2) Watching the Daisies Grow

3) Golden Music of the Brain

4) Golden Cycles & Organic Spacetime

5) Golden Geometry of

6)

7) Water Structured in the Golden Ratio

**Science of the Organism**

*The golden ratio orchestrates all of nature’s cycles to create organic spacetime. Dr. Mae-Wan Ho*

*Circle map Wikimedia*

Nature is replete with cycles and oscillations, from subatomic vibrations to planetary motion, solar cycles and beyond. Some say even the Universe cycles through deaths and rebirths. Not just Buddhists, but distinguished British theoretical physicist Roger Penrose at Oxford University and his co-author Armenian cosmologist Vahe Gurzadyan at Yerevan Physics Institute have drawn the same conclusion, based on data collected by NASA’s WMAP (Wilkinson Microwave Anisotropy Probe) satellite [1].

Gurzadyan analysed seven years of microwave data from WMAP as well as data from the BOOMERanG balloon experiment in Antarctica, and found concentric circles in the microwave background in which the range of the radiation’s temperature is markedly smaller than elsewhere. Penrose and Gurzadyan said these circles enable one to “see through” the Big Bang into the previous aeon; they are the signature left behind by the spherical ripples of gravitational waves generated when black holes collided.

These ripples or gravitational waves have been confirmed 4 years later by the BICEPS (The Background Imaging of Cosmic Extragalactic Polarization 2) experiment at the South Pole, and announced to the press with great fanfare, promising the world “a whole new era” in physics [2].

Cycles may well be involved in organizing
the entire Universe; but it is in living systems that their pivotal role is
most clearly defined [3] The
Rainbow and the Worm, The Physics of Organisms (ISIS publication). Cyclic
activities create a nested fractal hierarchy of organic space times that simultaneously
maximizes global cohesion *and* local freedom, thereby enabling organisms
to transfer and transform energy most rapidly and efficiently, as and when
required.

In [4] (Golden Music of the Brain - Story of
Phi Part 3, *SiS *62), we saw that the golden ratio is
prominent in the ‘resting’ rhythms of the brain, thereby ensuring the highest
degree of *a*synchrony for diverse activities (freedom from interference),
providing the maximum possibility for spontaneous diverse interactions between
rhythms and rapid transition from resting to active state. This suggests that
the golden ratio is involved in organizing cyclic activities for maximum local
freedom and global cohesion, as in ‘quantum jazz’ [3].

In this article, we delve more deeply into cyclic activities and how the golden ratio works its magic for organic spacetime.

Cycles are intimately tied to the study of dynamical systems, beginning with celestial mechanics. Planets move around the sun in yearly cycles; and the great English physicist Isaac Newton (1642-1727) tried to describe the planetary cycles in terms of his laws of motion more than 300 years ago.

Newton wrote down differential equations for dynamical systems consisting of massive bodies interacting through gravitational forces. If there are only two bodies, these equations can be explicitly solved and one finds that the bodies revolve on ellipses around their centre of mass. If there is a third body – the ‘three-body problem’ – no exact solution exists even if, as in the solar system, two of the bodies are much lighter than the third [5]. So, why is the solar system stable? Or is it? We shall come back to this pressing question later.

Dynamical systems of planets can
be treated mathematically as oscillators. A harmonic oscillator has a certain
natural frequency. When subjected to an external force of the same frequency,
resonance occurs and the motion of the oscillator becomes unbounded or
unlimited. For a typical nonlinear oscillator, whenever the perturbing force
has a frequency that is a *rational *multiple of the natural frequency of
the oscillator, i.e., in whole number or fractions of whole numbers (such as ½,
2/3, 5/8, etc.) resonance will occur. Similarly, in the motion of planets
around the sun, one planet exerts a periodic force on the motion or a second,
and if the orbital periods of the two are rational multiples, this can lead to
resonance and instability.

In 1954, Soviet mathematicians Andrey Kolmogorov (1903-1987) suggested a way of solving the problem. First, linearize the problem about an approximate solution and solve the linearized problem, then, inductively improve the approximate solutions by using the solution of the linearized problem as the basis of a Newton’s method argument. (A linear problem is one where a small change in one variable produces a correspondingly small change in another continuously.) These ideas were fleshed out over the next decade by Russian mathematician Vladimir Arnold (1937-2010) and German-American mathematician Jürgen Moser (1928-1999), resulting in the Kolmogorov Arnold and Moser (KAM) theorem. The KAM theorem is very important for understanding how cyclic activities interact with one another.

The three mathematicians were
investigating the behaviour of integrable (solvable) *Hamiltonian *systems
defined by momentum (mass x velocity) and position. The trajectories (paths of
motion) of Hamiltonian systems in phase space - mathematical space representing
all possible states - are confined to a doughnut-shaped surface, an invariant
torus. Different initial conditions will trace different invariant concentric
tori in phase space, separated by unstable chaotic regions, where the motion is
irregular and unpredictable.

The KAM theorem states that if
the system is subjected to a weak nonlinear perturbation, some of the invariant
tori are deformed and survive, while others are destroyed [6]. The ones that
survive are those that have “sufficiently irrational frequencies” (the
non-resonance condition, so they do not interfere with one another). The golden
ratio being the most irrational number is often evident in such systems of
oscillators. It is also physically significant in that circles with golden mean
frequencies are the last to break up in a perturbed dynamical system, so the
motion continues to be *quasi-periodic*, i.e., recurrent but not strictly
periodic or predictable.

An important consequence of the
KAM theorem is that for a large set of initial conditions, the motion remains
perpetually quasi-periodic, and hence *stable*. KAM theory has been
extended to non-Hamiltonian systems and to systems with fast and slow
frequencies.

The KAM theorem become increasingly
difficult to satisfy for complex systems with more degrees of freedom; as the
number of dimensions of the system increases, the volume occupied by the tori *decreases*.
Those KAM tori that are not destroyed by perturbation become invariant Cantor
sets, or* Cantori*; and the frequencies of the invariant Cantori
approximate the golden ratio.

The golden ratio effectively enables multiple oscillators within a complex system to co-exist without blowing up the system. But it also leaves the oscillators within the system free to interact globally (by resonance), as observed in the coherence potentials that turn up frequently when the brain is processing information (see [4] ). To get a better picture, we look at the circle map.

Cycles are often represented by the circle map, a graph that maps the circle onto itself. The simplest form is [7, 8]:

*q _{n}*

Where the variable *q _{n}*

The most studied circle map
involves a ratio of basic frequencies *w* = *f* = (√5 – 1)/2, the ‘golden mean critical point’ at *K*
=1 and *W* = *W*_{c }= 0.60666106347011 (≈
*f* = 0.618033989…), reported in many experiments in which
universal numbers associated with the golden ratio were observed and
documented.

Circle maps contain some key features.

*Arnold tongues*

Arnold tongues, named after Arnold of the KAM theorem, are
regions in the phase space of circle maps with locally constant rational
rotation (winding) numbers between the driver and the natural oscillator frequencies,
*p*/*q*. They were first investigated for a family of dynamical
systems defined by Kolmogorov, who proposed this family as a simplified model
for driven mechanical rotors described by equation (1). The circle map of the
equation exhibits certain regions in the parameters space when it is locked to
the driving frequency (phase-locked, or synchronized). Here, *q* is interpreted as the polar angle such
that its value lies between 0 and 1; the two parameters are K the coupling
strength between the driver and the rotor, and *W* is driver frequency. A typical map with Arnold tongues is
given in Figure 1.

*Figure 1 Some
Arnold tongues in the standard circle map e
= K/2p versus W*

For small to intermediate values of *K*
(0-1) and certain values of *W*,
the map exhibits phase locking. In the phase-locked region, *q _{n}* advances essentially in a rational multiple
of

The phase-locked regions, called Arnold
tongues, are coloured yellow in Figure 1; while the quasi-periodic regions are
white. Each yellow V region touches down to a rational value of *W* = *p*/*q* in the limit of *K*
_{→}
0. The values of (*K*, *W*)
in one of these regions will all result in a motion with rotation number *w*
= *p*/*q*. For example, all values of (*K*, *W*) in the middle V region correspond to *w*
= ½. In other words, the sequence stays locked on to the signal despite
significant noise or perturbation. This ability to lock on in the presence of
noise is central to the utility of phase-locked loop electronic circuits.

The circle map in Figure 1 is invertible or symmetrical around the mid line.

*The golden mean critical point*

For values of *K*>1, the circle map is no longer
invertible. In Figure 2a [9], the circle map is extended to *K* = 4. Arnold
tongues of synchronization are in grey with winding numbers indicated inside
the tongue. The white regions are quasi-periodic, and the stippled regions appearing
beyond the line K = 1 represent* chaos*. (See the fully coloured version
of this map at the beginning of this article.) This map also depicts fractal
self-similarity on different scales. Fractal self-similarity and chaos are
closely related. A chaotic system has a fractal dimension, i.e., a fractional
dimension between the usual integer values, 1, 2, 3, or 4, and exhibit
self-similarity over many scales.

It is important to note that *chaos*
most definitely does not mean random. Mathematically, *chaos is locally
unpredictable and sensitive to initial conditions, but it is globally
determined or bounded by ‘strange attractors’* (see later). There is no
universally accepted or rigorous definition of chaos, and there are many
interesting attempts at describing it (see [10]).

*Figure 2
Extended circle map (see text for details)*

The golden mean critical point
(GM) is where the curve of constant irrational winding number *f* = (√5 -1)/2 terminates on the
line *K* = 1 (see Fig 2b), and quasi-periodic behaviour undergoes
transition to chaos. This point is marked by an infinite sequence of unstable
orbits with periods given by the Fibonacci numbers (see [11] The Story of *Phi*
Part 1, *SiS* 62) for the
intimate connection between the Fibonacci sequence, where the ratio of
successive numbers converges to the golden ratio).

The golden ratio is thus located at “the edge of chaos”, and has a role in keeping the system of oscillators active without interfering with one another as well as away from the state of chaos. Unfortunately, the term chaos has a much undeserved negative connotation, because of its association with the breakdown of order and total randomness, which it definitely does not involve.

So, is our solar system stable? There is good evidence that the planetary orbits around the sun exhibit golden ratios or ratios according to the Fibonacci sequence numbers, as many people have commented (see [12, 13] for example). The question is whether it will remain stable as such, at least for billions of years, or in transit to chaos much sooner than that. Some astrophysicists claim however, that the planetary orbits are chaotic and sensitive to initial conditions, hence unpredictable for longer than 100 million years into the future [14]; so there is still no immediate cause for alarm.

Edward Norton Lorenz,
MIT mathematician and meteorologist is the generally acknowledged ‘father’ of
chaos theory [15]. During the winter of 1961, Lorenz was running a climate
model on the computer described by 12 differential equations, when he decided
to repeat one of the runs with a small change. Instead of calculating to six
decimal digits, he rounded that to three to save computing time, fully
expecting to get the same results. But he didn’t. That was the beginning of his
discovery of the “sensitive dependence on initial conditions” of chaotic
systems, which he described as the “butterfly effect”. It makes long term
weather prediction impossible. His ‘toy’ equations produced the ‘Lorenz
attractor’ (Figure 3) (the prototype of ‘strange attractors’ associated with
chaotic systems), which bears serendipitous resemblance to butterfly wings, and
became the emblem of the chaos era that followed. Numerous ‘strange attractors’**
**have been** **created, mostly as computer artwork, beginning with the
book by Clint Sprott [16]. But chaos theory has found applications in meteorology,
physics, engineering, economics, biology and medicine.

*Figure 3 The
Lorenz attractor by Clint Sprott*

The Lorenz attractor is a fractal, with self-similar structure on different scales. It has a fractal dimension of 2. 06215 - almost a 2-dimensional surface but not quite, as it cannot be flattened out - and lives in a space of at least 3 dimensions. It contains unstable periodic orbits that can be identified using various mathematical procedures [17], and can also be regarded as twisted or ‘knotted’ periodic orbits [18].

This brings us back to the importance of
cycles in understanding natural processes. The Lorenz attractor (Figure 3) is a
*Poincare section* of the dynamical system’s phase space, which gives a
picture of how the trajectory of the system intersects with the section or the
surface of the phase space. Poincare section is named after French polymath
Henri Poincare (1854-1912), who excelled in all fields that existed during his
lifetime, and was the one who laid the foundations of modern chaos theory with
his research on the ‘three body problem’ (see earlier). Poincare also
emphasized the importance of periodic orbits.

Chaos theory has been taken up enthusiastically in every field including quantum physics, in the form of ‘quantum chaos’, which tries to build a bridge between chaos in classical mechanics and the wavelike motion of electrons in atoms and molecules. Martin Gutzwiller, a leader in quantum chaos wrote [19]: “Phase space for a chaotic system can be organized, at least partially around periodic orbits, even though they are sometimes quite difficult to find.”

*Figure 4 Turbulent
flow generated by the tip vortex of the aeroplane wing shown up by red agricultural
dye, Wikimedia*

Chaos is typical found in turbulent flows of fluids, gases, and the atmosphere. Turbulence is traditionally regarded one of the most intractable problems in physics and mathematics. Mary Selvam, now retired as deputy director of Indian Institute of Tropical Meteorology in Pune, first proposed a theory of turbulent fluid flow based on fractal space-time fluctuations in 1990 (see [20]).

Selvam treats the fractal fluctuations on all space-time scales as a superimposition of a continuum of eddies or vortices. Large scale fluctuations result from the integration of smaller scale fluctuations within. The larger scale eddies grow by enclosing smaller scale eddies and the growth trajectory traces an overall logarithmic spiral flow path with the quasi-periodic Penrose tiling pattern for internal structure.

The ratio of radii or circulation speeds
corresponding to the successive growth steps of the large eddy generating the
geometry of the quasi-periodic Penrose tiling pattern is, of course, equal to
the golden ratio *f *=
1.618..(see [11]). Interestingly, the turbulence created by the tip vortex of
the aeroplane in Figure 4 does trace out a logarithmic spiral approaching the
golden ratio.

Treating turbulence as a continuum of discrete eddies or cycles with Penrose tiling pattern of growth captures key features of organic spacetime. Cycles imply perpetual return, a regeneration conferring identity, stability and autonomy, the very signatures of life. My artist impression of fractal organic spacetime (Figure 5) could be readily interpreted in terms of larger eddies (activity cycles) enclosing smaller ones within; except for the quantum entanglement that takes it away from a perfect nested hierarchy.

*Figure 5 Artist
impression of organic fractal and entangled spacetime (from [2])*

English mathematician philosopher Alfred North Whitehead (1861-1947) was in no doubt that the entire Universe is a super-organism consisting of organisms on every scale from galaxies to elementary particles [21], and argued it is only possible to know and understand nature both as an organism and with the sensitivity of an organism. I have taken his words to heart, and benefited a great deal in my continual quest for the meaning of life.

This series will end with an amazing new vision of the
universe [22] (Golden
Geometry of E Infinity Fractal Spacetime, *SiS* 62) that brings us much closer to our
intuitive feeling for organic spacetime.

*Article first published 24/03/14*

- “Our universe continually cycles through a series of ‘aeons’”, the Daily Galaxy, 26 September 2011, http://www.dailygalaxy.com/my_weblog/2011/09/we-can-see-through-the-big-bang-to-the-universe-that-existed-in-the-aeon-before-.html
- “Primordial gravitational wave discovery heralds ‘whole
new er’ in physics”, Stuart Clark,
*The Guardian*, 17 March 2014, http://www.theguardian.com/science/2014/mar/17/primordial-gravitational-wave-discovery-physics-bicep - Ho MW.
*The Rainbow and the Worm, the Physics of Organisms*, World Scientific, 1993, 2^{nd}edition, 1996, 3^{rd}enlarged edition, 2008, Singapore and London. https://www.i-sis.org.uk/rnbwwrm.php - Ho MW. The golden music of the brain. The story of phi part 3. Science in Society 62 .(to appear) 2014.
- Eugene WC. An introduction to KAM theory. Preprint January 2008, http://math.bu.edu/people/cew/preprints/introkam.pdf
- Kolmogorov-Arnold-Moser theorem. Wikipedia, 24 January 2014, http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem
- Effect of noise on the golden-mean quasiperiodicity at the chaos threshold. http://www.sgtnd.narod.ru/science/noise/2noise/eng/2noise.htm
- Arnold tongue. Wikipedia, 4 February 2014, http://en.wikipedia.org/wiki/Arnold_tongue
- Ivankov NY and Kuznetsov SP. Complex periodic orbits,
renormalization, and scaling for quasiperiodic golden-mean transition to
chaos.
*Phy Rev E*2001, 63, 046210. - Chaos theory. Wikipedia, 29 January 2014, http://en.wikipedia.org/wiki/Chaos_theory
- Ho MW. The story of phi part 1. Science in Society 62 2014.
- Lombardi OW and Lombardi MA. The golden mean in the solar system. The Fibonacci Quarterly 1984, 22, 70-75. http://www.fq.math.ca/22-1.html
- Phi and the solar system. f Phi 1.618 The Golden Number, 13 May 2013, http://www.goldennumber.net/solar-system/
- “Is the solar system stable?” Scott Tremaine, Institute for Advanced Study, Summer 2011, http://www.ias.edu/about/publications/ias-letter/articles/2011-summer/solar-system-tremaine
- Sprott JC. Honors: A tribute to Dr Edward Norten Lorenz. EC Journal 2008, 55-61, http://sprott.physics.wisc.edu/lorenz.pdf
- Sprott JC. Strange Attractors: Creating Patterns in Chaos, M&T Books, New York, 1993.
- Viswanath D. Symbolic dynamics and periodic orbits of the Lorenz attractor. Nonlinearity 2003, 16, 1035-56.
- Birman JS and Williams RF. Knotted periodic orbits in
dynamical systems – 1. Lorenz’s equations.
*Topology*1983, 22, 47-82. - Gutzwiller M. Quantum chaos. Scientific American January 1992, republished 27 October 2008, http://www.scientificamerican.com/article/quantum-chaos-subatomic-worlds/
- Selvam AM. Cantorian fractal space-time fluctuations in
turbulent fluid flows and the kinetic theory of gases.
*Apeiron*2002, 9, 1-20. - Whitehead AN.
*Science and the Modern World*, Lowell Lectures 1925, Collins Fontana Books, Glasgow, 1975. - Ho MW. Golden geometry of infinite spacetime. Science in Society 62 .(to appear) 2014.

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**www.gold-dna.de** Comment left 25th March 2014 01:01:50

Thank you very much for spreading the word of PHI. I am looking forward to the last part in this series.
Anyone interested in PHI on an even deeper level can visit http://www.gold-dna.de/phi.html ... unfortunately in German, but worth the trouble ;-)
Guido Vobig

**Scott Collimgwood** Comment left 26th March 2014 18:06:05

Just in passing I think it's important to note that Buddhist cosmology is taken from the Vedic literature.