I would be interested in the simplest form of self-replication, that is, the simplest automata machine, and the simplest initial set of states...
Interestingly, Phi spirals -- seem to construct themselves in nature at many different scales, and without (apparently) running inside of an automata machine of any sort...
I nominate Phi spirals as universal (sans apparent automata machine) constructors...
If someone could explain what type of automata machine Phi spirals are running in, and what this automata machine's rules are, then I think that would go a long way to understanding the universe, but at this point in time, I cannot determine a universal automata machine, nor its rules...
I only know that Phi spirals are automatically formed by nature, at a variety of scales, and in a variety of mediums...
What are some examples of Phi spirals (assuming you mean golden spirals?) outside of biology? We see them all over the natural world but off the top of my head I can't think of any non-biological examples. A quick read of Wikipedia points to logarithmic spirals occurring occasionally (although not specifically golden spirals).
(Also if we're just accepting geometric shapes, I'd nominate ellipses as being far more common, since they show up all the time in orbital mechanics.)
Spiral galaxy’s which occur due to decreasing orbital velocities (edit: in radians) as you move from the center. Whirlpools and hurricanes are shaped by similar rotational effects.
However, their close approximations not exactly the correct shape.
That’s not enough to offset the increased distances from longer orbits. So you still get a spiral as the velocity in in radians decreases. But, it looks like a Golden spiral because the velocities in m/s are almost constant due to dark matter.
PS: Should have said orbital period to be more clear.
First paragraph of the Purpose section includes some simpler examples. And they are links, so check them out as well!
> However, it is clear that far simpler machines can achieve self-replication. Examples include trivial crystal-like growth, template replication, and Langton's loops. But von Neumann was interested in something more profound: construction, universality, and evolution.
You also get beautiful spirals from Belousov–Zhabotinsky reactions. They can be simulated by cellular automata, and are manifested in nature by chemical reactions, slime molds, and reefs of tube worms!
I don't think they're Turing complete or self replicating per se, but you can start them on a random configuration, and they will form several spiraling "attractors" around oscillating cores ("nucleation"), that send out concentric spiraling waves, which meet waves from other attractors (or boundaries in the environment like a maze) and reinforce or cancel each other out, and also they can solve mazes and climb gradients and find food! (Plus, slime molds are not only beautiful, but make great pets, and they're easy to care for!)
>Plasmodium of Physarum polycephalum is a large cell, visible by unaided eye, which exhibits sophisticated patterns of foraging behaviour. The plasmodium's behaviour is well interpreted in terms of computation, where data are spatially extended configurations of nutrients and obstacles, and results of computation are networks of protoplasmic tubes formed by the plasmodium. In laboratory experiments and numerical simulation we show that if plasmodium of Physarum is inoculated in a maze's peripheral channel and an oat flake (source of attractants) in a the maze's central chamber then the plasmodium grows toward target oat flake and connects the flake with the site of original inoculation with a pronounced protoplasmic tube. The protoplasmic tube represents a path in the maze. The plasmodium solves maze in one pass because it is assisted by a gradient of chemo-attractants propagating from the target oat flake.
Dictyostelium discoideum, axenic strain, aggregation on a petri dish
>The aggregation of Dictyostelium discoideum amoebae after starvation provides one of the best examples of spatiotemporal pattern formation at the supracellular level. This transition from a unicellular to a multicellular stage of the life cycle occurs by a chemotactic response to cyclic AMP (cAMP) signals emitted by aggregation centers in a periodic manner. Amoebae are capable of relaying the signals emitted periodically by a center located in their vicinity. This excitable response to periodic signals explains the wavelike nature of aggregation over territories whose dimensions can reach up to 1 cm: within each aggregation territory, the amoebae move toward a center in concentric or spiral waves with a periodicity of the order of 5 to 10 min. Waves of cellular movement correlate with waves of cAMP; the latter present a striking similarity to waves observed in oscillatory chemical systems such as the Belousov--Zhabotinsky reaction.
Rudy Rucker's CellLab Rule Documentation: ZHABO, ZHABOF, and ZHABOFF (the same rule that appears on the cover of Margolus and Toffoli's classic book on the CAM-6 hardware, "Cellular Automata Machines: A New Environment for Modeling"):
>A picture of Zhabo appears on the cover of [Margolus&Toffoli87].
>Margolus and Toffoli make a interesting simile between the Zhabotinsky reaction and a reef of tubeworms. When a tubeworm feels safe, it sticks a plume out of its shell to seine the water for food. If a feeding tubeworm senses any disturbance nearby (e.g. the presence of several other feeding tubeworms), it retracts its plume and waits for a few cycles before feeding again.
Margolus and Toffoli's Cellular Automata Machines: A New Environment for Modeling:
Don Hopkins' Cellular Automata and Video Feedback Demo Reel (Showing three variations of the CAM-6 "WORMS" rule: Yuppie Worms, Middle Class Worms, and Bohemian Worms, rendered with an AfterEffects plug-in I developed)
If we think about a the Belousov–Zhabotinsky reaction, it's sort of different than most other chemical reactions, that is, most other chemical reactions are unidirectional, Belousov–Zhabotinsky reaction is bidirectional, or perhaps we'd call it cyclical...
What would be fascinating, I think, would be to attempt to figure out what it would take to stabilize a Belousov–Zhabotinsky reaction in one state... like what chemical or chemicals, and how much of them would do that?
Even more interesting... try to stabilize it in one state using electricity... or sound... or other electromagnetic wave phenomena...
If it could be stabilized in one state using any wave phenomena, then perhaps we might unlock some new understanding about this reaction, and Chemistry in general...
Anyway, my apologies, the above thought was unrelated to this discussion, but I needed to write it down someplace, and this was the most convenient place...<g>
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Interestingly, Phi spirals -- seem to construct themselves in nature at many different scales, and without (apparently) running inside of an automata machine of any sort...
I nominate Phi spirals as universal (sans apparent automata machine) constructors...
If someone could explain what type of automata machine Phi spirals are running in, and what this automata machine's rules are, then I think that would go a long way to understanding the universe, but at this point in time, I cannot determine a universal automata machine, nor its rules...
I only know that Phi spirals are automatically formed by nature, at a variety of scales, and in a variety of mediums...