Why the Universe is Fractal
Chris King – Mathematics
Department University of Auckland
Abstract: Life exists in the universe
because the laws of nature arising from cosmic symmetry-breaking are
intrinsically fractal and embrace chaotic dynamics in a manner that permits new
structure to emerge on increasing scales, as we move from the level of
fundamental particles to organisms. In a fundamental sense this is a
cosmological property of the universe, because complex living systems represent
the most complete interactive consummation of the four fundamental forces of
nature. If the laws of nature permitted only ordered periodic, or stochastic
solutions to molecular aggregation, as in crystals, or amorphous glasses, the
complex structures of tissues and life would remain impossible.
Fig 1: (a) WMAP survey. The cosmic
background shows evidence consistent with cosmic inflation, (b) fractal
inflation model4 (c) Life on the cosmic equator - despite being
dwarfed by annihilating forces such as black holes, life occurs as a S of complexity on the cosmic equator
in space-time, an interactive peak as significant as the a of big bang or W of ?heat death? or ?big crunch?.
Symmetry-breaking
and Cosmic Inflation
The
source of this complexity also lies in cosmological processes right at the
source of the ?big bang?. The
basis of the cosmic inflation concept is symmetry-breaking, in which the
fundamental forces of nature, which make up the matter and radiation we relate
to in the everyday world gained the very different properties they have today.
There are four quite different forces. The first two are well known -
electromagnetism and gravity - both long-range forces we can witness as we look
out at distant galaxies. The others are two short-range nuclear forces. The colour
force holds together the three quarks in any neutron or proton and indirectly
binds the nucleus together by the strong force, generating the energy of stars
and atom bombs. The weak radioactive force is responsible for balancing the
protons and neutrons in the nucleus by inter-converting the flavours of quarks
and leptons.
Particles come in two types, fermions, of
half-integral spin, which can only clump in complementary pairs in a single
wave function and thus, being incompressible, make up matter, and bosons of
integral spin which can become coherent and can all enter the same wave
function in unlimited numbers, as in a laser, and hence form radiation. As well
as virtual particles appearing and disappearing through quantum uncertainty,
bosons mediate the forces which act between the particles. We thus have another
fundamental complementarity manifesting as the relationship between matter and
radiation. The half integral spin of electrons was first discovered in the
splitting of the spectral lines of electrons in atomic orbitals into pairs
whose spin angular momentum corresponded to ?1/2 rather than the 0, 1, 2 etc.
of atomic s, p - orbitals. As spin states have to differ
by a multiple of Planck 's constant h, a particle of spin s
has 2s+1 components. A glance at the known wave-particles,
indicates that the bosons and fermions we know are very different from one
another in their properties and patterns of arrangement. There is no obvious
way to pair off the known bosons and fermions, however there are reasons why
there may be a hidden underlying symmetry, which pairs each boson with a
fermion of one-half less spin, called super-symmetry, because in
super-symmetric theories the infinities that plague quantum field theories
cancel and vanish, the negative contributions of the fermions exactly balancing
the positive contributions of the bosons. This would mean that there must be
undiscovered particles. For example corresponding to the spin-2 graviton would
be a spin-3/2 gravitino, a spin-1 graviphoton a spin-1/2 gravifermion and a
spin-0 graviscalar.
Fig
2: (a) The wave-particles are divided into two disparate groups of bosons and
fermions. The fermions, which make matter are divided between quarks which
experience all the forces including colour and leptons which experience only
the electro-weak and gravity. The bosons, which mediate the forces have integer
spin and freely superimpose, as in lasers and hence also make radiation.
Half-integer spin fermions only superimpose in pairs of opposite spin and hence
resist compression into one space, thus making solid matter. Each quark comes
in three colours (RGB) and pairs of flavours (up, down etc.) Electromagnetism
is first united with the weak force ostensibly through the spin-0 Higgs boson,
then with the colour force gluons and finally with gravity. (b) The forces
converge at high energies. (c) Force differentiation tree, in which the four
forces differentiate from a single super-force, with gravity displaying a more
fundamental divergence. (d) the scalar Higgs field has lowest energy in the
polarized state, (e) the stable atomic nuclei with their increasing
preponderance of neutrons are equilibrated by the weak force. This force is
chiral, engaging left-handed interactions, for example in neutron decay, as
shown. Weak interactions may explain the chirality of RNA and proteins.
The four fundamental forces appear to converge at very
great energies and to have been in a state of symmetry at the cosmic origin as
a common super-force. A key process mediating the differentiation of the
fundamental forces is cosmic symmetry-breaking. The short-range weak force
behaves in many ways as if it is the same as electromagnetism, except the
charged W+,W- and neutral Z0 carrier particles
corresponding to the electromagnetic photon are very massive. One can consider
this division of a common super-force into distinct complementary forces as a
kind of sexual division, just as the symmetry-breaking division into male and
female is a primary division. In this respect gravity stands apart from the
other three forces, which share a common medium of spin-1 bosons and broke
symmetry first. Originally the particles had zero rest mass like the photon,
but some of the boson force carriers, like the W, changed to mediate a
short-range force by becoming massive and gaining an extra degree of freedom in
their wave function (the freedom to change speed) by picking up an additional
spin-0 particle called a Higgs boson. The elusive Higgs may also explain why
the universe flew apart.
The universe begins at a temperature a little below
the unification temperature - slightly supercooled, possibly even a result of a
quantum fluctuation. In the early symmetric universe, empty space is forced
into a higher-energy arrangement than its temperature can support called the
false vacuum. The result is a tremendous ?negative? energy of the Higgs field,
which behaves as exponential anti-gravity, inflating the universe in 10-35
of a second to something already close to its present size. This inflationary
phase becomes broken once the Higgs field collapses, breaking symmetry to a
lower energy polarized state, rather like a ferro-magnet does, to create the
asymmetric force arrangement we experience, to form the true vacuum. In this
process, the Higgs particles, which are zero spin and have one wave function
component, unite with some of the particles, such as W+/- and Z0
to give them non-zero rest mass by adding their extra component, allowing the
additional longitudinal component of the wave function associated with a
varying velocity. Because the true vacuum is at a lower energy than the false
one it grows to engulf it, releasing the latent heat of this energy difference
as a shower of hot particles, the hot fireball we associate with the big bang.
Gravity has now reversed to become the attractive force we are familiar with.
Two energies, gravitational potential and kinetic, which previously cancelled,
now add. An insignificant universe - almost nothing - becomes one of almost
incalculable proportions. The end result is a universe flying apart at almost
exactly its own escape velocity, whose kinetic energy almost balances the
potential energy of gravitation.
Symmetry-breaking can leave behind defects if the true
vacuum emerges in a series of local bubbles which join. Depending on whether
the symmetries, which are broken are discrete, circular, or spherical,
corresponding anomalies in the form of domain walls, cosmic strings or magnetic
monopoles may form. In addition other weakly-interacting particles may emerge,
such as the axions which some researchers associate with cold dark matter. In
some models, inflation is a fractal branched structure like a snowflake which
is perpetually leaving behind mature universes like ours4. Recently
it has become clearer that, even with additional dark matter, possibly
comprising neutrinos and other exotic particles, there may not be enough mass
to stop the expansion, which may even be accelerating. Various hyperbolic forms
of inflation and an additional repulsion called quintessence involving a
long-range repulsive dark energy have both been invoked to address this
problem.
Life?s Emergence as Symmetry-broken Interaction
Life is the ultimate
cosmological consequence of the four forces of nature acting in hierarchical
sequence, the colour and weak forces binding the quarks to form protons and
neutrons, then atomic nuclei, then atoms, and finally the electromagnetic force
becomes dominant in forming molecules. Complex molecular matter organized in
the fractal form we find in tissues is the ultimate expression of the
interaction of all the forces, which emerged in the symmetry-breaking at the
cosmic origin1.
Fig 3: Interactive
quantum bifurcation and wave nature in chemical emergence (a) Bifurcation
diagram of the periodic table shows how the key bioelements arise from primal
quantum interactions resulting from cosmic symmetry-breaking. (b) Atomic,
hybrid and molecular orbitals are wave functions determining molecular
geometry. Below energy diagram of the non-linear charge interactions causing
molecular orbital formation. (c) The hierarchical interactive structure of a
molecule (LiH) illustrating how the colour, weak and electromagnetic forces
combine to form a complex polarized structure.
It is the twisted nature of cosmic symmetry-breaking, which makes
the makes the combined action of the nuclear and electromagnetic forces capable
of forming around a hundred different types of stable nuclei, through the
mutual interaction of strong force attraction, electromagnetic repulsion of the
protons, mediated by weak force conversion to neutrons. The fact that the
stable nucleons are neutral and positively charged polarizes the entire
electromagnetic make up of atoms. The positive charges of the protons clumped
together in the nucleus give atoms their unique highly polarized structure of
orbital negatively charged electrons. Without this uniquely polarized
situation, which is itself a direct consequence of cosmic symmetry-breaking,
complex molecular life would be impossible.
Moreover the non-linear interaction does not stop at the major
bonding types, for molecules admit a whole cascade of non-linear bonding
interactions from covalent and ionic through the hydrophobic and hydrogen bond
interactions that shape nucleic acid and protein structure, to the long-range
cooperative weak polar and van der Waal interactions that together make the
global cooperativity of enzyme action and cellular organelles, including the
excitable membrane, possible. It
is the unresolved nature of the electromagnetic force in molecular bonding and
the transition from strong bonds to global cooperative weak interactions that
make the fractal dynamics and fractal supramolecular associations of complex
molecules into organelles, cells and tissues possible. Symmetry-breaking has
thus caused molecular matter to adopt a fractal structure, which ultimately
becomes tissues and organisms on the planetary surface held together by the
last force, gravity, energized by the negentropic surfeit of incoming solar
radiation bathing our photosynthesis-based biosphere with its diversity of
plant, animal fungal, protist and prokaryote forms.
Fig 4: Global
structures of t-RNA and protein enzymes are mediated by cooperative long-range
weak bonding,
in association with water structures. The hairpin loops of RNAs
give them a similar tertiary structure to globular enzymes and permit catalytic
activity, as well as the molecule being intrinsically capable of complementary
replication through A-U and G-C H-bonding specificity in the same manner as DNA.
We owe to the unique twisted symmetry-breaking of the forces of
nature the very capacity for molecular life and with it biological complexity
and the tree of evolution to ramify. In effect biological complexity and with
it the conscious brain becomes the ultimate cosmological interactive result of
cosmic symmetry-breaking, the S at the cosmic equator, representing the
fulfillment of both a and W.
Chemistry is often portrayed in terms of ball and stick models, as
if only the particle properties of atoms are of significance in chemical
bonding. Indeed the atomic nature of matter is one of the principle foundations
for a reductionistic explanation of all living processes in terms of the simple
actions of atoms as the ?building blocks of the universe?. However
wave-particle complementarity is at the very core of all chemical interactions.
The periodic table of the elements has periodicity only because the wave
properties of the electronic orbitals give rise to a series of s, p, d, and f orbitals of
increasing spins of 0, 1, 2 etc. Thus the second row of the periodic table,
after H and He, as illustrated in fig 3, consists of the second layer of one 2s and three 2p orbitals.
Because electrons are fermions, they can only enter a given wave orbital in
pairs of opposite spin, these four orbitals allow eight electrons in to
complete the shell, corresponding to the eight elements in the second row,
running through C, N and O - carbon, nitrogen and oxygen. In fact the energies
of these orbitals equilibrate to form hybrid sp orbitals by wave
superposition, resulting in the planar sp2 and
tetradedral sp3 arrangements we find in molecules,
from water to diamonds.
Fig 5: Recombinase involved in genetic
crossing-over. Complex molecules and supra-molecular complexes are possible
because the electromagnetic force involves a cascade of cooperative weak
bonding effects, which result in a fractal structure of the active
conformations and dynamics of large protein and nucleic acid molecules. Ion
channels for example display fractal time-scale dynamics as well as fractal
structure due to primary sequencing, secondary structures such as a-helices and
tertiary conformations.
From here the process is highly non-linear. Electrons are capable
of forming molecular wave orbitals, such as s and p, which orbit
around more than one atom - often two, but sometimes, in the case of conjugated
single and double bonds, a whole molecular ring. Because of non-linear charge
interactions between the electrons and with the nucleus, which, like gravitation,
obey an inverse square law in 3-D space, the lowest energy molecular orbital is
lower in energy than either atomic orbital and the electron enters it, binding
the atoms together into a molecule such as H2. This is the basis of
the covalent bond. The ionic bond arises from a similar lowering of energy by
electron transfer from one atom to another, resulting in net attraction between
the resulting positively and negatively charged ions, such as Na+
and Cl-.
These non-linearities of charge interaction are also manifest in
the ?periodic? table, which is not actually periodic, because charge
interactions on the inner orbitals make the properties of corresponding
elements in successive rows, such as those between O and S or between C and Si,
qualitatively very different. Chemists love to describe chemical bonding as a
simple ball and stick arrangement that, given appropriate energetic reagents in
an artificial ?closed? system, can ?stick? almost any pair of atoms together in
any arrangement we wish. This is central to the mechanistic atomic view of
chemistry and biology. This mechanist picture begins to seriously unravel,
however, when we consider what happens when we ask a very different kind of
question - "What will happen if we simply let the chemical elements go in
the kind of situation we find in the universe at large? - What structures will
emerge in the free interaction of the elements under energetic
stimulation?"
Fig 6: Tissues (right) have a fractal structure as a consequence
of charge non-linearity in chemical bonding: (a) molecular level (b) cellular
organelles (c) organs (skin). This fractal structure is similar to that arising
from the non-linear quadratic dynamics of the Mandelbrot set (left). By
contrast with the classical Mandelbrot set, which arises from a continuous
global iterative map, the genetic process forming tissues is
quantum-interactive and results from non-linearities of orbital charge
interaction. In this respect it resembles the capacity of cellular automata,
which also display fractal and chaotic dynamics, for universal computation6.
We can see a first part of this answer lies in a series of quantum
bifurcations that arise from cosmic symmetry-breaking. The backbone of life
arises from the strongest covalent bonds of all among the elements, -C
N -C
C- and >C=O. Given the ubiquity of H this gives the
interaction of the 1s orbital of H with the 2sp3 hybrid of
carbon, nitrogen and oxygen primary status. Here we are using 'bifurcation' as
a qualitative change in the interactive system caused ultimately by the
underlying variables of cosmic symmetry-breaking. Despite suggested
alternatives such as silicon-based life there is abundant evidence for this
primary interaction being the central 'royal route' to living systems.
Molecules containing chains of conjugated multiple C bonds have been detected
in interstellar space. Clouds of cyanide HCN and formaldehyde HCHO have been
discovered in the Orion nebula where new star and solar system formation is
taking place and huge galactic gas clouds containing molecules such as the
two-carbon sugar glycoaldehyde and the simplest amino acid, glycine. HCN and
HCHO are also key energized intermediates in primal chemical simulations.
A second key interaction arises from the increasing
electronegativity, as we move from C to O. Electronegative oxygen binds its
electrons very tightly because of the larger number of positive protons in its
nucleus for the same electron shell. Oxygen is actually more electronegative
than corrosive chlorine. The C-H bond is covalently neutral while the N-H and
O-H bonds are successively more polarized. Water H2O has the highest melting point of any
hydride because of its very strong polar and ionic interactions. This is why
oily hydrocarbons don't dissolve in highly polar water, effectively separating
the entire biological milieu into two distinct polar and non-polar domains,
typified by the division between the fatty lipid membrane and aqueous
cytoplasm, the 'micelle' or oil-droplet structure of globular protein enzymes
and the stacking of nucleic acids such and RNA and DNA in their double helices.
Water also has one of the highest specific heats of any substance because of
its many internal quantum modes and effectively forms the quantum substrate of
all living molecules. Further complexifying bifurcations of the orbital
interactions involve second row P and S, splitting between the ionic properties
of Na+/K+ and Ca++/Mg++ and involve
orbital properties of transition elements, including Mn, Fe, Zn, Co and others.
Fig 7: Polymerization of
HCN can produce many molecular components of living systems including nucleic
acid bases, amino acids, polypeptides, and porphyrins used in photosynthesis
and respiration.
From here, the 'incompleteness' of the reductionistic description
really begins to bite. A simple molecule like HCN, although it contains a
mammoth triple bond is unstable to self-polymerization, because the triple
bond's p orbitals are at a higher energy than the s orbitals and
opening them up to form only single or alternate (conjugated) double bonds as
in adenine and other molecules illustrated below reduces the energy. We thus
find that HCHO and its sister molecules can polymerize to form a vast array of
sugars and HCN can polymerize, particularly in association with ubiquitous
simple molecules like urea (NH2)2 to form the purine and
pyrimidine nucleic acid bases A, G, C and U, various amino acids and
polypeptides and the porphyrins we associate with chlorophyll and hemoglobin.
The nucleic acid base adenine for example is simply (HCN)5 and
ribose, the backbone of RNA is one of the forms of (HCHO)5.
Although the final transition stages of the origins of life still
remain to be elucidated, it is clear that ribonucleic acid, or RNA, which
unlike DNA can form weak bonds between its backbone and its bases, can form
complex 3-D tertiary structures based on hairpin loops of double helix and is
thus capable of both participating in complementary replication and acting as a
catalyst in a similar manner to the coded protein enzymes of living cells. RNA thus in a single molecule, whose
components are relatively easily generated in free energy polymerizations,
contains the combined capacity for autocatalysis and complementary genetic
replication. Indeed the metabolism of higher eucaryote cells is dependent on
extensive RNA processing in the nucleus, ribonucleotide coenzymes remain a
evolutionary fossils in key metabolic pathways, and the ribosome which is responsible for translating DNA
generated messenger RNAs into proteins consists of a functional core of RNA
molecules.
Fig 8: (a) ATP one of the
four fundamental nucleotide units of RNAs, and the energy currency of the cell
consists of pentamers of HCN and HCHO joined and energized by tri-phosphate.
(b,c) Simple RNAs can produce autocatalytic reactions. (d) A symmetry-breaking
model of the genetic code (King).
Ultimately this fractality and non-linear interaction leads to the
evolutionary tree of life and to the paradigm of natural selection, mutation
and a variety of forms of sexual recombination and the increasing algorithmic
complexity of genetic life as a generator of fractal organismic form and
function and to the ultimate peak of interactive complexity so far discovered
in the universe, the sentient human brain, consisting of some 1011
neurons comprising some 1015 synaptic junctions and hence to the
unsolved problems of subjective consciousness, free will and the possible role
of conscious observer in collapsing the wave function of the universe into the
unique trajectory of history we experience, rather than the overlapping quantum
probabilities of the Schrodinger cat paradox experiment.
Fig 9: Left: Evolutionary tree has a hot origin Centre top: Five layers of neurons in the cerebral cortex. Centre bottom: Pyramidal cell. Neurons use their fractal dendritic structures to make many-to-many functional connections. Right: The author's brain performing a simple cognitive task under functional magnetic resonance imaging.
Fractality and Chaos in the Universe at Large
The inverse square law of gravitation in 3D space leads to chaotic
dynamics as a natural feature of stellar and planetary interactions. The sheer diversity of both the planets
in our solar system, and the moons of Jupiter, which do not have a basis in
differential compositions of the major rocky and gaseous planets with
increasing distance from the Sun, yet nevertheless have very different
chemistries and dynamics, illustrate the sensitivity of planets to a variety of
formative conditions in the early phases of solar system formation.
Fig 10:
Both the major planets and the moons of Jupiter display a high degree of
diversity, illustrating the sensitivity of solar system formation to the
initial conditions leading to such systems.
It is clear from research into chaos in conservative orbits, the
loss of strongly mode-locked asteroid orbits and the Henon-Heiles system, and
recent prediction of possible major instabilities of the orbits of Mercury,
Earth and Mars, leading to a possible planetary collision of Earth within 500
million years5, that the periodicity of planetary orbits is only a
temporary phase in a longer-term chaotic many-body process.
Fig 11:
Hubble's image of Galaxy cluster Abell 1689, situated two thousand million
light-years away, is one of the most massive objects in the Universe.
On a larger scale, solar systems, galaxies and galaxy clusters
form a fractal distribution of matter in the universe at large, also molded by
other forces such as dark matter and dark energy. Calculations of the fractal dimension of such clusters3
suggest these are fractal with a fractal dimension approaching 2.
At the very largest scales there is controversy2 as to
whether the fractal structure gives way to the homogeneity cosmological
theories predict should arise from the isotropic nature of the ?big bang?
perturbed only by possible quantum fluctuations that have become inflated to
the scale of galaxy clusters, as deduced from the maps of the cosmic background
radiation (fig 1a).
Fig 12: Left:
Large scale structure of the universe showing clumping of galaxy clusters,
possibly associated with streams of dark matter. Right: Super-computer
simulation of the same fractal process.
References:
1.
King C C 2002
Biocosmology http://www.math.auckland.ac.nz/~king/Preprints/pdf/biocos1.pdf
2.
Gefter A 2007 Is the Universe a Fractal? New Scientist 9
March.
3.
Montuori M et. al. 1997 Statistical properties of galaxy
cluster distribution Physica A 246 1-17.
4.
Linde A 1998 The Self-reproducing inflationary universe
Scientific American.
5.
Shiga D 2008 Solar system could go haywire before the Sun dies New
Scientist 23 Apr.