To begin with let's look at what is known as the coastline paradox. Briefly it goes like this: If you measure the circumference of Britain with a 1km long stick, and then do the same with a 100m long stick, you will find that with the shorter stick you obtain a larger measurement. This will happen again if you use a 1m stick, the shorter the stick the longer the coastline. The simple reason of course is that the smaller stick can capture more detail.

Next imagine rising above the surface of Britain in a hot air balloon. At the lowest altitude you will see most detail. As you rise your image of the island will have less and less definition. Another thing you notice is that similar physical patterns appear, over and over, at different size scales. This phenomenon can be found all around us, even out into the universe at large and the very fabric of physics.

Now, keeping all that in mind, imagine progressing, from the big bang, through time, and looking at the overall complexity of the universe. {wild speculation follows} As time elapses we find increasing complexity evolving from simpler forms. More variety to its structure, more detail to its content (enter Life). Greater definition to the communications and even thoughts of beings like us.

At this point we should take stock of some core concepts. Firstly "entropy", that which we immediately call to mind when we think of the arrow time. These days Entropy is understood more in terms of energy dispersal, and equilibrium state than the older, order-disorder conception. The minimal composition whereof being that: As time progress (in the universe), less and less energy is available for doing work. This is of course the genesis of the heat death hypothesis for the Universe. What we need to be cognizant of here is: 1) global entropy always increase in a closed system, and we have no reason the classify the Universe as such. 2) Heat death is only one of several theories for the 'end' of the Universe.

Next let's think about 'complexity', it is in fact a distinct concept from order/disorder/chaos, the notable feature of which is that it increases over time. The apparent increase of overall, combined complexity, together with the observed proliferation of fractal structures, would suggest that the Universe has an overall fractal structure. But how could we leverage that?

There are certain metrics that can be employed in respect of fractals: for instance Kolmogorov complexity in regard of information complexity, and Hausdorff dimension relating to the iterative depth. The most promising one being fractal dimension which "...is an index for characterizing fractal patterns or sets by quantifying their complexity."

With the universe continually growing in complexity, the fractal dimension gives us a completely different means to measure Time. The usual way is by the interval of regular events. The new way evaluates the purely physical relation called the Fractal dimension, which gives us a time base fixed to the beginning of the Universe. An Absolute Time frame.

Question: Has this or anything similar been the subject of a serious treatise?

Bonus question: What would it mean if overall complexity starts diminishing?


Edit: Let me reiterate, the question is not about whether the universe has a fundamental fractal structure. The question is: What consequences would follow if the universe is fractal, specifically would it entail a universal time scale, starting from the origin of to universe, and measurable with fractal dimension. And of course, who has said what for or against it.

Why this question? An absolute time frame would have unimaginably far reaching impact on Science. (and with this my penchant for understatement has reach a new... extremity)

However it seems the notion that the universe could be fractal and/or that fractal dimension could be used as a measure of universal complexity, seems to big an obstacle for serious consideration of the question. So in this section I will try to make the premises a bit less ridiculous.

First a note on "complexity", it is different things to different scientists, as such it doesn't have a canonical definition in Science. But here we are talking about the entire universe, every conception of complexity must be taken into account. Here is an example from biology that links fractal dimension with complexity:

Fractal Dimension as a Quantitative Measure of Complexity in Plant Development John D. Corbit and David J. Garbary

Abstract

The shapes of 51 fronds from three species of brown algae (Fucus vesiculosus, Fucus serratus and Ascophyllum nodosum) were evaluated by computing the fractal dimensions (D) of their outlines. There was no difference in fractal dimension among mature fronds of the three species, and D was highly correlated with both developmental stage and structural complexity. With increasing age the plants grew not only larger but also more complex in form. Fractal dimension increased systematically with increasing complexity of shape from about 1 to 1.6. Fractal dimension thus provides a useful quantitative measure for the elaboration of shape complexity during plant development.

Note that complexity, and fractal dimension, increased with age Now recall "growth" is iterative, and the universe is growing.

https://www.thenatureofcities.com/2017/06/25/effect-iteration-urban-form-part/

So called 'fractals' are everywhere, to the extent that the word has been popularized. It now means almost any self similar hierarchical pattern. But they are not only geometric.

Fractal geometry, Turing machines, and divide-and-conquer recurrences [pdf] S. Dube, Informatique théorique et Applications/Theoietical Informaties and Applications

Quoted from: https://cstheory.stackexchange.com/q/16965

These results show that for every Turing machine there exists a fractal set which can be viewed, in a certain sense, as geometrically encoding the complement of the language accepted by the machine. One can build a fractal-based geometrical model of computation which is computationally universal. Secondly we survey the results which show how fractal geometry can be fruitfully used to solve divide-and-conquer recurrences. A recursive algorithm possesses temporal self-similarity and there is a natural connection with spatial self-similar objects (fractal images). This approach yields a new and gênerai way of solving such divide-and-conquer récurrences.

Fractals are in our bones and in our minds. They were expressed in the universe before Live began, are found at every scale and in every dimension...

It is no stretch to say that if a subsystem contains iterative structures its superior system should be the source. That is because in the self similar aspect of a fractal precludes subsystems from having new information. I'm not fan of Occam's razor but:

What is more plausible, a fractal universe or a universe of fractals?

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    @Gordon Every kind of complexity. The apparent increase of overall, combined complexity, together with the observed proliferation of fractal structures, would suggest that the Universe has an overall fractal structure. That would allow a timescale based on fractal dimension. It has some problems that l'll explain with the similar concept "disorder": Let's say the absolute amount of disorder in the universe could be measured, that would give us a time measure since the big bang. But... – christo183 Nov 7 at 14:48
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    Why identify complexity (or fractal dimension) with time? Even assuming that it did monotonically grow, which it does not, see heat death. We already have the concept of complexity and means to measure it, what we want is a concept accounting for experienced duration. So what does this renaming accomplish? At least clock time does grow monotonically. – Conifold Nov 7 at 19:53
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    @Conifold The reason would be that it can provide an absolute, nonlocal time reference. Non-Monotonic asymmetries in either time reference, could quite possibly be undetectable to us. - see replies to Gordon. Also note, while heat death have more acceptance, it is likewise a speculative concept. – christo183 Nov 7 at 20:37
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    I do not understand what "Non-Monotonic asymmetries in either time reference" are. People do talk of thermodynamic arrow of time, which would be the arrow of decreasing "complexity". But it is at best a correlation, it tells us nothing about time itself. – Conifold Nov 7 at 22:13
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    @Conifold A Bang, very shortly after there was no complexity in the universe. 9 billion years later the universe was full of complex systems, including one planet (at the very least) that is today absolutely crawling with some particularly complex chemistry. - A timescale is made by counting seemingly regular periodic events then we assume that scale can be extrapolated indefinitely. 'X-time' simply takes a snapshot of the surroundings and does a calculation on it. - The current Big Whoop is based on the accelerating influence of 'dark energy', we know not whence it came or when it will leave. – christo183 Nov 9 at 8:06

When one looks at a spatial fractal pattern one can zoom in or out and see a similar repeating pattern. Consider what that would have to mean if, instead of a spatial pattern, we were looking at a temporal pattern. Looking at a temporal pattern would mean we are looking at changes.

One change that might come to mind are the seconds changing on a clock, but that is an artificial and uniform pattern. Zooming in or out of that pattern would repeat the artificial uniformity, but not provide the interesting structure seen in a fractal spatial pattern.

To find a more interesting temporal structure would mean that one has identified a more interesting pattern in measurable changes that one sees repeating at different time (rather than space) intervals. One fractal pattern than has been identified is called Elliott Waves. Here is how it is described by the Investopedia Staff for market prices:

Elliott proposed that market cycles resulted from investors' reactions to outside influences, or predominant psychology of the masses at the time. He found that the upward and downward swings of the mass psychology always showed up in the same repetitive patterns, which were then divided further into patterns he termed "waves."

Elliott's theory is somewhat based on the Dow theory in that stock prices move in waves. Because of the "fractal" nature of markets, however, Elliott was able to break down and analyze them in much greater detail. Fractals are mathematical structures, which on an ever-smaller scale infinitely repeat themselves. Elliott discovered stock trading patterns were structured in the same way. He then began to look at how these repeating patterns could be used as predictive indicators of future market moves.

Socionomics carries this fractal perspective on change beyond markets. Here is Robert Prechter relating the Elliott Wave patterns from markets to claiming that the patterns in much of nature are similar.

R.N. Elliott’s discovery of the Wave Principle fifty years ago was a major breakthrough in sociology. His observations reveal that social psychological dynamics create the same pattern of “waves” in aggregate stock price movement from the smallest to the largest degree of trend (see Figure 1). In fact, there is a new science, the science of fractals, indicating that much of nature is made up of the kind of patterns and relationships that Elliott recognized and described.

Alan Hall sees the pattern in various forms of evolutionary change:

The Elliott wave model suggests that growth in the diversity of life on earth has unfolded in a five-wave pattern spanning 600 million years. Socionomist Alan Hall’s 2015 Social Mood Conference presentation reveals the ubiquitous fractal and spiral patterns in nature, solar luminosity, atmospheric evolution, mineral evolution and species extinctions.

Here is the question: Has this or anything similar been the subject of a serious treatise?

The Elliott Wave fractals are taken seriously by market traders. The patterns, although not deterministic, provide suggestions of what one might expect the markets to do in the future at various time-frames, hence, the justification for calling these wave patterns fractal.

For a treatise on this topic see Robert Prechter's The Socionomic Theory of Finance.


Reference

Investopedia Staff, "Introduction to Elliott Wave Theory" Investopedia April 27, 2018 https://www.investopedia.com/articles/technical/111401.asp

"Bye, Bye Birdies" Socionomics Institute https://www.socionomics.net/2017/07/mood-riffs-bye-bye-birdies/

Prechter, R. R. "The Fractal Design of Social Progress" https://www.socionomics.net/2014/11/article-the-fractal-design-of-social-progress/

Prechter, R. R. (2016). The Socionomic Theory of Finance. Socionomics Institute Press.

  • (Un)fortunately, in the markets, whenever a pattern is definitively identified, profiteers will immediately latch onto it and thereby change the system dynamic. But market analysis is still a good example, since from the historical processing power applied to it, we should be able to discern a time-complexity relation. – christo183 Nov 7 at 11:27
  • @christo183 The reason why markets are important is because of the quantity of data. This allows the pattern to be verified at different scales. The pattern is not deterministic, just like the spatial fractal of the British coastline is not perfect like a mathematical chart. Profiteers can take advantage of it, but their behavior is also part of the effects of whatever is causing the pattern (perhaps "social mood"). What is surprising (to me) is that this pattern is not cyclical but spiral. – Frank Hubeny Nov 7 at 14:26
  • I wonder, is the "quantity of data" ever used as a metric? Does it always increase? What would a contraction in data amount, or variance in growth slope, mean for the market? – christo183 Nov 7 at 14:59
  • @christo183 Without the data, the Elliott wave pattern would not be visible. This is a more complicated pattern than the thesis-antithesis-synthesis pattern although it is a similar spiraling pattern. At the moment we may be at the peak of a "grand supercycle third wave", but how would we know for sure without data assuming social mood which creates the pattern is real. Having data does not mean knowledgeable traders will be able to distort the overall market and change the pattern. They may individually be able to get out of the way or take advantage of the trend. – Frank Hubeny Nov 7 at 15:42

There are a number of problems in those premises. What time means. And what an objectively 'fractal' ordering in the universe might mean.

The search for large scale fractal behaviour is called fractal cosmology. The key marker of fractals is having a fractional dimension in the index of change in complexity with scale. Indications are that the universe as a whole does not have a well defined ratio of complexity change with scale.

However, it is interesting to note that black hole surfaces are thought to be fractal through analysis based on the fluid-gravity correspondence. This relates to the holographic principle, in which the organisation in volumes is related to surfaces, and relating higher dimensional objects to projections of them into lower dimensions. I can't find anything reliable about whether that opens up scope for fractal relationships, in particular there might be a fractal index of complexity over the dimensions of string theory, rather than those visible to us. There is also talk about black holes acting as mirrors which is a natural extension of gravitational lensing. I can't say if these things fit together, but it seems tantalising.

What do you mean by time? What you suggest, some measure of complexity, could perform some functions but not others. Cosmologically, you could see it as a kind of clock, the emergence of large scale order from the Planck scale Big Bang. But what after the end of the stelliferous era when only black holes are left? Most crucially, how objectively could you measure it and relate it to other things? The light-clock is fundamental to time in phtsics because it gives fundamental information about what could have impacted what, so on time-ordering and causality. A less complex state emerging can't tell us time has gone backwards. However, time is not thought itself to be fundamental, byt emergent.

Time dissappears in the Wheeler-DeWitt equation, one of our best stop-gaps for combining relativity and quantum ideas. According to the developing field of Entropic Gravityit may be that gravity is an emergent phenomena in a similar way that temperature is. Time's arrow, the subjective sense for us that caysality is absolute, time is irreversible and so on, can be pictured as created by the rise in correlation of quantum states, described in the 'purification principle'. Again, it may be fractal behaviour isn't where you are looking for it, but could still be discovered at some other level or in another way of picturing things.

  • Even if the universe is fractal I don't think a complexity clock would replace a light-clock, given the difference in energy scales observed for their respective 'ticks'. A complexity clock's accuracy and resolution would depend strongly on how much detail of the universe it can observe, and the subsequent processing power involved. Any actual clock I can imagine would have a tick period on the order of years, with insufficient accuracy for geological use. Making all this rather theoretical. – christo183 Nov 9 at 4:07
  • christo183 No doubt. Consider the higher dimensional space implied by this schema universetoday.com/48619/a-universe-of-10-dimensions There might be waves or other patterning in the larger probability space, and our experience of 'outputted' instantaneous-now and causality, just the shuffling of a zoom process into the data, iterating. – CriglCragl Nov 9 at 21:40
  • Another problem with the fractal universe: the fractal structure(s) may not even be accessible to scrutiny. Consider two eldritch iterative processes interacting somehow. It would be highly likely that there would arise an emergent phenomenon, let's call it the dimension of Time. Imagine several such interactions, so as to form all of our familiar dimensions, from emergence. Now Reality is an interaction of emergent, rather than fundamental, 'forces'. Then there can deep hierarchies, simultaneously interacting at many 'Hausdorff dimensions'.- there's your multiverse right there. Also new edits – christo183 Nov 10 at 11:59

I quite agree that the universe clearly has a fractal structure. But your suggestion of measuring time via complexity, while intriguing, seems to presuppose a reliable correlation between time and complexity that I don't see any support for. It's entirely possible for something to move from a state of greater complexity to one of less --consider the highly ordered, fractal shape of a snowflake melting into water. Your time arrow would seem to run locally backwards in the vicinity of the melting snow.

Conversely, if you're looking at total complexity, in the universe as a whole, the law of entropy suggests the arrow would be firmly and unavoidably pointed in the opposite direction to the one you suggest.

If you mean something other than these two possibilities, you'll need to do more work to elucidate what that is.

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    We are looking at universal complexity. Not sure why: " the law of entropy suggests the arrow would be firmly and unavoidably pointed in the opposite direction to the one you suggest." - Also some edits. – christo183 Nov 8 at 14:36

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