Those who like simple problems can try to demonstrate this relationship before moving on. It’s insultingly easy:
We owe this fortunate discovery to John Arioni. The elementary demonstration, along with other unexpected relationships, is on the site Cut the knot . The number φ is, naturally, the golden ratio (1+√ 5)/2, in decimal figures 1.6180339887…, and φ-1 is the reciprocal, 0.6180339887… . And since its infinite decimal places can be calculated by means of the simplest continuous fraction, here we will also call it the continuous ratio or continuous proportion, because of its unique role as mediator between discrete and continuous aspects of nature and mathematics.
This looks like the typical casual association of recreational math pages. One can get φ in many different ways with circles, but to my knowledge this is by far the most elementary of all, being the radius the unit of reference. In other words, this relation seems too simple and direct not to contain something important. And yet it has only recently been discovered, almost by chance.
Since Euclid and probably much earlier, the entire history of findings on this proportion has been derived from the division of a segment «in the extreme and mean ratio», and has developed with the construction of squares and rectangles. The most immediate cases involving the circle come from the construction of the pentagon and the pentagram, no doubt known to the Pythagoreans; but one does not need to know anything about mathematics to realize that the relationship contained in this symbol is of a much more fundamental order —just as, from the quantitative point of view, the 2 is closer to 1 than 5, or from the qualitative one, the dyad is closer to the monad than the pentad.
If the circle and its central point are the most general and comprehensive symbol of the monad or unit, we have here the most immediate and revealing proportion of reciprocity, or dynamic symmetry, presented after the division into two parts. The Taijitu has a double function, as a symbol of the supreme Pole, beyond duality, and as a representation of the first great polarity or duality. It is, as it were, halfway between both, and both are linked by a ternary relationship —precisely the continuous proportion.
A relation is the perception of a dual connection, while a proportion implies a third order relationship, a «perception of perception». Since at least the times of the Kepler’s triangle, we have known that the golden mean articulates and conjugates in itself the three most fundamental means of mathematics: the arithmetic mean, the geometric mean, and the so-called harmonic mean between both.
We could ask ourselves what would have happened if Pythagoras had known about this correlation, which would certainly have exalted Kepler’s imagination as well. It will be said that, like any other counterfactual, there is no point about it. But the question is not as much about the past that might have been as it is about the possible future. Pythagoras could hardly have been as surprised as we are, since he knew nothing about the decimal values of φ or π . Today we know that they are two ratios running to an infinite number of decimal places, and yet they are linked exactly by the most elementary triangular relationship.
Mathematical truth is beyond time, but not its revelation and construction. This allows us to see certain things with the insight of a Geohistory, as it were, in four dimensions. There has been speculation about what would have happened if the Greeks had known and made use of the zero, and whether they might have developed modern calculus. This is very doubtful, since they would have still needed to make a series of great leaps far from their conception of the world, such as the numbering system, the zero and its positional use, the idea of derivative, and so on. The double spirals were a common motif in archaic Greece, and the arithmetical speculations of the Pythagoreans, very similar in nature to those developed by the Chinese over time; but for whatever reason the Greeks did not intertwine the two spirals into one, and, in China itself, a diagram like the one we know today did not came into existence until the end of the Ming dynasty and only after a lengthy evolution.
Which is just another example of how hard is to see the obvious. It’s not so much the thing itself, but the context in which it emerges and in which it fits. Depending on how one looks at it, this can be encouraging or discouraging. In knowledge there is always a high margin to simplify, but as in so many other things, that margin depends to a large extent on knowing how to make it happen.
The Taijitu, the symbol of the supreme Pole, is a circle, a wave and a vortex all in one. Of course, the vortex is reduced to its minimum expression in the form of a double spiral. Characteristically, the Greeks separated their double spirals, and eventually turn them into squares, in the motifs known today as grecas. It is just another expression of their taste for statics, a bent that set the general framework for the reception of the golden mean in mathematics and art, and which has come down to us through the Renaissance.
The series of numbers that approximate infinitely the continuous proportion, known to us as Fibonacci numbers, appeared already long before in the numerical triangles consecrated in India to Mount Meru, «the mountain that surrounds the world», which is just another designation of the Pole. As it is well known, from this figure, called the Pascal’s triangle in the West, a huge number of combinatory properties, scales and sequences of musical notes are derived.
The polar triangle, known in other cultures as Khayyam’s triangle or Yang Hui’s triangle, is one of those «extraordinarily well connected» mathematical objects: from it one can derive the binomial expansion, the binomial and normal statistical distributions, the sin(x)n+1/x transform of harmonic analysis, the matrix esponential and the exponential function, or the values of the two great gears of calculus, the constants π and e. It is almost incredible that the elemental connection with the Euler number has not been discovered until 2012 by Harlan J. Brothers. Instead of adding up all the figures in each row, one only needs to extract the ratio of ratios for their product; the difference between sums and products is a motif that will emerge several times throughout this article.
The polar triangle looks like an arithmetic and «static» representation, while the Taijitu is like a geometric snapshot of something purely dynamic. However, the rich implications for music of this triangle, partially explored by the work of Ervin Wilson, largely circumvent the separations created by adjectives such as «static» and «dynamic». In any case, if the staircase of figures deployed in Mount Meru is an infinite progression, when we finally see the lines hidden in the circular diagram of the Pole we immediately know that it is something irreducible —the first offers us its arithmetical deployment and the second its geometric retraction.
The oldest known mention of the triangle, albeit a cryptical one, can be found in the Chandaḥśāstra of Pingala, where Mount Meru is shown as the formal archetype for metric variants in versification. It is also fair to say that the first Chinese author to deal with the polar triangle is not Yang Hui but Jia Xian (ca. 1010-1070), a strict contemporary of the philosopher and cosmologist Zhou Dunyi (1017-1073), the first author who publicized the Taijitu diagram.
Nowadays very few people are aware that both figures are representations of the Pole. It is my conjecture that all the mathematical relationships that can be derived from the polar triangle can also be found in Taijitu, or at least generated from it, although under a very different aspect, and with a certain twist that possibly involves φ. Both would be a dual expression of the same unity. Mathematicians will see what is the point of this.
Between counting and measuring, between arithmetic and geometry, we have the basic areas of algebra and calculus; but there is an overwhelming evidence that the latter branches have developed in one particular direction more than in others —more in decomposition than in composition, more in addition than in multiplication, more in analysis than synthesis. So the study of the relations between this two expressions of the Pole could be full of interesting surprises and basic but not trivial results, and it poses a different orientation for mathematics.
It can be seen that the arithmetic triangle has closed links with fundamental aspects of calculus and the mathematical constant e, while the Taijitu and the constant φ lack in this respect relevant connections —hence the totally marginal character of the continuous proportion in modern science. It has been said that the latter, unlike the intimate connection with change of Euler’s number, is a static relationship. However, its appearance in the extremely dynamic character of the yin-yang symbol already warns us of a general change of context.
For centuries calculus has been dissolving the relationship between geometry and change in favor of algebra and arithmetic, of not so pure numbers. Now we can turn this sandglass upside down observing what happens on the upper bulb, the lower bulb and the neck.
The appearance of the golden mean between the yin and yang in a purely curvilinear fashion not only is not static but on the contrary cannot be more dynamic and functional, and indeed the Taijitu is the most complete expression of activity and dynamism with the minimum number of elements. The diagram also has an intrinsic organic and biological connotation, inevitably evoking cell division, which in fact is an asymmetrical process, and, at least in plant growth, often follows a sequence governed by this ratio. In other words, the context in which the continuous ratio emerges here is the true antithesis of its Greek reception that has lasted until today, and this can have far-reaching implications on our perception of this proportion.
Oleg Bodnar has developed an elegant mathematical model of plant phyllotaxis with hyperbolic golden functions in three dimensions and with coefficients of reciprocal expansion and contraction that can be seen in the great panoramic book that Alexey Stakhov dedicates to the Mathematics of Harmony . It is an example of dynamic symmetry that can be perfectly combined with the great diagram of polarity, regardless of the nature of the underlying physical forces.
The presence of spiral patterns based on the continuous proportion and their numerical series in living beings does not seem mysterious. Whether in the case of a nautilus or vegetable tendrils, the logarithmic spiral —the general case- allows indefinite growth with no change of shape. Spirals and helixes seem an inevitable result of the dynamics of growth, by the constant accretion of material on what is already there. At any rate, we should ask why among all the possible proportions of the logarithmic spiral those close to this constant arise so often.
And the answer would be that the discrete approaches to the continuous proportion also have optimal properties from several points of view —and cell growth ultimately depends on the discrete process of cell division, and at higher levels of organization, on other discrete elements such as tendrils or leaves. Since the convergence of the continuous ratio is the slowest, and plants tend to fill as much room as possible, this ratio allows them to emit the greatest number of leaves in the space available.
This explanation seems, from a descriptive point of view, sufficient, and makes it unnecessary to invoke natural selection or deeper physical mechanisms. However, in addition to the basic discrete-continuos relationship, it contains implicitly a powerful link between forms generated by an axis, such as the pine cones, and the so-called «principle of maximum entropy production» of thermodynamics, which we will find later again.
Needless to say, we do not think this proportion has «the secret» to any universal canon of beauty, since surely such a canon does not even exist. However, its recurrent presence in the patterns of nature shows us different aspects of an spontaneous principle of organization, or self-organization, behind what we superficially call «design». On the other hand, the appearance of this mathematical constant, due to its very irreducible properties, in a great number of problems of optimization, maximums and minimums, and parameters with critical points allows us to connect it both naturally and functionally with human design and its search for the most efficient and elegant configurations.
The emergence of the continuous proportion in the dynamic symbol of the Pole —of the very principle- augurs a substantive change both in the contemplation of Nature and in the artificial constructions of human beings. Contemplation and construction are antagonistic activities. One goes top-down and the other bottom-up, but there is always some sort of balance between both. Contemplation allows us to free ourselves from the connections already built, and construction gets ready to fill the resulting void with new ones.
It is somewhat strange that the continuous proportion, despite its frequent presence in Nature, is so poorly connected with the two great constants of calculus, π and e —except for anecdotic incidences as the «logarithmic golden spiral», which is only a particular case of an equiangular spiral. We know that both π and e are transcendental numbers, while φ is not, although it is indeed the «most irrational number», in the sense that it is the one with the slowest approximation by rational numbers or fractions. φ is also the simplest natural fractal.
Until now, the most direct link with trigonometric series has been through the decagon and the identities φ = 2cos 36° = 2cos (π/5). It has not been associated so far with imaginary numbers, i being the other great constant of calculus, which is concurrent with the other two in Euler’s formula, of which the so-called Euler identity (eiπ = -1) is a particular case.
The number e, base of the function that is its own derivative, appears naturally in rates of change, the subdivisions ad infinitum of a unit that tend to a limit or in wave mechanics. The imaginary numbers, on the other hand, so common in modern physics, appeared for the first time with the cubic equations and pop up each time additional degrees of freedom are assigned to the complex plane.
Actually, complex numbers behave exactly like two-dimensional vectors, in which the real part is the inner or scalar product or and the so-called imaginary part corresponds to the cross or vector product; so imaginary numbers can only be associated with motions, rotations and positions in space in additional dimensions, not with the physical quantities themselves.
This is easier to say than to think of, since it is even more «complex» to determine what a physical quantity or a mathematical variable can be independently of change and motion. Both to geometrically interpret the meaning of vectors and complex numbers in physics and to generalize them to any dimension a tool like geometric algebra may be used —»the algebra flowing from geometry», as Hestenes put it; but even then there is much more to geometry than we may think.
Many problems become more simple on the complex plane, or so the mathematicians say. One of them, under the pseudonym Agno sent in 2011 an entry to a math forum with the title «Imaginary Golden Mean», which shows a direct connection with π and e : Φi = e ± πi/3 . Another anonymous author found this same identity in 2016, along with similar derivations, looking for fundamental properties of an operation known as «reciprocal addition», of interest in circuits and parallel resistances calculations. As refraction is a kind of impedance, it may also have its place in optics. The relation in the polar diagram may be associated right from the start with geometric series and hypergeometric functions associated with continuous fractions, modular forms and Fibonacci series, and even with noncommutative geometry . The imaginary golden ratio, in any case, reflects as in a mirror many of the qualities of its real part.
The Taijitu is a circle, a wave and a vortex all in one. The synthetic genius of nature is quite different from that of man, and she does not ask for unifications because not to arbitrarily separate is enough for her. Nature, as Fresnel said, does not care about analytical difficulties.
The diagram of the Taijitu becomes a flat section of a double spiral expanding and contracting in three dimensions, a motion that seems to give it an «extra dimension» in time. It is always a real challenge to follow the evolution of this process, both spiral and helical, within a vertical cylinder, which is but the complete representation of the indefinite propagation of a wave motion, the «universal spherical vortex» described by René Guenon in three short chapters of his work «The Symbolism of the Cross». The cross of which Guenon speaks is certainly a system of coordinates in the most metaphysical sense of the word; but the physical side of the subject is by no means negligible.
The propagation of a wave in space is a process as simple as it is difficult to grasp in its entirety; one need only think of Huygens’ principle, the universal mode of propagation, which also underlies all quantum mechanics, and which involves continuous deformation in a homogeneous medium.
In that same year of 1931 when Guenon was writing about the evolution of the universal spherical vortex, the first work was published on what we know today as the Hopf fibration, the map of the connections between a three-dimensional sphere and a sphere in two dimensions. This enormously complex fibration is found even in a simple two-dimensional harmonic oscillator. Also in the same year, Paul Dirac conjectured the existence of that unicorn of modern physics known as the magnetic monopole, which brought the same kind of evolution into the context of quantum electrodynamics.
Peter Alexander Venis gives us in a wonderful work a completely phenomenological approach to the classification and typology of the different vortices. There is nothing mathematical here, neither advanced nor elementary, but a sequence of transformations of 5 + 5 + 2, or 7 classes of vortices with many types and countless variants that unfold from the completely undifferentiated only to return to the undifferentiated again —or to the infinity of which Venis prefers to speak. The transitions from ideal points with no extension to the apparent forms of nature seems quite arbitrary without the aid of vortices, hence their importance and universality.
Venis does not deal with the mathematical and physical aspects of such a complex subject as vortices, and of course he does not apply to them the continuous proportion; on the contrary he gives us the privilege of a new fresh vision of these rich processes, in which the insight of a presocratic naturalist and the capacity for synthesis of a Chinese systematist meet together effortlessly.
Even if the Venis sequence admits variations, it presents us a morphological model of evolution that goes beyond the scope of ordinary sciences and disciplines. The author includes under the term «vortices» flow processes that may or may not have rotation, but there is a good reason for that, since this is necessary to cover key conditions of equilibrium. He also applies the theory of yin and yang in a way that is both logical and intuitive, which probably admits a fairly elementary translation to the qualitative principles of other traditions.
The study of this sequence of transformations, in which questions of acoustics and image are closely linked, should be of immediate interest in order to deepen the criteria of morphology and design even without the need to enter into further considerations.
A metric-free description would be, precisely, the perfect counterpoint for a subject as badly affected by arbitrariness in the measurement criteria as the study of proportionality. Naturally, mathematics also has several tools essentially free of metrics, such as external differential forms, which allow the study of the physical fields with maximum elegance. Then, perhaps, the metrics that physics deals with could be used as a middle ground between both extremes.
Thus, in this search to better define the context for the appearance of the continuous proportion in the world of phenomena, we can speak of three types of basic spaces: the ametric or metric-free space, the metric spaces, and the parametric or parameter spaces.
By metric-free space we understand the different spaces that are free of metrics and the action of measurement, from the purely morphological sequence of vortices above to projective geometry or the metric independent parts of the topology or differential forms. The projective, metric-independent space is the only true space; if we sometimes speak of metric spaces it is only because of the different connections with metric spaces.
By metric spaces, we mean those of the fundamental theories in physics, not only mainstream theories but also other related, with a special emphasis on Euclidean metric space in three dimensions of our ordinary experience. They include physical constants and variables, but here we are particularly interested in theories that do not depend on dimensional constants and can be expressed in homogeneous proportions or quantities.
By parametric or parameter spaces we mean the spaces of correlations, data, and adjustable values that serve to define mathematical models, with any number of dimensions. We can also call it the algorithmic and statistical sector.
We are not going to deal here with the countless relationships that can exist between these three kinds of spaces. Suffice it to say that to get out of this labyrinth of complexity in which all sciences are already immersed, the only possible Ariadne’s thread, if any, has to trace a retrograde path: from numbers to phenomena, with the emphasis on the latter and not the other way around. And we are referring to phenomena not previously limited by a metric space.
Much has been said about the distinction between «the two cultures» of sciences and the humanities, but it should be noted that, before attempting to close this by now unsurmountable gap, we should begin first by bridging the gap between the natural, descriptive sciences and a physical science that, justified by its predictions, becomes indistinguishable with the power of abstraction of mathematics while isolating itself from the rest of Nature, to which it would like to serve as foundation. Reversing this fatal trend is of the greatest importance for the human being, and all efforts in that direction are worthwhile.