By Darryl Penney

 

Abstract: the golden ratio has intrigued scholars from all ‘walks of life’ and many books have been written on the subject over hundreds of years because of a ‘feeling’ about a ratio that a number of people say is important, and the reason is, I believe, that the ratio is a quest/solution to the fifth dimension. I also believe that this feeling of ‘something’ that has been ascribed as being of beauty/completeness/satisfaction is probably a resonance with the measurement/entanglement/ecstasy of the ‘fabric’ of the whole universe as well as locally. It seems that this effect is defined physically/measurably simply by a ratio, but logically as a feeling and the possibilities of beauty/completeness/satisfaction have been explored/quested by Life and used to that end simply because it can be used. In other words, it is possible that Life has taken a ‘resonance’ of our universe [absolute] and has ascribed to it the concept of beauty/completeness/satisfaction, defining an absolute for one of the large number of concepts that Plato had problems defining, such as justice, for which an absolute must be assigned. Beauty/completeness/satisfaction can be incorporated into our reality because the ‘resonance’ is common to all members and so can be used as a selection ‘tool’ and ‘the primary focus’ of Darwin’s major work Descent of Man, ‘two-thirds of the text was devoted to sexual selection’, but does not say how or why it exists.

 

The following quotation is from the book, The Story of Measurement by Andrew Robinson (p 42) and sums well the extreme feelings that the golden ratio has invoked for hundreds if not thousands of years and sets the scene for, I believe, an explanation of why those feelings have been generated. The vagueness of the feelings that such a simple ratio can invoke is, I believe, a window into the workings of the universe and how Life has made use of what might be only an un-needed off-shoot of the fifth dimension and shows the measurement/entanglement of a ratio (measurement) and a feeling (entanglement). Notice that there is a local effect (a+b) and a universe-wide effect (a and b) of measurement/entanglement in a similar way to local measurement/entanglement of the ‘bounciness of matter’ (a+b)=1 and universe-wide measurement/entanglement of gravity (a and b)=1 [chapter 77].

 

But first, a digression is necessary , ‘the Nobel laureate Eugene Wigner, gave a celebrated lecture with the title, “The unreasonable effectiveness of mathematics in the natural sciences”…. The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it.’ (p 45) I need to say that there is a rational explanation and that is that mathematics ‘works’ because mathematical/logical relationships are the fifth dimension and quests of a mathematical relationship (a+b)=1 are the basis of our universe. Further, it will be shown that one of the quests may or may not be necessary to the functioning of the universe, but Life, I believe, has used it to enrich our lives and provide a means of sexual selection.

 

‘It is often claimed that the face in Leonardo’s Mona Lisa reveals the proportions of the golden ratio. That is, if you draw a rectangle around her face, the ratio of the rectangle’s height to its width is close to 1.618:1 (just over 8:5) Leonardo never noted that the ratio was in his mind, but it is a fact that he was a close friend of Luca Pacioli, who published a three-volume treatise on the ratio, Divina Proportione. Pacioli particularly believed that the ratio’s ‘divine proportion’ was to be found in the human face.’

 

‘The ratio has been alleged in many other contexts, such as the Greek Parthenon, the two-column type area of Gutenberg’s Bible, the spirals of the chambered nautilus shell, and even the proportions of modern credit cards. It has fascinated mathematicians from Euclid (who first defined it in writing) to Kepler and Roger Penrose. ‘”Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal.  In fact, it is probably fair to say that the golden ratio has inspired thinkers of all disciplines like no other number in the history of mathematics”, writes Mario Livio in the Golden Ratio. But there is no proof that any of these apparent incidences are intentional.’

 

‘Mathematically, the golden ratio, now known as phi – after Phidias, the Greek sculptor said to have used it in the Parthenon – is best defined in terms of the division of a line into a larger part a and a smaller part b. The golden ratio is the ratio when the whole line is to the larger part as the larger part is to the smaller part. To be more precise: phi = (a+b)/a=a/b. It equals (1+ square root 5)/2, an irrational number approximately equal to 1.618.’

 

In a probability space the physical dimensions are space-time and  (a+b)=1 as an illustration, for measurements/observers a and b and this fifth dimension can be unfolded to see the component parts that the probability space quests in the manner of quantum mechanics. Questing is continual and means that measurements remain indeterminate, as they must, until a measurement is required by an iteration or mind/brain and that measurement is current only at that time of measurement.

 

Firstly, (a+b)=1 can be unfolded to show (1) from the equation, measurement/observer a and b, (2) from the equation, entanglement (a+b), (3) the logical part of measurement/observer a and b, so there is an independent/orthogonal logical duality to measurement, (4) there is the logical part of entanglement (a and b), and there is only one solution/absolute (5/6) to this equation and that is that the speed of electromagnetic radiation is a constant/absolute. Note that this solution has a physical side and a logical side that occurs because the dimensions [space-time, energy] produce the space to time relation for all energies, and the other two combinations produce the Heisenberg uncertainty relations energy-space and energy-time. The physical side is quested and returns a particle, wave, magnetic or electrical effect as sought, and it is an interesting point that the electromagnetic effect is taken to mean that the electric and magnetic fields are orthogonal. The lack of a solution other than for the speed of light leads to Plato’s problem (7/8) of no absolutes unless assigned for justice, beauty etc.

 

Secondly, an illustration of the (logical) questing that is carried out infinitely fast [product of the space] compared to the constant slow speed of light, relativisation of the dimensions allows the law of conservation of energy to maintain the energy at zero [negative potential, the others energies positive]. However, the questing that is the ‘organizational power’ behind quantum mechanics quests every possibility and we have yet to consider another possibility, and that is (9/10) (a+b).

 

(a+b) is very similar to the interval of the golden ratio and bearing in mind that (a+b) represents measurement and logic as independent/orthogonal entities in the physical world, they will necessarily appear as concept/context to our mind/brain. (a+b) contains a possibility that must be quested, so let’s do it, as above, that it is an interval, and an interval has a solution and that solution is the golden ratio. To expand, (a+b) would contain a duality/orthogonality of measurement and logic, both of which are essential to appreciation of music, paintings, scenery etc. and, as we have seen, logic is a ‘rag bag’ of entanglement/logics of space/machine as well as world O.

 

It could be that (a+b)/a=a/b represents the experience of the measurement operation to the thing being measured equates to the relation of the measurement to the observer, but this might be reading too much into the fifth dimension and it might just be a piece of luck that brightens our world. On the other hand, it does seem to make sense to a (logical) degree, but that does lead into the question of the difference between logic and mathematics. ‘In the Principia, Russell and Whitehead defended the view that mathematics was basically an elaboration of the laws of logic, with no clear demarcation between them.’ (Is God a Mathematician?, Mario Livio, p 189) The derivation below shows, bottom-up, a foreshadowing that the universe is based on mathematics, but we, Life and the bio-computer of evolution derive logic, but that will have to wait for the future.

 

The Golden ratio may not be necessary to the functioning of the universe and is a mechanical logic quested at the same time as the necessary parts and we have made use of it in world O. In fact, as an example, there may be no absolute for justice, beauty etc. (Plato’s problem), but we still need a reality of administration from, say, no punishment to capital punishment that has a consensus in concept and context etc., or, another example, the reality of beauty giving a ratio/measurement (golden ratio) as well as consensus of agreement because consensus is necessary in a reality if it is to be useful.

 

Thirdly, notice that a and b have been considered as measurement/entanglement in the physical universe and can be written as (a+b+c …)=1, but the mental field of the mind/brain leads to concept/context because our mind/brain evolved a new and different field that is determinate, not universe-wide because our reality must be definite and defined if we don’t want to end as something’s dinner. Our mind/brain uses (11/12) the mathematics of concepts that is immediately apparent from (a+b)=1. In other words, Life is bound by the dimensions of the universe, but Life has created its own universe that is the ‘struggle of life’ bounded by its own reality and we have, by necessity, changed our outlook into the Newtonian mode.

 

In the physical universe, everything goes to completion, but Life has ‘built’ on this by using the units of survival [flight distance] – speed of attack and distance to escape or catch the prey and this requires a measurement by the eye (usually) that coalesces (necessarily) into what we want to see. In other words, a force is an impulse that is halted and requires a mind/brain to determine how much force is to be applied.  Newtonian physics is built on forces, ‘as we now know, his faith in forces was justified to such an extent that it is widely regarded as Newton’s most important scientific innovation.’ (Let Newton be!, John Henry, p 59, edited by John Fauvel)

 

Further, ‘a slightly naughty thought can come to one’s mind here. Is it due to the nature of physics that the mathematical tools started by Newton are so essential for many parts of physics through the centuries? Or is our selection of worked-on parts of physics still largely Newtonian? (p 244) This quotation requires/legitimises the search for enigmas, such as the golden ratio, constant speed of light, logic of gravity etc. that are hidden by Newtonian physics because the logic is subsumed into the concept of force.

 

To digress, the above is a derivation/explanation that I am comfortable with, but how does it jibe with history over thousands of years? Surprisingly, what I am saying is not much different to the ideas of antiquity to Newton’s day. ‘In the sixth century A D the term quadrivium was coined by Boethius to designate the four mathematical sciences dealing with numbers and magnitude: astronomy, geometry, arithmetic, and music. (p 102) ‘What was important to the Pythagoreans was the arithmetical relationship between the pitches … the arithmetic mean ½(a+b) … the harmonic mean 2ab/(a+b) … the geometric mean square root (ab). These numerical relationships were thought to provide a key to the structure of the entire universe. Indeed, the same ratios were thought to be found in the distances between the planets, and it was taught by the later Pythagoreans that the planets, or the spheres that carried them around a central fire, also produced sounds – the famous harmony of the spheres.’ (p 106)

 

Is the contraction/illustration (a+b)=1 and (a+b) very much different to the above? Also, (a+b)=1 is a simplification that is intimately-linked/derivable-from the dimensions that underlie the structure of the universe in a similar way as was believed, above. The continuity of reality is apparent because 0 [certainty of non-existence] and 1 [certainty of existence] are infinitely unlikely, but still exist and we have no way of knowing whether we exist, do not exist or exist in some ‘twilight zone’. For those people that are more comfortable with existence, there is the possibility of existence, but we need the dimensions of a probability space to exist. On the other hand, Life can exist here, and through evolution, determines a logic of possibilities that is open to us through studying fossils and history and that makes us agents of logic.

 

To return to (11/12) the mathematics of concepts, this is immediately apparent from the dimensions (a+b+c+d etc.)=1, where I have used ‘etc.’ to indicate that it is our choice to limit the number of terms because this is the basis of the evolved mind/brain and not universe-wide entanglement. Notice that both measurement and logic are available and we use both in decision-making because as we increase the number of concepts and contexts, the accuracy of our decision should increase. This decision-making is five dimensions, but the predator/prey situation requires (13) forward-planning and a future time, and that could be a sixth dimension that Life uses and is a fundamental part of staying alive. Contrast this to: future-time as required by Life, Newtonian-time is a fixed time interval, Einstein-time varies with relative speed according to the Lorentz transformation, the universe-time is continually changing as part of relativisation and time in a mathematical probability space is time-passing.

 

I will repeat a quotation from chapter 77, ‘it seems that … knowledge cannot spring from experience alone but only from a comparison of the inventions of the intellect with the facts of observation’, wrote Einstein, (The Story of Measurement, Andrew Robinson, p 10) because the golden ratio has caused such questing from scholars for a long time. There is an explanation, as above, that the fundamentals of the universe ‘speak’ to us and that happens much more than is often appreciated. Our world O contains/subsumes world (physical) P and we often don’t realize/appreciate the basis of measurement behind sight, hearing etc., entanglement behind society etc. and ecstasy behind pleasure/appreciation as well as the combinations. In other words, the fifth dimension (world P) is measurement/entanglement/ecstasy, but whilst the effect is there in world P, the interpretation is world O, concept/context/pleasure and the sixth dimension [forward-planning] is purely world O.

 

Further, quantum mechanics and relativity do not affect us, usually, but we don’t recognise that the questing, behind quantum mechanics is the familiar evolution, business and organizational practices. Relativity and relevance have a basis in love, family life, reality as well as evolution and business etc. The physical world intrudes in small ways, but usually our reality, having evolved over 3,000 million years disguises them because our mind/brain has built a determinate space on the entanglement space of the universe. The physical manifestations are there in the diffraction of water waves, the Michelson-Morley experiment, our seeing is measuring, our forward-planning produced the Newtonian units of speed, force and distance that we use for hunting etc.

 

Questing, I believe, involves all possibilities that necessarily remain indeterminate until measured at some instant in some way and using Einstein’s quotation and looking at the description of the golden ratio, the golden ratio is a unique solution and an absolute of beauty/love etc., but it is entwined with the other two to make combinations even though independent/orthogonal. This links in with the mathematics of concepts and shows that the proverb ‘handsome is, as handsome does’ (a simple solution of the mathematics of concepts) needs the new attractor, that is, the golden ratio (an absolute) to make it more useful and accurate, especially to make a reality [consensus] for sexual selection.

 

Given that the female half of the population aspire to beauty by painting their faces, hanging jewellery from their ears, walking on high heels etc. and the other half would like to possess that ‘beauty’, it can be seen that beauty in all its forms from music, art, bush-walking etc. form a large part of our lives. The relationships can take many forms, such as harmonics, but absolutes are unique solutions, such as the constancy of the speed of electromagnetic radiation, instantaneous entanglement across the universe for the conservation of energy, gravity and other ‘nuts and bolts’, and now we can assign the appreciation of a concept of beauty/correctness/satisfaction to world O.

 

Over the years, I have gained the impression that sexual selection was a small part of evolution and it came as a surprise that the ‘primary focus’ of Darwin’s major work Descent of Man, ‘two-thirds of the text was devoted to sexual selection. Darwin believed that the more attractive an individual animal is to potential mates, the more likely he or she is to be successful in the “struggle for reproduction”.’ (Evolution in the Antipodes: Charles Darwin and Australia, Tom Frame, p 38)

 

As mentioned above, and I am sure that Darwin would have agreed that colour in fish, birds etc. play a large part in sexual selection, as well as the beauty of the pelt, ratio of limb size are all sexual signals and we should not be surprised by the fact that the Golden ratio, as an absolute, is so all-pervasive. It is well known that ladies wear long hair, to be identifiably feminine, and as a measure of their personal health. My dancing partner spends an inordinate amount of time worrying about her hair, its length, thickness, condition etc. However, it must be recognised that sexual selection is more than a (Newtonian) concept because it also generates a local logic/feeling [context, within the self, (a+b)] but also a resonance with the universe [context, across the reality, (a and b)] both measurably and is a concept that can be used logically and practically as a ‘target’ for evolution.

 

Conclusion: Every space and what is in it must be a product of the dimensions of that space and the test is simply to derive everything from the dimensions and if anything cannot be explained, we need to add another dimension to cater for it. I am, at present, content that the five physical (and the sixth in world O) are adequate to generate our universe. I have always wondered how science could use/accept logic, beauty etc. as part of the four space-time dimensions, and I believe that the above, using five dimensions is far more believable [also it was/has been expected/anticipated since the 1920s].

 

‘Bohr argued that we needed to go back to the drawing board to understand the quantum world and could not use concepts that were familiar in everyday life’ (50 physics ideas, Joanne Baker). The irony is, and I believe that I have shown it above, that the everyday world is the same as the quantum and the relative world when viewed in the correct way through the fifth dimension. The ‘roots’ are questing and relevance and they pervade literally everything, such as organization, evolution, family etc. and they are concept, context and satisfaction and underlie the questable-part of the fifth dimension.

 

Finally, placing the above within an over-arching unity, I believe that every ‘transaction’, in the mode of quantum mechanics to derive a determinate answer requires answering each of:

 

measurement/forward-planning/quest/relevance/beauty/entanglement

 

References: all quotations are fully referenced in the body and earlier chapters can be found on   darrylpenney.com  if required.