Affiliation:  ^{a} J. Heyrovsky Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Prague, Czech Republic 
Abstract This work arose from the author's finding that the ratio of the radius of hydrogen, estimated recently (C.H. Suresh, N. Koga, J. Phys. Chem. A, 105, 5940 (2001)) by density functional methods, to the ground state Bohr radius is the Golden ratio, which operates in a variety of natural phenomena. It is found that the Golden ratio indeed plays a quantitative role in atomic physics. The interesting results are (1) that it arises in atomic dimensions due to the electrostatic forces between negative and positive charges; (2) that the energy of atomic hydrogen is actually equivalent to the energy of the simplest atomic condenser with the Golden mean capacity; (3) that the origin of two terms in the Rydberg equation for absorption and emission is in fact in the ground state term; (4) that all atoms can be assigned definite values of cationic and anionic radii based on the Golden ratio and covalent radii; (5) that these radii are additive and explain quantitatively bond lengths like those of alkali halides, of hydrides of many elements and of many other bonds, whether covalent or ionic or both; and (6) that the work functions of alkali metals can be evaluated using the bond lengths.

We also reprint here for your convenience from: http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/goldenmean.htm

further on this plumbing the physics of Golden Mean... :
goldenmean.info/goldenphysicsvindicated
goldenmean.info/budapest08/physicsoverview.html

Finally  we reprint here from
goldenmean.info/newscientistfractalsolutions
Physics SOLUTIONS based on fractality from Dan Winter..
New Scientist Published Article Comment from Dan Winter: HOW Fractality Causes Gravity& Life Force: PHASE CONJUGATION ( and review of links) 
Exerpt from new New Scientist Magazine Article: Can Fractals Make Sense of the Quantum World? http://www.newscientist.com/article/mg20127011.600canfractalsmakesenseofthequantumworld.html?full=true 
(at this moment near the final published comment  on New Scientist web site from Implosion Group reads:) Fractality:Electrical Mechanism Of Gravitypaper 2006 Tue Mar 31 09:05:03 BST 2009 by dan winter more depth: (from both international conferences on unified field physics of budapest) below we reprint for your convenience  FRACTALITY highlights from goldenmean.info: Golden ratio perfected fractality  or phase conjugation creates the CHARGE IMPLOSION  which is behind ALL centripedal and self organizing universal forces especially including: *gravity goldenmean.info/budapest08/physicsoverview.html *life force ( goldenmean.info/biophoton *origin of color ( goldenmean.info/fractalcolor *perception and consciousness ( goldenmean.info/budapest08 *bliss / peak experience / ecstasy ( goldenmean.info/clinicalintro , goldenmean.info/matrix *DNA's electric and immortalizing purpose and destiny ( goldenmean.info/dnamanifesto *origin of alphabet and symbol ( goldenmean.info/dnaring , spirals.eternite.com ) Fractal Field a COMPLETE FRACTAL COSMOLOGY: goldenmean.info/completefractalcosmology PHASE CONJUGATION (fractal optimized) as the wave mechanism in principle of PERCEPTION: goldenmean.info/perception Fractal / nondestructive (golden ratio optimized / implosive) CHARGE COMPRESSION is the essential symmetry and CAUSE of all: 1. mass creation, 2. gravity, 3. biology, 4. healing, 5. self organization from chaos, 6. symbol making, 7. consciousness, and 8. all BLISS / PEAK EXPERIENCE / ENLIGHTENMENT. The FRACTAL FIELD  Revolutionary Research Frontier  with Radically Powerful Solutions to Virtually ALL Major Global Issues.. 1. Peacemaking: How The FRACTAL FIELD Can Make Teaching PEACE Into a New and Powerful SCIENCE  more: goldenmean.info/peaceuniversity 2. Agriculture: How the FRACTAL FIELD Can Revolutionize Farming by creating Phase Conjugate Environment which properly allow DNA to Communicate and Thrive electrically  more: goldenmean.info/germination , and goldenmean.info/phaseconjugation Be smart:food storage in a refridgerator is NOT fractal goldenmean.info/fractalfridge 3. Architecture: How the FRACTAL FIELD Can ReInvent Architecture now that the Principle of Building a BIOLOGIC CAPACITOR Allows us to Create Structure to Truly Create Healing and Bliss in Biology  more: goldenmean.info/architecture 4. Genetics: How the FRACTAL FIELD Can ReDefine Success in Genetic Research / and Engineering  based on ReDefining DNA COHERENCE, and DNA 'Radio', and DNA Ability to Absorb and Radiate the Electric Field of LIFE!  more: goldenmean.info/decode , and goldenmean.info/12strands , goldenmean.info/phaseconjugation 5. Psychology of BLISS / Ecstasy and PEAK EXPERIENCE: How the FRACTAL FIELD Allows Us Now to Define, Measure and most of all TEACH  Peak Experience , Bliss, Ecstasy  True Enlightenment  more: goldenmean.info/clinicalintro 6. Urban Design: How the FRACTAL FIELD Allows us to ReInvent URBAN DESIGN Based on Fractal Charge Compression to Attract People, Money, Charge and LIFE FORCE Back into Urban Design more: goldenmean.info/lightcity , goldenmean.info/rosycross 7. Weather / Climate / Rainmaking: How the FRACTAL FIELD Allows us to Attract Rain  and Regulate Precipitation  more: goldenmean.info/rain , and The Yantra Pics at goldenmean.info/callingallangels 8. Energy: How the FRACTAL FIELD Can Allow Us to PROPERLY Use Charge SelfSimilarity to COHERE and Couple the Gravity Field for Electric Charge  WITHOUT Destabilizing the Earth Grid  more : goldenmean.info/gravitycause , goldenmean.info/notfree How the (Phi) FRACTAL FIELD provides radical new solutions to GLOBAL WARMING: goldenmean.info/warming , and WHEN COLD FUSION gets COLD: goldenmean.info/fusion 9. Another example of FRACTAL FIELD  self similar charge field revolutionizing medicine.. note how Negative Ion Wind (a fractal field) radically sweeps infection and parasites from the body ( see: ref 1) perfectly consistent with using fractality to measure ANY liquid's ability to support life ( ReDox Potential measure at goldenmean.info/healingphase ) and  measuring fractality in Air to find and measure life's electric signature ( see IGA at phaseconjugation link). + Fractal Field is Electrical Engineering Instructions for SUCCESSFUL DEATH!  goldenmean.info/death 10.FRACTALITY in the HEARTsolution to EKG+ virtually ALL disease resistance: goldenmean.info/holarchy , goldenmean.info/heartmathmistake 11.FRACTALITY in TIME (time=measurement of charge rotation) charge systems emerge from chaos only this wayfractality (how to generate synchronicity): goldenmean.info/TIMESTAR , goldenmean.info/timewave Just as Bruce Cathie showed nuclear critical mass varied with grid position, in space AND time  fractality (constructive charge compression) is the OBVIOUS principle so capacitors properly arranged in a golden spiral firing in TIME  WILL reduce critical mass! (sorry CIA  the principle of FRACTALITY {fear's opposite} is PUBLIC information). 12.Fractality in the CELL: Bruce Lipton of "Membrane Mediated Biology"+ "What the Bleep" discusses how the cell membrane gets infinite surface area, infinite foldedness (DW adds Harmonic Inclusive by Golden Ratio) and therefore potentially infinite information by getting FRACTAL in his latest paper/book:"FRACTAL EVOLUTION: 
Can fractals make sense of the quantum world?
QUANTUM theory just seems too weird to believe. Particles can be in more than one place at a time. They don't exist until you measure them. Spookier still, they can even stay in touch when they are separated by great distances. Einstein thought this was all a bit much, believing it to be evidence of major problems with the theory, as many critics still suspect today. Quantum enthusiasts point to the theory's extraordinary success in explaining the behaviour of atoms, electrons and other quantum systems. They insist we have to accept the theory as it is, however strange it may seem. But what if there were a way to reconcile these two opposing views, by showing how quantum theory might emerge from a deeper level of nonweird physics? If you listen to physicist Tim Palmer, it begins to sound plausible. What has been missing, he argues, are some key ideas from an area of science that most quantum physicists have ignored: the science of fractals, those intricate patterns found in everything from fractured surfaces to oceanic flows (see What is a fractal?). Take the mathematics of fractals into account, says Palmer, and the longstanding puzzles of quantum theory may be much easier to understand. They might even dissolve away. It is an argument that is drawing attention from physicists around the world. "His approach is very interesting and refreshingly different," says physicist Robert Spekkens of the Perimeter Institute for Theoretical Physics in Waterloo, Canada. "He's not just trying to reinterpret the usual quantum formalism, but actually to derive it from something deeper." That Palmer is making this argument may seem a little odd, given that he is a climate scientist working at the European Centre for MediumRange Weather Forecasting in Reading, UK. It makes more sense when you learn that Palmer studied general relativity at the University of Oxford, working under the same PhD adviser as Stephen Hawking. So while Palmer has spent the last 20 years establishing a reputation as a leading mathematical climatologist, he has also continued to explore the mysteries of his first interest, quantum theory (see "Quantum ambitions"). "It has taken 20 years of thinking," says Palmer, "but I do think that most of the paradoxes of quantum theory may well have a simple and comprehensible resolution." Arguments over quantum theory have raged since the 1920s, starting with a series of famous exchanges between Einstein and the Danish physicist Niels Bohr. Bohr and his supporters believed that the theory's successful description of atoms and radiation meant you should abandon old philosophical concepts, such as the idea that objects have definite properties even when no one is there to measure them. Einstein and his followers countered that such radicalism was wildly premature. They argued that much of the quantum weirdness was nothing more than a lack of adequate knowledge. Find a quantum system's "hidden variables", Einstein suspected, and quantum theory might make common sense, a view that quantum enthusiasts thought was ultra conservative and out of touch. The argument rages to this day. Fractals unitePalmer believes his work shows it is possible that Einstein and Bohr may have been emphasising different aspects of the same subtle physics. "My hypothesis is motivated by two concepts that wouldn't have been known to the founding fathers of quantum theory," he says: black holes and fractals. Palmer's ideas begin with gravity. The force that makes apples fall and holds planets in their orbit is also the only fundamental physical process capable of destroying information. It works like this: the hot gas and plasma making up a star contain an enormous amount of information locked in the atomic states of a huge number of particles. If the star collapses under its own gravity to form a black hole, most of the atoms are sucked in, resulting in almost all of that detailed information vanishing. Instead, the black hole can be described completely using just three quantities  its mass, angular momentum and electric charge. Many physicists accept this view, but Palmer thinks they haven't pursued its implications far enough. As a system loses information, the number of states you need to describe it diminishes. Wait long enough and you will find that the system reaches a point where no more states can be lost. In mathematical terms, this special subset of states is known as an invariant set. Once a state lies in this subset, it stays in it forever. A simple way of thinking about it is to imagine a swinging pendulum that slows down due to friction before eventually coming to a complete standstill. Here the invariant set is the one that describes the pendulum at rest. Because black holes destroy information, Palmer suggests that the universe has an invariant set too, though it is far more complicated than the pendulum. Complex systems are affected by chaos, which means that their behaviour can be influenced greatly by tiny changes. According to mathematics, the invariant set of a chaotic system is a fractal. Fractal invariant sets have unusual geometric properties. If you plotted one on a map it would trace out the same intricate structure as a coastline. Zoom in on it and you would find more and more detail, with the patterns looking similar to the original unzoomed image. Gravity and mathematics alone, Palmer suggests, imply that the invariant set of the universe should have a similarly intricate structure, and that the universe is trapped forever in this subset of all possible states. This might help to explain why the universe at the quantum level seems so bizarre. For example, it may point to a natural explanation for one of the biggest puzzles of quantum physics, what physicists refer to as its "contextuality". Quantum theory seems to insist that particles do not have any properties before they are measured. Instead, the very act of measurement brings their properties into being. Or, put another way, quantum systems have meaning only in the context of the particular experiments performed on them. Ever since Einstein, many physicists have hoped that a new approach might go beyond quantum theory and find a way to restore belief in objective and independent properties. But in 1967, mathematicians Simon Kochen and Ernst Specker published a theorem showing that this dream, if possible, cannot be done in quite the way physicists would like. Central to Kochen and Specker's theorem is a thought experiment. Say you choose to measure different properties of a quantum system, such as the position or velocity of a quantum particle. Each time you do so, you will find that your measurements agree with the predictions of quantum theory. Kochen and Specker showed that it is impossible to conceive a hypothesis that can make the same successful predictions as quantum theory if the particles have preexisting properties, as would be the case in classical physics. This result has driven many physicists to reach a startling conclusion about how to interpret quantum theory. Either you have to abandon the existence of any kind of objective reality, instead believing that objects have no properties until they are measured, or you have to accept that distant parts of the universe share a spooky connection that allows them to share information even when the distance and timing means that no signal could have passed between them without travelling faster than light. Palmer's idea suggests a third possibility  that the kinds of experiments considered by Kochen and Specker are simply impossible to get answers from and hence irrelevant. The key is the invariant set. According to Palmer's hypothesis, the invariant set contains all the physically realistic states of the universe. So any state that isn't part of the invariant set cannot physically exist. Suppose you perform the KochenSpecker thought experiment and measure the position of an electron. Then you ask what you would have found if you repeated the experiment, only this time measuring the electron's velocity instead. According to Palmer, when you repeat the experiment you are testing a hypothetical universe that is identical to the real one except that the positionmeasuring equipment is replaced with velocitymeasuring equipment. This is where the fractal nature of the invariant set matters. Consider a place of interest you want to visit along a coastline. If you get the coordinates even slightly wrong you could end up in the sea rather than where you want to be. In the same way, if the hypothetical universe does not lie on the fractal, then that universe is not in the invariant set and so it cannot physically exist. Due to the spare and wispy nature of fractals, even subtle changes in the hypothetical universes could cause them to fall outside the invariant set. In this way, Spekkens says, Palmer's hypothesis may help to make some sense of quantum contextuality. "I think his approach is really interesting and novel," says Spekkens. "Other physicists have shown how you can find a way out of the KochenSpecker problem, but this work actually provides a mechanism to explain the theorem." Following on from this, Palmer believes that many other features of quantum theory also fall into place. For example, quantum theory is famous for making only statistical predictions  it can only tell you the probability of finding an electron with its quantummechanical spin pointing up. This arises naturally, suggests Palmer, because quantum theory is blind to the intricate fractal structure of the invariant set. Just as our eyes cannot discern the smallest details in fractal patterns, quantum theory only sees "coarse grain approximations", as if it is looking through fuzzy spectacles. Other physicists seem inspired by the novelty of Palmer's approach. "What makes this really interesting is that it gets away from the usual debates over multiple universes and hidden variables and so on," says Bob Coecke, a physicist at the University of Oxford. "It suggests there might be an underlying physical geometry that physics has just missed, which is radical and very positive." Coecke points out that very few scientists working on fundamental physics have explored how fractals might be incorporated into the theory, even though they are commonplace in other parts of physics. Palmer is hoping that will change. In a paper submitted to the journal Proceedings of the Royal Society A, he shows how the basic idea can account for quantum uncertainty, contextuality and other quantum puzzles (www.arxiv.org/abs/0812.1148). Many details still need to be fleshed out, says Coecke. "Palmer manages to explain some quantum phenomena," he says, "but he hasn't yet derived the whole rigid structure of the theory. This is really necessary." Palmer accepts the criticism and is hopeful that he will be able to improve his theory over time. In the best of worlds, he thinks his framework may provide a way to finally reunite the warring parties of Einstein's and Bohr's followers. After all, the theory backs Einstein's view that quantum theory really is incomplete. It is, Palmer says, blind to the fractal structure of the invariant set. If it wasn't, it would see that the world is not only deterministic, but it never exhibits any spooky effects. On the other hand, it also agrees with the view of Bohr and his followers: the properties of individual quantum systems are not independent of the entire world, especially the experiments we humans use to explore them. We are stuck with the disturbing fact that how we measure always influences what we find. For now, quantum theory remains mysterious but its air of mystique may not last forever. Quantum ambitionsWhen Tim Palmer finished his PhD in physics at the University of Oxford 30 years ago, he had the opportunity to work as a postdoc with Stephen Hawking at the University of Cambridge. The hot topic in theoretical physics back then was supergravity, a theory that aimed to include gravity in a universe with 11 dimensions. Despite Hawking's enthusiasm for the idea, Palmer remained lukewarm. Supergravity takes quantum theory as an unquestioned starting point and then tries to bring gravity within its fold, an approach Palmer found unappealing. "I felt that quantum theory was at best a provisional theory," Palmer recalls. Instead, he switched to climate science where he rapidly established an international reputation. Today Palmer is known for pioneering a method called ensemble forecasting, which incorporates the role of chaos to create climate forecasts that include specific estimates of their own accuracy. But even as Palmer's work became widely influential  so much so that he has taken a key role on the Intergovernmental Panel on Climate Change  he could never forget the quantum puzzles that so occupied him before. What is a fractal?Fractals are geometrical shapes that aren't smooth like circles or rectangles. They are irregular structures with the same structure repeating on ever finer scales. No matter how much you blow up a picture of a fractal, it will always look the same. The natural world contains many examples of fractals, including ferns, broccoli, river networks, blood vessels and coastlines. Mark Buchanan is a writer based in Cambridge, UK Fractals like this one exhibited by the Romanesco cauliflower could help explain the wackiness of quantum theory (Image: fishmonk / stock.xchng)  
