ABSTRACT: From subatomic particles to superclusters of galaxies, nature has a nested hierarchical organization. There are also suggestive hints that self-similarity, the idea of similar form on different size scales, might be a fundamental property of the cosmological hierarchy. These features are the hallmarks of fractal structure. Could nature, as a whole, be a fractal system?
By now everyone with an interest in science or computers has stared in wonder at pictorial representations of the amazing fractal called the Mandelbrot set. If you are one of the few remaining holdouts, then do yourself a favor and explore the full color illustrations of this infinite mathematical labyrinth (one readily available source is James Gleick's popular book Chaos1). These images, depicting seemingly endless layers of complexity and geometric inventiveness, strike deep intellectual and emotional chords. A large part of the magic is due to the fact that as you plunge deeper into the Mandelbrot set you encounter ever-more-tiny copies of itself within the riot of detail. This "worlds within worlds" aspect, or self-similarity, calls to mind the famous poem of William Blake:
"To see a World in a Grain of Sand...
And a Heaven in a Wild Flower
Hold Infinity in the Palm of your hand
And Eternity in an hour."
A truly remarkable thing about the Mandelbrot set is that it is not generated by long strings of incomprehensible equations, but rather by a simple recursive algorithm that can be embodied in a few lines of computer code. So the seemingly infinite complexity of the Mandelbrot set has a simple underlying order. Here one is reminded of Coleridge and his "unity in variety." Using this fractal as an archetype, one can say that two hallmarks of fractal systems are: (1) inherent hierarchical organization, and (2) self-similarity, i.e., the copies within copies within...motif.
Another notable example of a fractal structure2 was presented by E.E. Fournier d'Albe in 1907, long before the current wave of interest in self-similarity. On the largest scale, Fournier's fractal is a "snowflake" made up of 5 parts. But each of these parts is a miniature copy of that "snowflake," and these miniature copies are in turn composed of still smaller "snowflakes," and so on in an infinite hierarchical series. Fournier also had his readers imagine the fractal continuing infinitely to ever-larger "snowflakes." This fractal structure is simple (only one shape is involved) and very regular ("snowflakes" at different scales are separated by a constant scale factor of 1/7), but again the mind tends to find its boundless self-similarity intriguing.
Fractals are not limited to the realms of mathematics and computer graphics, but rather abound nearly everywhere you look in nature. Tree branching, cloud structures, galaxy clustering, fern shapes, the veins of human and leaf, music, coastlines, fluid turbulence, crystal growth patterns, and very much more, involve self-similarity. Given that self-similar structure is ubiquitous within segments of the cosmological hierarchy, it is surprising that little attention has been given to the possibility that nature as a whole might be a fractal system. Let us explore this unorthodox but reasonable idea.
The cosmos certainly has the first hallmark of a fractal system: nested
hierarchical organization. From elementary particles to superclusters of galaxies
we see parts collected to form "wholes'" and the latter collected into larger
"wholes," and so on. This hierarchical organization is remarkably stratified,
which means that a few classes of objects populating limited segments of the
hierarchy dominate in terms of mass and numbers. For example, virtually all
observable mass is locked up in atoms (neutral and ionized states). Virtually
all of this mass is, in turn, lumped into stars, and virtually all stars are
collected into galaxies. In between atomic scale objects and stellar scale objects
there are many interesting classes of objects, including ourselves, but they
are quite rare relative to the dominant classes of objects. So, to a first approximation,
the portion of nature that can be directly observed is organized into a discrete
hierarchy: ... which are composed of galaxies, which are composed
of stars, which are composed of atoms, which are ....
The more difficult question is whether nature's hierarchy manifests a physically
meaningful degree of self-similarity. Certainly we would not expect to find
the exact self-similarity seen in Fournier's fractal. But, for all their apparent
differences, might not atomic, stellar and galactic systems be more self-similar
than is currently thought? Einstein once commented: "It has often happened in
physics that an essential advance was achieved by carrying out a consistent
analogy between apparently unrelated phenomena."3 Let
us pursue this slightly unorthodox line of thought and see where it leads us.
We begin by looking for regular scaling relationships that would be indicative
Given the strong stratification of the cosmological hierarchy, one at least has a very rudimentary self-similar organization: galactic "particles" are composed of stellar "particles", which are composed of atomic "particles." However, this essay proposes that there are more meaningful scaling relations among these "particles" that populate such enormously different size scales. Atoms have radii on the order of 10-8 cm while the typical radius for a star would be 5x109 cm. So we adopt 5x1017 as our tentative scaling factor for lengths, and call it K. Since space and time are treated nearly equivalently in relativistic physics, it will be assumed tentatively that K is also the appropriate scale factor for temporal "lengths."
A promising place to begin the search for self-similarity would be with the
well-known analogy between atomic nuclei and neutron stars. In fact the latter
have often been likened to "gigantic atomic nuclei." Atomic nuclei typically
have radii on the order of 10-12 cm and multiplying that value by
K gives an estimate on the order of 5 km for neutron star radii. Since
neutron stars are believed to have radii of roughly 10 km, the first test of
our approximate scaling relation is encouraging. Classical spin periods for
atomic nuclei are on the order of 10-19 sec to 10-20 sec.
Scaling these values by K gives expected spin periods for neutron stars
on the order of 10-2 sec to 10-1 sec, which are nicely
within the observed range of typical spin periods for neutron stars: 10-3
sec to 100 sec. Atomic nuclei also undergo vibrational oscillations
with periods of roughly 10-22 sec to 10-21 sec. Do neutron
stars oscillate with periods that are about K times longer? In fact,
observed neutron star oscillations are in the range of 10-4 sec to
10-3 sec, which is in good agreement with our expectations.4
So for the case of atomic nuclei and neutron stars we find a fairly reasonable
degree of self-similarity between the proposed analogues. Both involve extremely
dense and rapidly spinning objects whose radii, rotational periods and oscillation
periods scale by a factor of K, as expected.
An important question is whether or not the scale factor K will continue to hold good for extensions to the galactic scale. Galaxies have "peculiar velocities" (random motions in addition to the general Hubble expansion) that average roughly 400 km/sec. Since the galactic "particles" are undergoing such enormously energetic random motions, the only reasonable atomic scale analogues would be fully ionized atomic particles, i.e., nuclei or electrons. Since galaxies are clearly extended objects with considerable substructure, electrons are ruled out as potential analogues. If we scale our nuclear radius (about 10-12 cm) by K2 (since we are "going up" two scales), we get a predicted radius on the order of 1023 cm for a typical galaxy. Again this is correct; galactic radii typically are on the order of 1022 cm to 1023 cm. Moreover, if we scale typical nuclear spin periods by K2, we get the correct order of magnitude for galactic spin periods: on the order of 108 years.
Bearing in mind that self-similarity need not be exact, and that analogues
from different scales are not expected to be exact copies, let us return
to the comparison between atoms and stars. Atomic radii range from about 10-9
cm for the ground state of small atoms to 10-4 cm for very highly
excited Rydberg atoms. Scaling by K, we would expect most stars to have
radii ranging roughly from 109 cm to 1014 cm. This is
in fact the case; dwarf stars typically have radii on the order of 109
cm and supergiant stars have radii of up to 1014 cm. Excited atoms
can act like tiny oscillators and a typical range for the periods of their oscillations
would be 10-16 sec to 10-10 sec. Do stars undergo colossal
oscillations with periods K times longer? Remarkably enough, yes; many
stars go through phases in which they become variable stars. Such stars are
characterized by regular pulsations with periods ranging from minutes to years,
in good agreement with the anticipated self-similar scaling. And even "constant"
stars like the Sun tend to have regular oscillations in this range, though of
Many who hear the foregoing evidence for cosmological self-similarity initially underestimate the restrictiveness of the model. They feel that one could choose other analogue pairs and/or different scaling relations and come up with an equally defensible model. If you think that this is possible, then I urge you to get out paper and pencil and actually try this experiment. Having considerable experience in exploring these matters, I think that you will be surprised at the uniqueness of the analogue and scaling results discussed above. In other models, the scaling relations get inadmissibly complicated and/or the derived analogue pairs do not match morphologically or kinematically. But do not take my word for it; try it yourself.
The major components constituting 99.99% of the observable universe:
nuclei, atoms, compact stars, main sequence stars and galaxies, have several
basic properties that are consistent with the hypothesis of discrete self-similar
scaling characterized by the scaling factor K. Are there further hints
that this apparent self-similarity might be a fundamental, but hitherto unappreciated,
symmetry principle of nature? Indeed there are a few more. A two-part paper
by the present author offers a more detailed review of the evidence for discrete
cosmological self-similarity.4 Therein it is shown
that the magnetic dipole moments of atomic nuclei and neutron stars are related
by the proposed self-similar scaling rules. In the same category, there is a
correlation between the angular momentum (j) of an atom and its magnetic dipole
moment (u). Stars have a corresponding correlation between their J and U values,
and the proportionality constant (d) that relates U/J and u/j is in agreement
with the value predicted by the fractal scaling rules. In a related example,
the angular momenta of stars (J) and atoms (j) are both related to their respective
mass values (M, m). Again the proportionality constant (k) that relates J/M2
and j/m2 is in agreement with the value predicted by the self-similar
scaling. Other supporting, though more circumstantial, pieces of evidence include
the well-known analogy between stellar scale and galactic scale jet phenomena,
and the intriguing rotational spin-up (or "glitch") phenomena that is seen in
both atomic nuclei and neutron stars. The author's review paper4
presents a total of 19 quantitative examples of self-similar scaling.
When astronomers recently discovered that the pulsar PSR 1257+12 was orbited
by two planets, they were quite surprised since such a system had previously
been considered a virtual impossibility. On the other hand, pulsar-planet systems
were anticipated by the self-similar cosmological model several years before
Although the idea of comparing atoms and stellar systems is repugnant
to many, probably owing to the demise of Bohr's first planetary model of the
low energy state atom (for which he won the Nobel prize), consider the
following description of a very high energy state Rydberg atom: "an ion
core and an isolated electron very far away, floating lazily around in a slow
orbit, much like a distant planet of the Solar System."6
The atomic nucleus/galaxy analogy finds further support in two recent findings:
that galaxies appear to have "quantized" redshifts,7
and that the statistics of galaxy clustering bear a notable resemblance to results
derived for subatomic particles.8 It can also be
noted that galaxies and atomic nuclei share the same basic spherical, oblate
and prolate shapes. Prolate shapes, discovered in nuclei and in galaxies only
within recent decades, came as twin surprises to physicists.
Bearing in mind that physically meaningful self-similarity does not require analogues on different scales to be exact replicas, but only to be similar in shape and motion, and related by consistent scaling relations, the case for discrete cosmological self-similarity looks reasonably promising.
THE CRUCIAL EXPERIMENT
I could understand a skeptic saying: "Well, these correlations are
provocative, but could it not be attributed to a combination of coincidence
and fortuitous choices of analogues?" I would respond that the analogue choices
are not arbitrary; they are self-consistent and constrained in shape, size and
motion. It also seems rather unlikely that we keep getting correct results by
chance, especially when we are jumping around by such outrageously large scale
factors. I think that it is more reasonable, and more scientific, to suspect
that we are seeing evidence for some form of cosmological self-similarity. What
we really need, however, is a dramatic and unambiguous test of the idea
that the cosmos has a discrete fractal organization. Happily such a test has
been identified, the initial experimental results are in, and definitive results
will soon be available.
The atomic scale is mostly populated by unbound protons and electrons; they
are at least 10 times more abundant than other atomic systems. If the cosmological
self-similarity hypothesis is valid, then the stellar scale would be similarly
dominated by two analogous classes of objects. The author's more detailed review
paper4 predicts that these analogues must be ultra-compact objects
(black holes) with masses of about 7x10-5 solar masses for the stellar
scale electron analogues and about 0.145 solar masses for the stellar scale
proton analogues. Galaxies should be teeming with these objects, but their "blackness"
would make them very difficult to observe directly. On the observational front,
it is known with a high degree of confidence that some mysterious population
of "dark matter" objects does indeed dominate the masses of galaxies by at least
a factor of 10. So the hypothesis of discrete cosmological self-similarity leads
naturally and unequivocally to the prediction that the observed, but enigmatic,
dark matter is primarily composed of our predicted proton and electron analogues.
In the summer of 1992 a very important experiment, conducted by a collaboration
of American and Australian scientists, began to search for stellar scale dark
matter objects.9 This research group, referred to
as the MACHO (MAssive Compact Halo Object) collaboration, is using an Australian
telescope to monitor millions of stars in the Magellenic Cloud galaxies in the
hope of detecting gravitational microlensing events caused by stellar scale
dark matter objects in the halo and bulge of our galaxy. If the mystery objects
have masses in the range of 10-6 solar masses to 102 solar
masses, then they will cause distinctive brightening of distant stars via gravitational
microlensing. The numbers, durations and amplitudes of the events can then be
used to determine the mass spectrum of the dark matter objects. Two other teams,
the European EROS collaboration and the American-Polish OGLE collaboration,
are currently conducting similar experiments and therefore any results can be
independently cross-checked. At least two other groups (the DUO and AGAPE collaborations)
are preparing for additional microlensing experiments.
The first two reported microlensing events came from a slightly different
type of experiment in which distant quasars are microlensed by objects
in intervening galaxies. Two research groups conducting these experiments have
identified microlensing events, and estimated lens masses of about 5.5x10-5
solar masses in one case10 and about 0.1 solar
masses in the other case.11 These estimates have
large error bars, but it is remarkable how close those first tentative results
are to our predicted masses of 7x10-5 solar masses and 0.145 solar
Then the MACHO collaboration recorded a major event with the classic signatures
of microlensing.12 The source star in the LMC brightened
by 2 magnitudes, about a factor of 7, and then faded to its normal luminosity
in a time symmetric and achromatic manner. The MACHO group derived a best fit
lens mass of 0.12 solar masses, which is statistically indistinguishable from
our prediction of 0.145 solar masses. The EROS group reported two further microlensing
events and, when a method of mass moments analysis was applied to the MACHO
and EROS data, a mass peak at 0.144 solar masses was derived.13
As of this writing more than 100 candidate events, mostly involving microlensing
of sources in our galaxy's bulge by intervening objects, have been reported.
The general consensus is that the main mass peak for the halo MACHOs
recorded so far is on the order of 0.1 solar masses, although we are still dealing
with low number statistics. At this writing no additional planet-sized microlenses
have been identified, but when the predicted abundance of these objects is considered,
none would have been expected yet. Given a sufficiently long observation program,
planetary-mass MACHOs should be seen and the full dark matter mass spectrum
should be revealed. The definitive predictions of main peaks at 7x10-5
solar masses and 0.145 solar masses, singled out from a mass range that extends
over 70 orders of magnitude (from 10-64 solar masses
to 107 solar masses), will have been unambiguously verified or falsified.
One other prediction that appears to have been verified is the strict requirement that just below the 0.15 solar mass peak there must be a sharp cutoff in the number of stellar scale objects, just as the atomic scale mass spectrum has a sharp cutoff just below the proton mass. New observational results, much to the surprise of astrophysicists, appear to have confirmed this unique cutoff at the predicted value.15,16 Particularly definitive is the low-mass cutoff for microlensing events (now more than 100) observed in the galactic bulge.17
POSSIBLE IMPLICATIONS OF COSMOLOGICAL SELF-SIMILARITY
If the dark matter is composed of ultra-compact stellar scale objects
with a mass spectrum that is approximated by predictions of the self-similar
hypothesis, then it would appear that discrete self-similarity is a newly identified
global property of nature. This would certainly change our current understanding
of the cosmos. Firstly it would provide a new approach toward a more unified
understanding of nature, since cosmological self-similarity implies analogous
physics on all observable scales. It would also imply that the usual assumption
that the universal hierarchy has cutoffs at about our current observational
limits, an assumption that has always seemed suspiciously anthropocentric,
should be questioned. If cosmological self-similarity is verified, then it would
seem more likely that additional scales underlie the atomic scale and encompass
the galactic scale. According to the new paradigm the Big Bang does not involve
the expansion of the entire universe, but rather just one metagalactic
object with dimensions far exceeding our current observational limits. Also,
a new fractal geometry of space-time-matter would appear to be called for.
The self-similar cosmological paradigm does not throw out well-tested fundamental
physics such as quantum mechanics, general relativity or even the Big Bang approximation.
Instead it views most of the current theories of atomic physics, stellar astronomy
and cosmology as valuable approximations that can now be reinterpreted, refined
and incorporated within the broader and more unified context of an unbounded
If microlensing experiments verify the unique predictions mentioned above, however, we would still be faced with some important and very difficult questions. How many scales are there in all, a finite number or "worlds within worlds" without end? How strong is the degree of self-similarity between analogues? Why is nature self-similar, and why are scales separated by a factor of about 5x1017? Like past discoveries, this one too would come wrapped in enigmas.
Some might argue that the self-similar cosmological paradigm is too fantastic to be true, that it is too speculative to deserve serious attention. But is it more fantastic or speculative than Alice In Wonderland theories like cosmic strings, shadow matter, Higgs bosons, the "many worlds" inter-pretation of quantum mechanics, etc. Probably not, if judged objectively, and at least the self-similar model can make definitive predictions and point to actual observational support. It is possible that nature really does involve the "worlds within worlds" structure of a fractal system. Certainly there is enough supporting evidence to warrant serious consideration of discrete cosmological self-similarity. And soon, via microlensing experiments, we will learn nature's own verdict on this hypothesis.
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