Atheism/The Blind Watch Maker
Why are there patterns (sometimes mathematical), symmetry, and fractals in nature?
Why aren't same to exact visual hallucinations as in Folie A Deux accepted by skeptics as telepathy?
Thanks for the question! I kind of got carried away with my answer here, hopefully you find the extended discussion interesting!
Mathematics in nature:
All things obey patterns. Patterns that are inconsistent or indecipherable are just patterns that are more complicated. For example a black-and-white checkboard is a pattern and it “feels” like a pattern because all I have do to describe it is say that it is composed of a grid of black and white squares, with (orthogonally) adjacent grid squares colored oppositely. I can also a describe a 2x2 grid composed of a red square, a blue square, a yellow square, and a green square with the red and green squares diagonal from each other. That objects doesn't “feel” like a pattern as much because almost everything there is to know about that “pattern” is contained in the description, no inference or mathematical deduction is needed.
All patterns are mathematical. Mathematics is nothing less than the study of all possible patterns and how those patterns can possibly interact with each other. For instance both the checkboard and the four-color board I mentioned earlier were specific solutions to a larger class of mathematical objects known as a “coloring problem”, which related to “coloring-in” a network so that no adjacent units have the same color. The checkboard pattern stands out because it is a solution with the additional property that it uses the minimum number of colors necessary to solve the problem, wheras the same cannot be said about the four-color board.
So there is nothing in nature, indeed, nothing that can possibly exist, which is not bound by the rules of mathematics. What we typically mean when we talk about mathematics in nature, though, are examples of natural objects whose mathematical properties are simple, familiar, and universalizable enough that we can immediately recognize the presence of mathematics. In other words, the set of natural objects for which simple mathematical models have great explanatory power. Fractals and symmetry are great examples of that.
So why, at the level of fundamental physics, is nature based off of a few “simple” mathematical rules? My many AllExperts posts speculating on the origin of the universe have come back to the idea that the simplest explaination is not only the best explaination, but the most likely type of universe to come into being. But why I suppose that is so is a much more difficult answer to articulate. Perhaps if we full understand the best way to fomulate the laws of physics it would become immediately clear why they are the only laws of physics that could ever have existed. Perhaps there is some sort of law-generating entity that our universe comes from and it would start to take-off (like Inflation Theory) as soon as it reached the sufficient complexity. In any event, our historical progress in physics has led us to simpler theories even as we understood about more things in the world (here's where we are now, that's all well-known particles, forces, and rules of physics). Here is a video where physics Nobel laureate Murray Gell-Mann describes in detail how the laws of physics are elegant and why he supposes the universe works that way.
But of course when looking at macroscopic objects and living things in nature, we also find simple mathematics descriptions at that level. No one attempts to understand the growth of a tree be specifying where the protons, neutrons, electrons, and photons that make it up are, instead we describe it in terms of mathematical relationships between biological, chemical, and thermodynamic structures. This property is known as emergence. There is a statistical source of emergence in systems of many similar objects, stemming from the law of large numbers. Thermodynamic concepts such as energy, entropy, temperature, and density are governed by conservation laws of physics that guide the interactions between the macroscopic parameters. There is also a biological emergence steming from evolutionary principles. Here is a particularly engaging evolutionary demonstration in which you might observe evolutionarily select models becoming simpler and more functional. In order for living things to survive, they must adopt strategies that are simple, effective, and consistent. A strategy that fits those characteristics is interesting and likely to be of some mathematical significance.
There is one last idea to discuss here. Our study of nature is necessarily a mathematical one and is also biased towards simpler results. To properly test a natural model, it must be quantifiable and specific. Biological models attempt to be general ways to describe many specimens, and therefore must ignore many of the differences between individual specimens. To say a biological feature is explained, the explaination should be parsiminous – explain the most about the feature while making the fewest assumptions. These practices are generally helpful in making progress on biological models while minimizing errors. However there is the side effect of a selection-bias, that could create the illusion that biology is more mathematical than it is.
Biomathematics is a very interesting subject and an active field of research that uses a variety of mathematical tools to illuminate emergent principles in nature. Population dynamics, cellular modeling, epidemiology, and reverse engineering are examples of biomath problems that come to mind.
What is a fractal anyway? A fractal is a mathematical object marked by self-similarity. A part of a fractal can be rescaled, translated, and rotated to approximately fit over itself. Some fractals come from mathematical results discovered to be self-similar, but one can always created a fractal explicitly using an interated function system – a series of geometrically constructions manufactured recursively.
Recursion is an effective strategy whenever the same strategy can be used in the same way on many different scales (almost complete scale invariance). Take, for example, the problem in which a tree must grow leaves in a pattern that maximizes leaves in the sunlight (and not in the shade of other leaves), while still being connected by branches to a common trunk. The same strategy that it needs to implement as a sapling (separating branches from the trunk) is also functional as an adult (dividing branches to maximize leave cover). And indeed, a simple recursive model generates a structure that captures the essence of tree architecture and also exhibits functional features (first avoidly overlap while falling in an upward arc, then as space becomes tighter putting some overlap at the regions with the most sunlight as well as putting leaves in an even wider arc).
As you imagine that the code or procedure for drawing such trees is pretty simple. Branch has width A, branch grows length B, Branch splits into two at angle C, Each split-off branch has width A/D and grows a length B/D, Each split-off branch splits into two at angle C with split-off split-off branches that has width (A/D)/D and grow length (B/D)/D etc etc. The amount of information that has to be evolutionarily developed and genetically transmitted is relatively low, and the divesity of forms that can be achieved with small modifications to the parameters is pretty incredible.
Real trees of course don't follow exact deterministic geometric shapes. In fact the “fractal” programs that draw more realistic trees typically follow functions that use probability distributions instead of fixed values for their parameters (so these models are only self-similar in a statistical sense). Of course, tree's don't measure distance and look up random number tables either. Trees must instead use chemical signals and nutrient levels as proxies for distance. To convey nutrients from the roots or leaves to the rest of the tree necessarily requires that the tree respond and correct the different nutrient needs of different parts, so it's not implausible to think that the trees can implement a simple cyclic pattern of growth and branching. Growing patterns responding to nutrient need can also allow plants to adapt better to their environment and implement subtle branching differences at lower and higher levels. A more complete fractal-based plant simulation program looks more like this.
Thinking about organisms responding to their environment with feedback brings us to control theory . Control theory describes how signals can be made to interact with each other to implent elaborate analytical techniques, use information from a variety of sources, and generate signals to control a system response. Like a fractal composed by a recursive function, feedback loops function by implementing the same piece of “code” again and again, but unlike fractals feedback loops depend directly on the environment. In this way they “measure” and “correct” for the environment, and this is why engineers use them in electronic control devices such a thermostat or a navigation system. Again we have an example in which objects of mathematical function see implementation in the artificial and natural world alike. Why are things done with math? Math works.
I think many of the other pattern in nature are a special case of recursion, in which the same geometric shape is repeated or the similar structure operate at different scales.
Which would be expect from nature, for it to be symmetric or asymmetric? If you're asking me, I'm a physicist so of course I'm going to say symmetry is natural. It is the asymmetries of the nature that need explaining. In the origin of the universe, it was quantum mechanical fluctuations (ie the probability is symmetry but the results are not).
In nonliving things whether or not an object has symmetry has to do whether the forces that have shaped it are symmetric. A pebble in ocean or a ball of hail can become almost spherically symmetric because it's been weathered from every side and it keeps rolling over so no one side is “down” more than any other. Sedimentary rock forms by particles collecting on a flat surface and being compressed from above, therefore we expect the resulting rock to have distinct layers of translationally symmetric planes (only broken by later geological processes). The shape of a rock that breaks off from a cliffside and shatters on impact on the ground may have no symmetry at all, particularly if its amorphous rather than cystalline.
The ways in which biological symmetry is broken depend on the functional constraints of the organism (see Wikipedia for a couple examples of the types of symmetry found in nature). Plants, jellywish, sea anemones, and starfish may have radial or rotational symmetry. Plants may not look very symmetric, but if they are symmetric in a sense if they have equal probability to grow in every direction. The only thing to break the symmetries is the location of the earth and the location of the sun, which is why they are only radially symmetric instead of spherically symmetric. The only change in the symmetry of an sea anemones' environment may be the direction of the surface it clings to and the only direction in a jellfish's environment is which way it moves. The classic starfish has five-fold rotational symmetry because it can travel as easy in any direction between, but needs a finite number of legs with which to push itself.
Most organisms have bilateral symmetry – One direction of symmetry broken by gravity, another direction of symmetry broken by direction of movement, and the last direction of symmetry approximately preserved. Land mammals propell themselves over the ground; birds and fish have to reference gravity which they change altitude (or depth).
And of course there are examples of organisms with no approximate symmetry. Take, for example, the fiddler crab who has a big claw and a small claw. The big claw serves a role in male fiddler crabs for gesturing to females and fighting with other males (over females). In this way the asymmetry serves a role in sexual selection, whereby females use claws as a proxy for the overall genetic health and reliability of a mate. Having this additional function of sexual selection is more important for crabs than having both claws for sifting through the sand, so this system can be maintained. For humans, in contrast, asymmetry can be seen as a sign of genetic disorder or disease and therefore the sexual selection pressure it towards symmetry rather than away from it. Humans evolved with a need for both one-handed and two-handed tool use, so it makes some sense to have one arm slightly stronger and more coordinated than the other but only to a modest extent. Flatfish, such as flounder, also lack symmetry. In this case there is the direction the flounder travels in, the direction of the ocean floor for its ancesters, and the direction of the ocean floor for the modern flatfish. Rather than getting wider and shorter, the evolutionary history of the flatfish had it rotate towards the ocean floor. The asymmetry in flatfish does not appear to harm it, so there is no strong evolutionary pressure to correct the asymmetry generated by its evolutionary history.
So we see that symmetry of living and non-living things is largely shaped by the symmetry of their environment. Ocassionally there is another reason to break symmetry, usually to avoid redundant allocation of biological resources or for certain structural features (ie limbs, branches).
Folie-a-deux is defined by ICD-10 (UN version of DSM) to be quote: “A delusional disorder shared by two or more people with close emotional links. Only one of the people suffers from a genuine psychotic disorder; the delusions are induced in the other(s) and usually disappear when the people are separated”. This definition alone would appear to almost completely rule out telepathy. The two patients do not communicate psychically, rather they communicate by conventional means. The shared delusion stop when the individuals are separated, because the conventional coomunication has stopped and they do not communicate supernaturally.
Some people have tried to set up experiments to look for human telepathy, supernatural communication, or extra-sensory perception. The experiments are difficult to set up, because one has to completely isolate the subjects, use double-blind testing, use correct statistical methodology and prevent experimenters from fabricating results in order to make money off of the “breakthrough”. Every so often someone claims to have found an ESP effect, but it always turns out that there was a methodological problem with their experiment and that their results are not replicable. In addition to this history of experimental failure to demonstrate telepathy, to demonstrate a transmission of delusion would have the additional difficulty that you couldn't experiment on mentally ill patients and you couldn't permit mentally illness to transmit if you thought it might work. So I feel pretty confident in saying that no such experiment for folie-a-deux has been conducted or will be conducted. This leaves us with observational data, which makes conventional information transmission difficult to rule out.
Your question indicates that you think the degree of similarity between the shared delusion to be especially compelling. Probably the simplest thing I could say is that I don't find it compelling. People have been known to act weird for time to time, and even people who act exceptionally weird will never seem so implausible to me that it warrants a supernatural explaination. People have been known to uncritically accept outlandish ideas en masse. People have been known to enter states in which they are especially suggestible and susceptible. People have been known to enter states in which they are particularly prone to certain ideas or narratives. People have been known to break-down dealing with the mental illness of a loved one. People have been known to engage in mimicry and repeat things.
Perhaps there is a particular case of folie-a-deux which you believe makes a case for telepathy, but this skeptic remains unflappable.
(polar coordinates, exponentiation)
(sphere packing, tessalation)
(rotational symmetry, tessalation, no fractals despite the image filename)
(rotational and reflective symmetry)
(natural object for which simple geometry only roughly describes it)
 A course in statistical mechanics sets about to explain this in detail. If you have knowledge of mathematics, physics, or chemistry at about a college sophmore level, a great amount of specifics are more accessible. The subfield of physics know as condensed matter represents the cutting-edge of this subject, examining the properties of especially unusual substances or unusual environmental conditions.
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