From the Science of Complexity to Christian Faith Carlos E. Puente Rome, April 17, 2007

Transcription

1 From the Science of Complexity to Christian Faith Carlos E. Puente Rome, April 17, 2007 Cover. It is a great joy for me to be in Rome to share this talk that summarizes what I understand regarding the delicate encounter between science and faith. I wish to thank Fathers Rafael Pascual and Pedro Barrajón for the opportunity and for the hospitality I have received during this visit. I hope that you will find the talk interesting. Page 2. The thesis of this work is that we humans, with the gift of a soul, may learn from recent advances regarding natural complexity to deepen our faith in Christ and to build a better world. Page 3. The talk is divided into three parts: lessons from turbulence, lessons from chaos, and lessons from normality. Page 4. So, here we start the first part. Page 5. As you are going to note, this talk contains several simple games that illustrate how natural complexity happens. Here we find the first one of them. This is a game for kids that may be easily understood molding modeling clay. Drawn above is the bar of clay as it comes out of the box. The game starts cutting the bar by a given factor, say 70% from the left, as is shown by the vertical line. Then the game continues, piling up the largest piece to the left and enlarging the second piece, also to the left, so that they make two contiguous parts having the same horizontal length. Clearly, the first piece is higher than the original bar and the second piece is lower. The game continues repeating the process on each piece. At the next level there are four rectangles, whose masses are 70% of 70%, or 49%, 30% of 70% or 21%, 70% of 30% or the same 21%, and 30% of

2 30% that gives 9%. Clearly, 49 plus 21 plus 21 plus 9 gives 100%, in virtue of the well known principle of conservation of modeling clay, something that does not work well if there are little kids at home. The next level contains 8 pieces and the most massive rectangle continues to grow in height. As the base of the rectangle is half of half of half, or 1/8, and as the area equals 0.7 cube, the height gives 1.4 cube, which is 2.74 times higher than the original bar. Page 6. What the game produces may be calculated without major difficulty and for arbitrary partitions p and q. At the first level of the game, beneath the initial bar, the amount of mass is precisely p and q. At the second level one finds, in order, p of p or p squared, p times q, p times q and q squared, which is no more than the familiar expansion of p plus q all squared. At the next level one gets p plus q all cube for the masses from level to level are found multiplying. As may be noted, all is related to the famous Pascal s triangle and the binomial theorem and the game defines a properly named multiplicative cascade. Page 7. Here is shown what happens when the game is repeated many times. We obtain many rectangles with very small bases and the original bar is broken into many thorns. The vertical scale increases dramatically due to the successive pile-ups, 1.4 to the power 12 or units to the left, and certainly the object pricks us if we touch it from above. As is seen, the thorns are ordered in layers according to Pascal s triangle. The highest thorn, shown compressed for otherwise it would not fit on the page, occurs once and contains p to the 12 of the mass. The smallest rectangle on the right, and almost invisible as it always decreases, happens also once and contains q to the 12 of the mass. There are 12 large thorns that contain p to the 11 times q and also 12 small thorns (also invisible) that have p times q to the 11 and 2

3 there are 66 thorns that contain p to the 10 times q squared, and so on. The layers are finely intertwined and their densities increase as we enter into Pascal s triangle. Certainly it is not easy to walk this object, for to visit someone at the same level requires going up and down many times, for the thorns that belong to the same level tend to be separated by holes, and this is true for any level. If the game is played many more times, the additional fragmentation gives rise to infinite thorns that lack any cohesion whatsoever for they are supported by dust. Because of this reason, the object given by this game is known as a multifractal measure. To fully appreciate the empty structure found on every one of the layers here, it is pertinent to introduce another game for kids. Page 8. This one is also played molding modeling clay, but instead of cutting the original bar by a value of p equal to 70%, this time it is done by the middle, piling-up left and right so that there is a hole of size one third by the middle. As before, this game continues dividing each peace and piling-up following the same proportion. At the end and for each level, this process produces a multitude of rectangles that, by construction, never touch. Page 9. Clearly, this simple game is another multiplicative cascade that generates thorns of equal size that happen over an infinite and disperse dust. It happens that by varying the size of the hole of the second game, from 1/3 to an arbitrary size h, such adjusts the topological structure present on all layers of the first game. For example, while the denser layers require the propagation of smaller holes, those that are more disperse correspond to larger holes. 3

4 The moral is that the two games, although apparently different, are at the end intimately related to each other. Both are divisive cascades and the second game lives inside the first one on each one of its layers. Page 10. To appreciate even more the games and as they give rise to thorny objects whose scales grow to infinity, it is convenient to consider their accumulated masses from the beginning, zero, to a point x that varies from zero to one, and as a function of x. The two cascades on the left yield, following the dynamics of the games, the objects shown on the right. For the first game one obtains a cloud profile that contains a multitude of horizontal-vertical notches. The most notorious happens when x equals 1/2 and with a height of 0.7, for from the beginning to the middle of the object there is 70% of the original mass. Then, there is a notch at 1/4 with height 0.49 equal to 70% of 70%, and so on. For the second game one finds a host of plateaus that correspond to the holes on the cascade. The largest one from 1/3 to 2/3 has a height of 1/2, for the original mass was piled-up left and right. Then, there are two plateaus with length 1/9 and heights 1/4 and 3/4, and so on. Page 11. As may be seen, the accumulated sets are mathematical monsters that have many points where tangents may not be defined. While the first profile has no derivatives at any point, the second one does not have them at the extremes of the plateaus. As there are notches and plateaus everywhere, both objects are locally flat and their distances, from top to bottom, are equal to two units, one horizontal plus one vertical. It happens that such a property is universal. When imbalances p or holes h are propagated, such define thorns and dust that give rise to accumulated objects having maximal lengths of two units. The same happens when combining the games and also when chance is used to define a general 4

5 cascade with variable imbalances and holes from level to level. As the serrated profiles given by the cascades are locally flat everywhere, if one lands on them one would believe to have arrived to flat land. Because of this clear deception and due to the fragmentation produced by the games, such profiles are adequately known in Physics and Mathematics as the devil s staircases. Page 12. It happens that the first game for kids is related to the way turbulence happens in nature. When the Reynolds number, Re, is big, the inertia of the fluid (given by the product of the velocity and a characteristic length) subjugates the fluid s cohesion (given by its viscosity) and the fluid breaks up into an irreversible chain of eddies, that divide into more eddies, and so on. Those carry with them unequal amounts of energy that correspond precisely to the masses of the first cascade, with the imbalance p equal to 70%, and they eventually dissipate in the form of heat when the scale becomes sufficiently small. Notably and as reported by Meneveau and Sreenivasan in 1987, observations of fully developed turbulence along one dimension are consistent with the successive breaking of eddies with the energies given by the ratio For several flows, both natural and in the laboratory and that include atmospheric turbulence, boundary layer, wake of a cylinder and others, there are layers of energy that are horizontal rearrangements of what is produced by the first game for kids. So that the goodness of the universal fit found by such investigators may be appreciated, below is shown the relationship between the magnitudes of the layers and their respective densities. While the squares denote the observations of turbulence, the parabola corresponds to the mathematical cascade with densities that increase as we enter Pascal s triangle. Page 13. As the increase in entropy in natural turbulence happens 5

6 universally via a simple cascade, it is sensible to use the cascade processes to study how we humans create our own turbulence. After all, all of us, from Afghanistan to Zimbabwe, are confronted by inertial forces that break our internal cohesion and, when such happens, crossing the threshold of the Reynolds number produces turbulent and intermittent behaviors. For, even if we want to deny it, many times we make mistakes repeating the same error many times. While the first game may be used to vividly describe the proliferation of inequalities generated by our competitive and preferential instincts that result in marked cynicism, the second cascade may be employed to represent the appalling effects of discriminations and their related distrust and fear that result while imposing equality by force. Note how these simple ideas and their associated diagrams properly reflect not only the political systems that have governed the world, but also our own selfish postures and actions, for they sadly express: why the third world made of 2/3 of the people in the world, that is of all, live under conditions of poverty; why 6,000 kids die a day because of lack of water; and why we have been living for a long time in an era of violence and terror. Page 14. As history has proven that the second game does not work due to its emptiness, it is relevant to ask, even if such a question is inadequate to some, if the globalization of the first cascade is the solution to the problems we face. In this sense, it is useful to remember Pascal s triangle to do some calculations. If the imbalance p is set at 70%, as in natural turbulence, and if n = 20 levels of the cascade are considered, one may study where the modeling clay is located. As such, the 5, 10, 20 and 40% of the largest thorns contain, in order, 57, 70, 84 and 95% of the mass. 6

7 Page 15. This adjusts the skewed distribution of wealth of the most powerful nation on earth, the United States, for in 1998 the richest there had, for the same percentiles, 59, 71, 84 and 95% of the resources. Certainly, this is an undesired coincidence that provides however a truthful warning and a clear moral. If imbalances continue their propagation, the laws of physics and common sense assure us that the energies shall dissipate and that we shall all bite the dust. For the distribution of wealth of any nation on earth may be adjusted via a multiplicative cascade that gives a devil s staircase, even if such requires using variable partitions from level to level. Page 16. Based on these reflexions we may see that there is a common sense solution. This is: reverse the cascade direction in order to mend what is broken, live at low Reynolds numbers to avoid violence and the anxieties of our modern world and, to say it using the language of ancient prophets, 1 cut mountains and fill valleys in order to restore unity. For both in a graphical and mathematical way, unity is composed of an infinite number of spirals rotating outwards that are opposed to the negative spirals induced by the diabolic power of the air, a true redundancy, 2 that always go inside drawing at the end the 2/3 of a false promise that can not arrive at a totality. 3 For it is in fact the devil himself, the prince of the power of the air and ruler of the divisive and turbulent cascades, our universal enemy. For he is the one who falsely whispers in our ears that division is inevitable and that brotherhood is a utopia in this world where he is his prince of disorder and dust Lk 3:4-6, Is 40:4-5 2 Eph 2:2, Eph 6:12, Jn 12:31, Rv 12:9 3 1 Jn 3:8, Jn 8:44 4 Gn 3:14

8 Page 17. Based on these observations, and using simple geometry, we may see by ourselves that there is a only one possible solution. That is for us not to play divisive games and instead to maintain dynamically the uniformity of the original bar, that is, the unitive level zero based on the insuperable power of zero that universally yields unity: always practicing the proverbial without exceptions, that is without holes. 5 This is the only straight and solid condition that, by not containing thorns or dust, we may walk without fear, 6 that is, the way, the truth and the life as defined by Jesus, who my never sinning always kept the essential plain of God s law. 7 As the accumulation of the uniform bar of modeling clay clearly results in the one to one line and as such a ramp has a minimal distance from top to bottom of 2 in virtue to the Pythagorean theorem, we may see why the hypotenuse of the shown triangle, as it reflects radical and always unitive love, is the pathway of peace. For the maintenance of true unity travels securely and always with slope one, while the diabolic and divisive games produce rough devil s staircases that eventually are as long as the legs of the same triangle. The moral is that the most simple equation X = Y represents and provides the root of love. That is, clearly and once again, Jesus Christ our savior, beautifully symbolized by the geometry of the cross and his final posture on it. For as it may be appreciated via the truthful toboggan that takes us to the Origin, nobody may arrive to the Father but via that way, by him, 8 for it is impossible to slide by a devil s staircase. Page 18. To emphasize even more the uniqueness of true equilib- 5 Jn 13:35, Mt 5: Jn 4:18 7 Jn 14:6, Is 40:5, Mt 5:17 8 Jn 14:6 8

9 rium, here it is shown the improbable point in the midst of a sea of possibilities that express all cascades that combine imbalances p and holes h. 9 As there are devil s staircases everywhere, we may note that it is hypocritical to judge others if we are not in the point of balance, where by obedience we do not judge. 10 And we also note that it is true that it is easier for a camel to pass by the eye of a needle (one sufficiently large, of course) than to find the essential point that we should not miss. 11 At the end, and despite the difficulties, there is a faithful algorithm that guarantees our arrival to the point. If the cascade have been performed for a finite number of levels, there is not defined a plane with height 2 but a convex surface curved up. As such, balance may be achieved easily recognizing the gravity of our faults and accepting the incomparable sacrament of reconciliation. 12 For there is a marked difference between 2 and 2, as there is one between east and west, 13 as there is one between the divisive, negative and selfish spiral of number 6 and the always loving, unitive and positive of number 9, as there was darkness when Jesus died for us and crowned by our multiple thorns. 14 Page 19. As you may see, these ideas remind us of our personal and collective options: equilibrium or turbulence; calmness or violence; rectitude or wickedness; reconciliation or separation; integration and its symbol the slender letter s or division and its symbol that negates it for the love of money is the root of all evil; 15 unity and its positive spirals rotating outwards that symbolize the three times the apostle Peter acknowledged Jesus after the resurrection 16 or dust and the 9 Mt 19:24 10 Mt 7:4-5, Mt 7:1 11 Mk 10: Jn 1:9, Mt 6:9-15, Mt 11: Ps 103:12 14 Mk 15:33-37, Mk 15: Tm 6:10 16 Jn 21:

10 lying and diabolical fraction, it should be clear, 17 filled with negative and selfish spirals that also symbolize the three negations by Peter before the cock crowed twice; 18 completeness or emptiness; and life or dissipation that is death. Page 20. To summarize this first part, I like to share a poem-song with you. It is called pathways: Two options before us two pathways ahead, the one is the longest the other straight. We journey directly or go by the legs, we follow intently or end up in pain. By walking the flatness or hiking the spikes, we travel in lightness or take serious frights. The incentive is unity and the call proportion, the key is forgiveness and the goal true notion. In wandering wickedness there is never fruit, but in ample humbleness one encounters the root. 10 Page 21. The second part of this talk comprises some lessons 17 Rv 13:18 18 Mk 14:66-72

11 from chaos, a celebrated modern theory that, as you shall see, provides additional truthful symbols and profound and unexpected connexions. Page 22. To summarize the matter of chaos it is convenient to start with the prototypical non-linear equation X k+1 = αx k (1 X k ), that defines the shown logistic parabola. Such an equation describes the normalized evolution (between 0 and 1) of a population, say rabbits, as a function of time. When there are few rabbits, the curve tells us that there is a tendency to increase in the next generation, but if there are too many rabbits the tendency is to decrease, for the limited resources hamper a greater growth. As it may be seen, if the number of rabbits is equal to its possible maximum, the population extinguishes in the next generation. Here is shown the evolution of a population ruled by such an expression, when the value of the parameter alpha, that may be any number between 0 and 4, is equal to 2.8. As is seen, from an initial value X 0, and following the vertical and horizontal lines to the same hypotenuse X = Y, one arrives, after several iterations, to a unique fixed point X, that corresponds to the non-zero intersection of the parabola and the straight line. As you surely know it, X depends exquisitely on alpha, as it follows. Page 23. When the parabola is below the one to one line, that is when alpha is less than one, as in the upper left case, X converges to zero. This happens as the slope of the parabola at the origin is less than the one of the line. When the parabola crosses the threshold, the origin is no longer reached in any case, as it repels. For example, when alpha is between 1 and 3, as in the upper right case, the dynamics converge to the intersection shown on the previous page. And if alpha increases beyond 3, there are oscillations, first every two generations and then every four, in a chain of bifurcations, as shown below. 11

12 Page 24. All bifurcations happen very quickly, in powers of two, when alpha increases up to a value α, where there is a multifractal attractor, similar to the one encountered in atmospheric turbulence, filled with thorns and dust, as it shall be shown later on. When alpha exceeds α, there are sometimes periodic repetitions and, more commonly, non-repetitive behaviors and subject to extreme variation in initial conditions, the well named aperiodic strange attractors that describe the wandering forever of the also well defined chaotic behavior. Page 25. At the end, the famous diagram of bifurcations, and shaped as a tree if we rotate it 90 degrees, summarizes the incredible behavior of the simple logistic parabola. As may be seen, varying alpha indeed has profound implications. Notably, the diagram contains periodic behavior that encompass all natural numbers, something difficult to foresee without the technological advances of modern computation. Page 26. Here is observed in more detail the tail of the diagram. As is seen, strange attractors, plotted as vertical collections of dots, are very common after α. As such are aperiodic sets, they end up being dusty as in the games for kids. It also happens that in all periodic bands, in the buds of the tree, as in the notorious one by the middle and corresponding to period 3, there are small copies of the same tree, which reflect the exquisite fragmentation of the object that looks alike or self-similar at diverse scales. Page 27. The diagram of bifurcations includes in fact a multitude of thorns that correspond to multifractal measures in the chaotic tree. The first one occurs at a value of alpha equal to α, that, as may be seen, combines imbalances and holes. There are thorns galore in all accumulation points of buds corresponding to the infinitely many periods encountered in the tail of the diagram. 12

13 Page 28. As some of you surely know, the tree of bifurcations is also known as the Feigenbaum tree, the fig tree, in honor of Mitchell Feigenbaum who proved for the first time some universal properties about the object. It happens that the bifurcations occur in an orderly manner both in their openings and in their durations. As is seen in the diagram, the bifurcations cross the line X (equal to 1/2 for the logistic parabola) in an alternating fashion and the openings decrease in such a way that the ratio from bifurcation to bifurcation tends to the number F 1, the first constant of Feigenbaum. Similarly, the duration of the bifurcations decreases quickly and the quotient from one bifurcation to the next tends to the number F 2, the second constant of Feigenbaum. These assertions show that there is order in the pathway that leads to chaos, but such do not imply, as often it is affirmed erroneously, that chaos itself is ordered. Page 29. Feigenbaum s numbers are in fact universal, for such are valid for a countless number of equations that give rise to other chaotic trees. The iterations of uni-modal functions, as the ones depicted here left and right and related to the shown equations, always yield, by increasing the parameter alpha, a straight root, a tender branch, bifurcation branches, and, in an intertwined fashion, periodic branches and the dusty foliage of chaos, that then corresponds to fig leaves. In all cases, the openings and durations of bifurcations happen precisely at the speeds given by F 1 and F 2. Chaos theory turns out to be relevant in several fields that include Ecology, Chemistry, Physics, and Economics. Among the most pertinent results it is worth emphasizing the discoveries of Albert Libchaber and Jens Maurer in 1978 in regards to liquid helium, for the pathway to turbulence in the dynamics of convection for such a 13

14 fluid corresponds to the ideas here explained, when alpha denotes the heat added to the helium. Page 30. Now, close to the end of this lesson, it is pertinent to study carefully some subtle details that take place in the plenitude of chaos. When the heat is maximum, that is when alpha is equal to 4, it would appear that all the dynamics would wander by a dusty strange attractor that encompasses all the interval [0, 1], as the shown parabola appears to imply via a signal that travels everywhere and without apparent repetition. But this is not true. Depending on the initial value X 0, there are also other cases that give rise to oscillations forever and for any period, as illustrated in the shown diagram for an example having period 3. This figure shows the value of X k as a function of the generation k and includes not only the three values eventually repeated, but also the successive pre-images of all pathways, thirteen generations to the past, that end up in the highest of the three points. Page 31. All these is relevant because there are initial values X 0 that escape the dreadful consequences of chaos, for such return dynamically and vitally to the root of the tree. These are the pre-images of zero that find the way despite the most implacable heat and that, by arriving to the middle point, pass by the one to finally rest at the origin. Notice how these observations remind us of the parable of the weeds, for the wheat in the pre-images of zero is finely surrounded by undesired behaviors that unfortunately end up in the fire. 19 Note how the ideas also remind us of the three exalted friends of the prophet Daniel dancing in the highest heat and without consequences and the protection of the prophet himself in the lions den, 20 for as the 19 Mt 13: Dn 3:1-92, Dn 6:

15 Psalmist assures us and as we may see it geometrically, though a thousand fall at your side, ten thousand at your right hand, near you it shall not come. 21 Page 32. The notions so explained also suggest more common sense reflexions. Without any doubt, it is best to avoid chaos and its truly infernal turbulence, for missing the point, even if by a rather small amount epsilon, has, with all probability, tragic consequences. For the famous butterfly effect, that expresses the incredible variability of chaotic dynamics, does not provide us with good options, for such always leave us irremediably trapped in an empty strange attractor in which there is no rest. 22 So, the advice is (even if it looks like three) and not precisely in relation to our rabbits: come down the chaotic tree as Zacchaeus did, the famous little one, who by amending his errors diminished his intrinsic heat to find the root and there accepted God s salvation for him and his family, 23 the same symbolic root in which Jesus saw the future apostle Nathanael, that is under the fig tree; 24 abandon yourself to the chosen threshold, X = Y, 25 to Jesus himself who is the door and the narrow gate (and how narrow it is), 26 to arrive to the love of the Father in the gentle state X = 0 of the Origen; 27 and move away from non-linearities when alpha is greater than 1, so that by not amplifying without proportion, you may escape the dust that is death. 28 These ideas represent beautiful and accurate connexions between the science of chaos and Christian faith, ideas that are perhaps unex- 21 Ps 91:7 22 Jb 40:12-13, Mi 7:17, Hos 8:7 23 Lk 19:1-10, Mt 18: Jn 1: Mk 8:34-35, Lk 9: Jn 10:9-11, Mt 7: Mt 11: Rom 5:12 15

16 pected given their geometric symbols and their universal concepts. But there is even more, as I shall try to explain next. Page 33. Could it be that the scientific fig tree has a prophetic value? Could it be that via modern science we are being given a merciful clue that illuminates the ancient word? As you know, Jesus cursed an allegoric fig tree without fruit that in consequence dried up to the root. 29 The chaotic fig tree shown here happens not to have any visible fruit, it contains leaves of dust that properly remind us of the leaves used by Adam and Eve to cover their shame, 30 and it is consistently cursed above the root by the inherent disobedience of crossing the threshold X = Y. 31 As you may see it for yourselves, this fig tree (and the other chaotic trees previously shown) has a tender branch or branches that literally contain a great multitude of buds in its periodic bands. As the foliage in the trees symbolize the way we progressively move away from the root of goodness and as the trees also contain above multitudes of thorns equally symbolic, 32 it is normal to ask ourselves if these ideas are a preamble of a near summer, as Jesus expressed it on his eschatological discourse. 33 For in the diagram it may also be appreciated why the ax lies at the root of the trees, as expressed by John the Baptist, 34 and why Jesus said to his astonished disciples that they could also curse the fig tree, 35 no doubt in the same way that he rebuked the wind, 36 for both cases reflect his triumph against the evil one and his deeds Mt 21:18-22, Mk 11:12-23, Lk 13: Gn 3:7 31 Dt 30:15-20, Ps 37:22 32 Mt 13:22 33 Mt 24:32-35, Mk 13:28-31, Lk 21: Mt 3:10 35 Mt 21:21 36 Mk 4: Mk 16:17-18

17 Page 34. Although by definition the notions herein and other ancient signs do not allow us to set exact dates, 38 they help us to sketch more options that heighten the goodness of being prepared and vigilant for the triumphant return of Christ. 39 Such options are: order or disorder; peace or chaos, to diminish as John the Baptist did when he knew about Christ or to augment believing that we are more than him; 40 obedience or rebelliousness; to be under the threshold X = Y or above; to receive in consequence blessings or curses; to rest eternally in heaven or to wander in hell. Page 35. To summarize this lesson, I like to share a fragment of another poem-song. This one is called a modern fig tree: Foliage of disorder trapped in empty dust, jumps astir forever enduring subtle thrust. Crossing of the outset leaving faithful root, looming tender offset failing to yield fruit. Cascade of bifurcations, increasing heat within, inescapable succession of branches bent by wind. Sprouting of dynamics attracted to the strange, o infinity reminding at the origin: the flame Mt 24:36, Acts 1: Mk 13: Jn 3:30

18 Page 36. The third and last part of this talk explains other lessons from normality. Page 37. As you surely know it, there is a beautiful and harmonic curve called the Gaussian bell that is always found when adding independent events among themselves. In virtue of the famous central limit theorem, such a curve, given by the exponential formula shown with argument x 2 /2, adjusts very beautifully, for example, the distribution of heights of those visiting Rome today, like me. As I shall explain next, many natural processes and others related to the violence induced by mankind do not exhibit normal or Gaussian traits, but rather adjust themselves to power-laws as the celebrated Pareto distribution shown here, in which the density does not decay exponentially, but as a function of the argument raised to apowera +1. As is illustrated here, there is a marked difference between these two types of curves, which have been drawn in arithmetic scales above and doubly-logarithmic below to distinguish them better. While the Gaussian bell, drawn here only in its second have, has a middle value precisely at the rounded peak shown, the power-laws do not have such a middle ground and, as such, they have no inherent scale that defines them. As it is seen, the normal curve decays quickly to zero, put the power-laws, having heavy tails, do so slowly, for they always have a non-trivial amount of mass in their extremes. Page 38. In the past few years it has been discovered the ubiquitous presence of power-laws in diverse processes related to natural complexity. Here you observe the one that is perhaps the most famous of them and which defines the Richter scale of earthquakes, with the events of higher magnitude having a lower probability of being exceeded and with the linear graph in doubly-logarithmic scales 18

19 indicating that there are no earthquakes of characteristic sizes. Such power-laws are also found in other manifestations of utter violence such as avalanches, hurricanes, floods, volcanic eruptions, forest fires, and more. Page 39. In addition to natural evil, power-laws also appear prominently in diverse situations related to our own actions as human beings. As observed for the first time by the Italian Vilfredo Pareto, such laws are found in the skewed wealth distributions in the world, both inside every nation and in the world as a whole. Here is seen the income distribution in the world, from its percentiles 30 to 85, in which it may be appreciated the painful widening of inequalities in two lines with negative slopes and in doubly-logarithmic scales from 1960 to This is truly sad for, as we mentioned it earlier, two thirds of our brothers and sisters live under conditions of poverty. Page 40. As noted by the physicist and pacifist Lewis Fry Richardson immediately after world war two, the distribution of human conflicts, from those associated to gangs to the major wars, also give a power-law. Incredibly, or maybe not, all the data reflecting our malign violence gets aligned in one sole expression that reminds us potently that to truly avoid the next great conflagration me must end the little conflicts, included in them, and in virtue to a lack of a characteristic scale, those that happen inside our hearts. Page 41. From all these ideas arise more elements that support our common sense. Asviolence generates power-laws, it becomes relevant to study how such are produced so that, by avoiding such mechanisms, we may find harmony and peace. Such divisive laws are found via the already studied multiplicative cascades, via preferential connexions on human networks, via the so-called highly optimized tolerance that reflects the disorganization in our modern 19

20 world, and via the celebrated self-organized criticality sketched here, that is, via the implacable accumulation of energy that grows a pile of sand at the beach until its excessive slope gives rise to predictable avalanches of unpredictable sizes. All these simple and modern ideas regarding complexity allow us to see clearly that the evil we face is not due to an invisible hand that manipulate us, although maybe yes if we allow it, but rather to our own actions that we ought to rectify. For as may be appreciated, humans without scale abuse their essential power and by their selfishness and greed yield power-laws. For at the end, the commonly encountered power-laws reflect three negations: in the negative of their slopes and in their doubly-logarithmic scales. Page 42. This last assertion may fully be appreciated understanding that the exponential function, the inverse of the logarithm, is intimately related with love, for such is the only solution to the mandate by Jesus to love one another, which translated into the language of calculus says integration without differentiation. Although this may appear to be strange, note how the number e properly denotes the supreme power of our communion with Jesus, for after reading carefully the eloquent story on the fifteenth chapter of the Gospel according to John we see him, the true vine, in the number one and us accepting the cross in 1/x, and, by maintaining ourselves with him and him with us, we aquire the power of the same cross, that in the glorious limit of a complete pruning, allow us to love as he does, to everyone and without differences. Page 43. Now, almost to finish up this talk, I like to share with you some results from my own research that provided an unexpected seed to the improbable jigsaw puzzle I am showing you between modern science and Christian faith. It happened that trying to comprehend the complex structure of 20

21 river networks in nature, one good day I was experimenting with two simple equations as the ones shown here, w 1 and w 2, that have z as a parameter. It occurs that by iterating such linear functions arbitrarily, with a positive effect of z on the first function and a negative effect on the second, one finds successive points (x, y) that give rise, after many such little points, to a wire, that, as you see on the left, looks like the profile of a mountain when the parameter z takes on a value of 0.5. My original idea was to study if the notions could be extended in order to construct not profiles but mountain surfaces that would allow me to study, varying proper parameters, how a drainage network related to such a topography evolved. Even though such an idea did not produce fruit, in another good day it occurred to me to study the shadows produced by the wires over the x and y axes, not only as a function of the parameter z but also in relation to the way the iterations are carried. It happens that if a biased coin is used to guide the iterations, for example one that defines the use of w 1 70% of the time and function w 2 the remaining 30%, projections dx and dy, as shown here, are found. As you may see, dx is no more but the multifractal object found earlier, but dy depends on the value of the parameter z. On the left is found an intermittent and highly complex object that appears to be guided by chance, but such a graph, that reminds us of geophysical data, does not depend on chance for it comes from a couple of deterministic objects, the multifractal and the wire. As may be seen on the right, when z tends to its maximum value of 1, the wire grows in dimension and, by filling up the plane due to its multiple oscillations, yields a Gaussian bell curve as a projection. This result turns out to be surprisingly universal, for changing the bias on the coin always gives rise to a bell, not the same one of course, 21

22 and all of these happens via the same limiting wire. These ideas sketch a Platonic approach to natural complexity, for in the same spirit of the allegory of the caveman found at The Republic, what we observe, dy, may perhaps be just a shadow defined from higher realities, that is the wire from x to y, via a suitable illumination, in this case dx. The ideas are also Platonic in a romantic sense for from them it has been arisen the desire to elucidate if natural complexity could in fact be just a shadow. Page 44. It happens that the notions may be extended to higher dimensions. This requires equally simple equations as the previous ones but with more parameters, in this case the scalings r s and the angles θ s that replace the parameter z. As you may see, based on these ideas one may generate wires from x in the vertical to the plane (y,z) in the horizontal, and this allows us to calculate highly complex projections or shadows but now over the plane (y,z). Page 45. As an example, here is a pollution pattern found as the shadow of a wire defined over four dimensions, from x to the space (y,z,w), and its respective shadows in diverse planes. Similarly, and this is something we try to do in detail, the ideas allow us to define diverse geophysical processes that include rainfall, and all of this is done in a Platonic way and without the need of invoking chance. Page 46. In the limit, when the magnitude of the scalings r s tend to 1, the wires fill up the volumes in which they live and in this superior dimension there are found bells independently of the illuminations dx. Here you see how based on a multifractal dx and via a wire from x to (y,z), drawn to the right, one finds a bell over two dimensions over (y,z), as shown from above in dyz and by the sides on dy and dz. 22

23 As you can see, these cases transmute arbitrary thorns and dust into the harmonic and smooth bell. As the bell is related to the conduction of heat via the process of diffusion and Fourier s law, the wires that fill up space end up inverting the prescribed physical order, for they vividly transform the dissipation implicit in the turbulent multiplicative cascades and its multifractals into the maintenance of energy. These results, when they appeared for the first time, urge me to think what could those wires represent. It was meditating on their universality that I concluded that such had properties related to those of love. For, what else could transform dust and dissipation into something harmonic and conducting but love? With the passage of time, we were able with my collaborators to prove mathematically the truthfulness of the Gaussian limit in the one-dimensional case, but the two-dimensional case remained elusive. It was that way how on another good day it occured to us to study how the circles were formed, drawing not the final summary of 15 million points as shown here, but successive groups of iterations, for example from 2,000 to 2,000. Page 47. What we found is surprising. The iteration of simple functions defines in the limit exquisite decompositions of the twodimensional bell. The two images shown here are just examples of an infinity of patterns that as they superimpose each other from perfect circles and truthful bells, and whose specific and unexpected geometries depend on the precise sequence used in the iterations, that is, in the succession found while tossing a coin. As you can see, in the central limit there is hidden order in chance. When the angles θ are synchronized, 60 degrees to the left and 90 degrees to the right, one travels as from glory to glory and a vital 23

24 beauty manifests itself that incites us to praise. 41 Page 48. Today we know that all ice crystals are found inside the bell. The ones shown here were found by my lovely wife with good patience, filling templates of known crystals step by step, building up sequences of suitable iterations via groups guided by fair coins and using two functions with angles equal to 60 degrees. Page 49. The bell also includes rosettes that adjust diverse biochemical compounds. Prominently, the one shown to the right is found using two functions with angles equal to 36 degrees and iterating them according to the binary expansion of the omnipresent number π. To the left you observe the rosette with 10 sides corresponding to the DNA of life. This way, the bell provides an improbable design of such geometry, one that depends on the precise succession of the first 40,000 bits of π. Page 50. But there is more. When the parameter z has a positive effect on both functions (before it was negative on the second one), one gets a wire shaped as a cloud and not a mountain, and now the limit, when z tends to 1, defines a universal bell centered at infinity. In a mystical manner, this powerful transformation, maximum positive, takes it all to the clouds and via the rhythm of the bell that evokes complete liberty, and this is done in such a way that it filters all disorder and annuls any entropy existing in any non-discrete illumination dx. How not to appreciate in the vital transformation from thorns and dust to the unitive bell and from obscure dissipation to luminous conduction an essential call to our Christian faith? For in the directionality of the diagram from x to y we may exclaim with Saint 41 2 Cor 3:18, Ps 139:

25 Paul: Where, O death, is your victory? Where, O death, is your sting? 42 Page 51. In the same spirit and close to finishing, I wish to show you with humility and joy the one that I believe is the most beautiful of diagrams I know. Here you observe the same wire of the previous page but illuminated by equilibrium, to complete a majestic trilogy. Here I appreciate, symbolically of course, the Father, powerful in heaven and denoted by π in the geometry of zero and the origin; the Son, always perfect and positive until the cross, expressed by the 2 of the straight hypotenuse; and the Holy Spirit that comes from the Father and the Son and denoted by e, who transforms us towards love if we allow him. Here there are the three infinite and irrational numbers present in the equation of the beautiful bell that vividly remind us of the invincible light of our sublime freedom. In the diagram one may also appreciate, from x into y, key events in the life of Jesus that include, his baptism with spirit and fire, 43 the power of his miracles, the gift of the Eucharist, 44 his transfiguration, 45 his glorious resurrection, 46 and his ascension into heaven. 47 And clearly, there we can see also the assumption of the blessed virgin Mary to heaven, which augments our faith that, despite our thorny multifractals, we would arrive one day to our best destiny, and perhaps raptured if the Lord returns when we are still alive. 48 Page 52. From this lesson there appear yet more options. Those are, the bell or power-laws, meekness or stubbornness, the normality of the sons of God or violence, conduction or dissipation of energy, 42 1 Cor 15:55 43 Mt 3:11 44 Mt 26: Lk 9:29 46 Lk 24: Lk 24: Thes 4:

26 plenitude or loneliness, light or darkness, what is positive or what is negative, truth or lies, love or selfishness. Page 53. To summarize this section here is a fragment of another poem-song. This one is called the amazing bell: The bell peals silent, o o reflecting its peace, and inside it gathers lovely masterpiece. Symmetric pure beauty, o o o mighty delight, this limit in fullness stores life s designs. Such vessel contains, o o alephs of all tastes, diatoms and crystals including DNA. But there is a case, o o reason to this song: the forward selection that raises it all. There is clear choice that rotates the 8, by loving sincerely we surely converge. Page 54. To summarize, this is what I have shown you. In consonance with Sacred Scripture, simplicity is always much better than complexity. We ought to refrain from doing what is natural, like bombing the enemy, and do instead what is normal, such as loving our enemies and praying for those that perse- 26

27 cute us. This means that if we required it, we ought to reclaim our inherent scale by rectifying our actions, that is, recognizing and amending our faults. So let us avoid accumulating energies living in serene equilibrium; let us not cross the chosen threshold remaining always in the root; and let us maintain the central limit always in the positive case. This gives p =1/2, α 1 and z 1, that guarantees, in Spanish and in any other language, peace. May the peace of the Lord be with you all. Page 55. Now to conclude what I hope has been a good day, I invite you to praise singing this song entitled. I shall sing a little bit and you will know how it goes: is justice that illuminates, is balance that fascinates:. is the incarnate alliance, is the established reliance:. is true word that matures, is a spiral that endures:. is the precious resting place, is the state of mighty grace:. 27

28 28 is smoothness that esteems, is a hummingbird that gleams:. is the short and precious root, is the weaving of the truth:. is a future that forgives, is crowned science that is:. is the ever tender tune, is the impartial tribune:. is all innocence that heeds, is a garden with no weeds:. is the simple clear sign, is the majestic design:. is independence that heals, is matrimony that shields:.

29 29 is the real chaste embrace, is the goodness of a yes:. is a smile that edifies, is a spin that rectifies:. is all gentleness in us, is the everlasting plus:. is inspiration that calls, is growing to be small:. is the forgotten territory, is the improbable story:. is revelation that nests, is surrendering the rest:. is the dustless short incline, is the faithful narrow line:.

30 30 is renouncing all spears, is experiencing no fears:. is the perennial giveaway, is pure life with no decay:. is the only perfect remedy, is loving, even the enemy:. For more information, please visit the web site: and in particular the pages: and complexity christianity.htm