Newcomb/Benford Law

[rloldershaw]

Thinking some more about the significance of Benford's Law, which was actually discovered by Newcomb in 1881 but ignored for 57 years, I think we can indeed make a deductive/empirical argument that we definitely live in a fractal world.

Many of those who have studied the Newcomb/Benford Law have been left with the feeling that the law holds some universal and very fundamental revelation about nature that has not been fully understood. For example, R. A. Raimi stated: "What remains tantalizing is the notion that there is still some unexplained measure in the universe ..".

Here is a simple argument that offers the explanation that Newcomb, Benford, Hill, Raimi, and many others have sought.

If nature's geometry is dominated by fractal geometry, i.e., if we live in a fractal cosmos, then we would expect to find our world dominated by the logarithmic distributions that follow the Newcomb/Benford Law. Since that is exactly the surprising discovery that Newcomb stumbled upon, and Benford verified empirically, and mathematicians like Hill and Raimi have refined, the conclusion seems straightforward and unavoidable. We do in fact live in a fractal universe.

Of course the next question would be: Why is it a fractal world? My answer to that next question would be that the natural extension of Einstein's relativity, which demonstrated the relativity of space, time, orientation, and states of motion, would be the extension to relativity of scale, which would require that the world must be fractal.

Here is a partial and very brief list of the natural distributions that obey the Newcomb/Benford Law.
surface areas of rivers
molecular weights
atomic element/isotope masses
E1 atomic transition lines in plasmas
universal physical constants
populations of 3,000 countries
surface areas of countries
full widths (lifetimes) of mesons and baryons
fibonacci sequence
half-lives of radioactive nuclei
exoplanet masses, radii, volumes, orbital periods,...
distances to galaxies
distances to stars in our galaxy
death rates
blackbody radiation
prime numbers
river lengths

In artificial systems: sizes of stored computer files, internet connections, sizes of bank accounts, and a vast number of additional things.

In fact it is exceedingly easier to list the things that do not obey the Newcomb/Benford Law than those that do. Examples of the former are square root tables, specific heat tables (restricted distribution), values of 1/n, digits of pi (I think). The law is not obeyed in cases of true randomness and narrowly restricted distributions.

Nature has been trying to tell us something fundamental about the cosmos. Are we ready to listen yet?

Robert L. Oldershaw
http://www3.amherst.edu/~rloldershaw

[Sockratease]

If nature's geometry is dominated by fractal geometry, i.e., if we live in a fractal cosmos, then we would expect to find our world dominated by the logarithmic distributions that follow the Newcomb/Benford Law. Since that is exactly the surprising discovery that Newcomb stumbled upon, and Benford verified empirically, and mathematicians like Hill and Raimi have refined, the conclusion seems straightforward and unavoidable. We do in fact live in a fractal universe.

Sorry, but I follow neither the logic nor the argument here.

First, what is this law of which you speak?  You give examples of things that follow it, but don't explain what it is that they are following.

And, just because we would expect to find such things were that law true, finding such things is not proof that the law is in effect since many other possible causes exist, and given that you list examples of things that do not follow this mysterious law - that implies that it is not Universal, and therefore calls into question whether it can be applied to the entire Universe.

Let's back up a bit.  Can you summarize what exactly this law is saying without resorting to external links?

As for Relativity Of Scale, that needs a lot of explaining too.  As I see it, scale invariance is more the norm.  No matter what scale you use as a frame of reference, relative proportions should remain unchanged.  Whereas with Relativity Theory, measurements of time and mass actually do change depending upon gravity and acceleration.  This has been proven and in fact your gps devices would fail if the effects of Relativity were not corrected for in the calculations.

What measurements would change with a change of scale?

[rloldershaw]

The law is commonly known as Benford's Law and it is well-known to mathematicians, scientists and natural philosophers.

If you do a Google search on Benford's Law, you will find a very large amount of reference material.

Wikipedia has a decent introduction. If you do searches on Benford's Law + Raimi, and a search on Benford's Law + Hill, you will be led to some of the best and most insightful research on what I call the Newcomb/Benford Law, so as to give proper credit to Newcomb.

As for relativity of scale, here is an arxiv.org paper that will get you started: https://arxiv.org/ftp/physics/papers/0701/0701132.pdf

One only gets somewhere in science by being willing to put in some serious time and effort.

Thanks for your comment.  RLO

[Sockratease]

The law is commonly known as Benford's Law and it is well-known to mathematicians, scientists and natural philosophers.

If you do a Google search on Benford's Law, you will find a very large amount of reference material.

Wikipedia has a decent introduction. If you do searches on Benford's Law + Raimi, and a search on Benford's Law + Hill, you will be led to some of the best and most insightful research on what I call the Newcomb/Benford Law, so as to give proper credit to Newcomb.

As for relativity of scale, here is an arxiv.org paper that will get you started: https://arxiv.org/ftp/physics/papers/0701/0701132.pdf

One only gets somewhere in science by being willing to put in some serious time and effort.

Thanks for your comment.  RLO

I happen to be a scientist with 19 years of research in my field, and I got a side degree in Philosophy, so I think I am qualified to say I know that if you can't summarize something you wish to discuss without external links that you may not know the material that well to start with.

If you want someone to take the time and put in the effort to research something, you have to give them a reason.

So far, nothing you've said motivates me to make the effort.  I have enough real science to do without adding on something I suspect is just pseudo science - but if you can actually discuss the subject rather than propose that I have not gotten anywhere in science because I don't care to research something I don't know anything about in a subject I find highly dubious - then that's fine with me.

If you don't know from my previous posts over the years - I do not believe fractals are anything more than a mathematical construct.  They have no corollary in reality.  Sure, many things use fractal geometry  and many things use iterative processes, but that does not make them fractals.

Fractal is a term without even a firm definition, so to say the universe is one counts as extraordinary.  As you know, extraordinary claims require extraordinary proof.

But honestly, while I find this subject interesting, you will need to do a lot more if you want to get a discussion going.  Your reluctance to summarize the concept does more to make me want to ignore the topic than research it.

Feel free to make that summary of this law if you like.  Should I find it interesting, I will surely read more.  But assuming I have heard of it is a bit much.  I am a Chemist working mostly in Corrosion Inhibition - I also am decades out of school so if I ever did hear of it, I guess my recall is less than perfect  (like everyone's).

Thanks for the effort, but I'll pass on the research until I know a little more background.

[rloldershaw]

Everyone is entitled to their opinion(s).

Nature alone is qualified to judge whose opinion(s) is(are) closer to reality.

That is why the predictions/testing parts of the scientific method are so important and why other criteria fall short of what is needed to successfully judge our models and understanding of nature.

[Tglad]

I think there is a perspective in which the universe is indeed fractal, and that is that most physics equations are conformal, meaning they are independent of scale. And therefore any object that is built up through a simple physical process like this will show fractal geometry.

Examples might be the distribution of sizes of asteroids in the asteroid belt or the distribution of craters on the moon by size. However more complex physics often breaks the scale symmetry (as it does for other symmetries).

The other reason one might say the universe is fractal is simply that fractal geometry generalises Euclidean geometry, so any random object is more likely to fit to a fractal model (over a scale and coordinate range) than to a Euclidean model of that range.

"the natural extension of Einstein's relativity, which demonstrated the relativity of space, time, orientation, and states of motion, would be the extension to relativity of scale, which would require that the world must be fractal"

Actually Einstein's relativity is already conformal, and therefore scale symmetric. That's why people like Roger Penrose experiment with conformal transformations of the universe to produce cyclic universes (see Cycles of Time). In fact most of the standard model is conformal, just the constants and certain aspects of some of the more complex parts are non-conformal.

This doesn't mean that the universe has to look like a fractal, conformal laws of physics can, and do produce emergent objects that don't appear fractal. A simple example is that gravity produces planets that are roughly spherical, which is closer to Euclidean geometry.

Lastly, one needn't over-stress scale symmetry and fractal geometry, there are other underlying symmetries that make the universe what it is, including Lorentz invariance, and the position and time symmetries of the Lagrangian of the physical laws which produce conservation of energy and conservation of momentum.

[rloldershaw]

We are in general agreement. For a long time I have argued that conformal geometry (continuous and broken) and fractal geometry (continuous and discrete) appear to have very much in common, and may be formally equivalent.

General Relativity has conformal properties for vacuum solutions (excluding matter) but when you include matter in the right hand side, then I think the conformal symmetries are thought to be lost. Various physicists have attempted to modify GR so as to recover full conformal invariance, but I do not think it is generally agreed that anyone has succeeded yet. I have tried a quite different (and very simple) approach and it can be seen in the following brief published paper: https://arxiv.org/ftp/physics/papers/0701/0701132.pdf .

While it is true that planets are roughly spherical, when you zoom in for a more accurate observation of the geometry, it is highly fractal down to the smallest measurable scales.

Physics often advances by discovering new symmetries in nature. I think the next major paradigm change will involve figuring out how nature manifests conformal/fractal geometries and conformal/fractal symmetries. I also think that broken/discrete symmetry will be a fundamental part of that advance.

Thanks for your excellent comment!

Robert L. Oldershaw
http://www3.amherst.edu/~rloldershaw

[Tglad]

Well if you scale distances by n and scale times by n then you get the same physics but with masses scaled by n. In other words, you still get a physically valid universe. So it is conformal if you scale mass too. Penrose applies conformal transforms on a universe with mass in it, and it is useful when looking at the evolution of black holes.

There is however a paper that I read once that tried to apply GR to a non-smooth manifold... i.e. a space-time that is fractal at small scales... something that is somewhat in line with the idea of the small scale quantum foam I suppose.

[Chillheimer]

this is becoming an interesting discussion.

However more complex physics often breaks the scale symmetry (as it does for other symmetries).

that's true - but not a showstopper.
Let's me use the mandelbrot-set as model to explain what I mean:
the deeper you zoom, the more complex your shapes become. and at some points they are shapes never before seen on your previous zoom path
like this http://www.fractalforums.com/index.php?action=gallery;sa=view;id=17497
of course you can look at this image (and otheres of similar complexity) and try to work out the rules in them, e.g. the angle of spiraling or density of patterns..
but the image still is based on nothing more then z->z²+c

so what I'm trying to say is, that complex physics is a phenomenon emergent from 'simple' recursive physics. And though the patterns might not be appear fractal at first sight, they still can be.
(I hope this was understandable, too little time for longer explanation, gotta finally start working here

[Tglad]

Yes, a great example is the eggs in the positive mandelbox, they contain these flat ribbons like they are prize winning eggs, but if you zoom in a long long way they too will become rough and fractal.

There is a hypothesis called the invariant set hypothesis that says that our universe is the one who's laws of physics remain invariant over eons (over thousands of big bags and crunches). In that sense we live in an invariant set of states (positions of all particles etc) just like the points on the border of a Mandelbrot set. It is worth reading if you can find it.

[rloldershaw]

More good and very useful comments - thanks!

The fractal invariant set theory is proposed by Prof. Tim Palmer and his main published paper on this subject can be obtained from arxiv.org.

When Einstein was working towards a unified theory that would further generalize General Relativity, he noted in his last publication (sent to an Italian Conference, 50 Years of Relativity, 1955) that his new equations seemed to have intrinsic self-similarity and scale invariance. However he was perplexed by the fact that atoms are observed to have definite lengths and masses, i.e., they did not seem to obey scale-free physics like globally conformal, i.e., scale-free models.

Discrete Scale Relativity [ https://arxiv.org/ftp/physics/papers/0701/0701132.pdf ] attempts to show how one can reconcile Einstein's insight on the scale-free nature of the equations with the physical observation of "absolute" atomic length scales. There is a way to have the cosmos we observe and also to have discrete, or broken, conformal invariance.

While lengths and times scale the same way and give scale-free velocities and the same c for all scales, as required by relativity, masses must scale differently and seem to require a non-integer exponent in the scale factor for masses. Hence, the physical matter in the cosmos has discrete fractal self-similarity.

RLO
http://www3.amherst.edu/~rloldershaw