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!)

https://en.wikipedia.org/wiki/Belousov%E2%80%93Zhabotinsky_r...

https://en.wikipedia.org/wiki/Attractor

https://en.wikipedia.org/wiki/Nucleation

Mould Time-lapse - The Great British Year: Episode 4 Preview - BBC One: Meet the Slime Mold

https://www.youtube.com/watch?v=GY_uMH8Xpy0

The Belousov-Zhabotinsky Reaction - Christmas Lectures with Ian Stewart

https://www.youtube.com/watch?v=o72GGxQqWt8

Sub-excitable Belousov-Zhabotinsky medium solves Reims maze: Oregonator model

https://www.youtube.com/watch?v=YWeSOEvVF7w

Can Slime Mould Solve Mazes? | Earth Lab

https://www.youtube.com/watch?v=HyzT5b0tNtk

Slime mould solves maze in one pass ... assisted by gradient of chemo-attractants

https://arxiv.org/abs/1108.4956

>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

https://www.youtube.com/watch?v=Yl3ESZ4XQLI

>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"):

http://www.rudyrucker.com/oldhomepage/celdoc/rules.html#Zhab...

>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:

https://www.amazon.com/Cellular-Automata-Machines-Environmen...

https://donhopkins.com/home/Tommaso_Toffoli_Norman_Margolus_... (by permission of the authors)

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)

https://www.youtube.com/watch?v=eCVJ08gK2o8

AfterEffects Aether Demo

https://www.youtube.com/watch?v=N1Y-29tX0Co