely due to the structure of the objects.

no_identd 8y ago
Yes. See here: https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.56.1505 https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.56.1505 Frisch, Hasslacher, Pomeau - Lattice Gas Automata for the Navier-Stokes Equation (1986) Citations (5k+) here: https://scholar.google.com/scholar?cites=5993777633272177716 https://scholar.google.com/scholar?cites=5993777633272177716 See also my other comment here: https://news.ycombinator.com/item?id=18757946 https://news.ycombinator.com/item?id=18757946

veddox 8y ago

Hm, sounds like mathematician giving fancy biological-sounding names to some procedurally generated abstract art forms. Frankly, I had hoped for more, given the title. There's some really cool work that has been done with digital artificial life that has real biological significance. For example, Richard Lenski's group at Michigan State University developed a platform called AVIDA (https://en.wikipedia.org/wiki/Avida https://en.wikipedia.org/wiki/Avida) that has been used for evolutionary biology research. Slightly different, 2012 saw the publication of the first whole-cell computational model (https://www.sciencedirect.com/science/article/pii/S0092867412007763 https://www.sciencedirect.com/science/article/pii/S009286741...). And in the past 30 years, computer simulations of ecosystems have slowly been gaining momentum and acceptance as a tool for addressing ecological questions that are very hard to study in real life (this happens to be the field I work in). Lenia, on the other hand, is nothing more than an abstraction of Conway's Game of Life. A pretty mathematical pastime that is fun to play around with and gives you some interesting patterns - but nothing that merits the description "biology". The whole paper just reeks with misapplied terms: from "taxonomy" to "ecology" and even "physiology". (They don't have anything to do with their real counterparts.) In short, I fail to find any significance for our knowledge of life on earth, or any other life, for that matter. Pretty graphs, though.

trevyn 8y ago
Molecular biology: https://youtu.be/_SGoDIkscXg?t=39 https://youtu.be/_SGoDIkscXg?t=39 Game of Life: https://youtu.be/xP5-iIeKXE8 https://youtu.be/xP5-iIeKXE8 Sure, one is 2D and discrete, the other is 3D and continuous... oh wait, Lenia is 2D and continuous. That’s interesting.
- nivertech 8y ago
  2D and discrete -> Conway's Game of Life 2D and continuous -> Lenia 3D and discrete -> Minecraft ? 3D and continuous -> Molecular biology
- tree_of_item 8y ago
  Sure, being continuous is cool, but your parent is absolutely right that calling anything about this "biology" is way out of line.

no_identd 8y ago

Pet Peeve Rant: 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 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 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_12 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 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/7696/7742 https://www.aaai.org/ocs/index.php/SSS/SSS14/paper/viewFile/... Kornai, András - Euclidean Automata Citations here: https://scholar.google.com/scholar?cites=2906127536931309801 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.pdf?sequence=1&isAllowed=y 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%2F978-1-4939-8700-9_58 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.pdf 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+of+Life+in+Hexagonal+and+Pentagonal+Tessellations%22 https://scholar.google.com/scholar?q=%22A+Note+on+the+Game+o... Citations of the 2009 version: https://scholar.google.com/scholar?cites=1062581570709413178&as_sdt=2005&sciodt=0,5&hl=en 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.pdf http://wpmedia.wolfram.com/uploads/sites/13/2018/02/08-2-4.p... Citations here: https://scholar.google.com/scholar?cites=9274761406110052964 https://scholar.google.com/scholar?cites=9274761406110052964 B: Research by Andrew Wuensche: http://users.sussex.ac.uk/~andywu/downloads/papers/self_rep.pdf 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=12260818150335116959 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: https://i4oc.org/ https://i4oc.org/ So tools like http://citationgecko.com/ http://citationgecko.com/ & http://cluster.cis.drexel.edu/~cchen/citespace/ http://cluster.cis.drexel.edu/~cchen/citespace/ can actually help put an end to this mess.