The crop circles, images in the wheat that appear mysteriously during the night, mostly in the British countries, fascinate and intrigue the scientists around the world from hundreds of years. Their origin is unknown, although they declare an evident geometric feature, which brings to mind an ancient symbolism, essential for pagan worship. There are many theories in this regard; including the one that arouses most interest it is that they have a symbolic form, that explicitly cites the concept of energy, i.e. that serves to represent and to tell the origin of all things: the energy that permeates the Earth. Crossing the symbolism of numbers, considered as (but not only) "signs", the present work intends to investigate the possibility of an analytical link between the physical concept of mechanical energy, as it is understood in classical mechanics, so in the sense of work done by a system of forces for a system of displacements (dual of a system stress -strain), with the evocative images suggested by the mysterious signs in the wheat, highlighting some of the most representative forms in this sense, with the aim to re-find (or re-invent) a new representation of the energy. Key-Words: crop circles, signs, numbers, energy, physic
1 Introduction It is known that from the point of view of Eastern philosophy, the cosmic energy is the source of all the living things, and it is the essence of the movement, processing and life. The primary energy is manifested in heaven (Yang) and in the earth (Yin). So the meeting of the cosmic energies with the nutrient energies of terrestrial, generates man and woman, mediator principle of Yin and Yang. In Western philosophy, the word "energy" derives from the late Latin energīa, in turn, from the Greek ἐνέργεια (energheia), a term used by Aristotle in the way of effective action, consisting of en, particle intensive, and ergon, capacity to act. During the Renaissance, inspired by the thought of Aristotle, the term was associated with the idea of expressive power. But only in 1619 Kepler [1] used the term in the modern sense of energy, namely as the ability of a body or system to do work. In classical physics, the energy is defined in various forms, each of which has its own energetic equation. The main forms of energy (each expressed through its own equation) are: mechanical energy, classically defined as the sum of potential energy and kinetic energy, chemical
energy, biological energy, electricity, light or radiant energy, thermal energy, electromagnetic energy, nuclear energy. Such forms of energy can be transformed one into another, but each time that such transformation occurs, a part of energy (more or less consistent) is inevitably transformed into thermal energy (that is, it produces heat); one speaks in this case of "dissipative effects". At the beginning of the twentieth century, Max Planck [2] introduced a new theory, based on the fact that some quantities of certain physical systems, such as energy or angular momentum, can vary only of discrete values, also called "quanta". So, if energy in classical physics is a continue scalar property stored in a system, in quantum mechanics the energy is "quantized", i.e. it can take a number of discrete values (or "energy levels"), all multiples of a quantum of energy, which represents the lowest amount of energy that can be stored in the system. These systems are called "linked", because the energy of the particle never exceeds the potential barriers.
2 Brief compendia of links between mathematic and signs Only two things are infinite, the universe and the human stupidity, and I'm not sure about the former (Albert Einstein) The infinite symbol was probably created by John Wallis, that used it in one of his first texts, De Sectionibus Conicis (1655), where is was called to express himself about the concept of "bent infinite", and then in Arithmetica Infinitorum (1656), when he ideated a simple infinite product able to find the π value
(1)
Between the various hypotheses before his death, the most reliable is that his inspiration was founded on a very similar image used by the Etruscan to indicate the number thousand, often used to say "many".
Fig. 1: Etruscan system of numeration But considering the various hypothesis, the most fascinating is the association of the infinite to the alchemic ancient symbol of , the snake that bites its own tail, from ever associated to the cyclical nature of all things, the neverending return, base of Gnosticism and Hermeticism [3]. The following picture shows a crop circle formation appeared Liddington Castle (Wiltshire) in June, 17 2001. It could represent the Uroboro around the Earth, and if we like to work with the imagination, we can see also the famous Lay Lines [4] that wrap around the earth. The Lay lines (or "pathways of the dragon", according to the principles of Feng Shui [5]) are currents of vital energy that cross the Earth in the direction of the parallels and meridians. In realty, the Ley Lines are alignments of a number of places of historical and geographical interest such as ancient monuments and megaliths, especially belonging to the British area, within which a large amount of energy is suspected or imagined that is given off. Flows of energy similar to rivers that intersect in a straight line on the Earth's surface, according to the Geobiology they have a variable width ranging from 3 to 30 or 40 meters, and an estimated length from 5 to 30 meters.
Fig. 2: Liddington Castle (Wiltshire), 2001. One of the many theories on the causes of location of crop circles, is the one linked to the presence of Lay Lines just in that position. In some pictures, the snake is shown half white and half black, recalling, in this way, the ancient oriental concept, pivotal of Tao philosophy, represented by Yin and Yang, that means the dualistic nature of all things, and above all the concept that the opposites aren't in conflict with each other, but their existence is necessary for the equilibrium of all creation, phenomenal and not. It's interesting also emphasize that the notion of infinite, in the "rational" sense of the term, rooted its raison d'être only in 1400, as dichotomy of finite. In fact, already from the Pythagoreans, the only concept that was taken into account was the idea of finite, as synonymous with perfection and completeness. So, extending this doctrine toward the numbers, only odd numbers were considered perfect, because their geometrical representation gives definite figures. With Democritus (450 b. C.), in Testimonies and Fragments, the idea of infinite began to take a form in the philosophical thought, and an overcoming of the ancient archaic vision has been beginning, and the cosmos has not been considering anymore as a finite and limited reality. But the real inclusion of the infinite notion as base of metaphysical philosophy will occur only in late Hellenistic period, of Neoplatonic tradition and culture, when the metaphor was used in religious matters with the aim to collimate the principles of faith and the rational human thinking. This concept became more significant through the ideas of Nicola Cusano, who in De Docta Ignorantia (1440), tried a reconciliation between finite and infinite, between man and God, stating with his theory of coincidentia oppositorum that it is possible to transform the infinite into an absolute dimension in which man will be able to increase his knowledge.
Naturally, man for his nature (finite), can be never fully understand the secrets and the knowledge of the absolute, but reasonably and potentially he will be able to have this perfection through the progression of one's spirit. With Giordano Bruno (1548-1600) the metaphysical and mathematical developments of the infinite concept are significantly radicalized in the naturalistic and pantheistic sense. In his famous Cosmological Dialogues, Bruno [6] processes his own idea, as infinite space and time. The Universe is not simply a finite region, but it extends itself without limits in all directions and in a-temporal manner. His conception had a great extension: he arrived to conceive the existence of infinite worlds, infinite solar systems, infinite living species and intelligences. These statements so daring for those times, caused many problems and theological debates. Already St. Augustine thought and stated that it was not possible to conceive that there was life on other planets, because if that were the case, we would have to accept the idea of a God willing to incarnate in other worlds, with a pietas and crucifixion repetitive in time, which would have declassified the idea of the divine in the service of men with the aim to redeem they from sin. This concept of a God subservient to the cross and with the ability to incarnate whenever there had been a need, was inconceivable for the Church to such an extent as to define heretic anyone would embrace such assumptions. It is true that even Cusano somehow came up at these theories, but he was a Cardinal, therefore equipped with a high power in the sphere of the Church, and this fact gave him the possibility to enjoy of some privilege. For Giordano Bruno was not the same, and therefore the idea of infinite condemned him to the stake at the dawn of the 1600s, however leaving, despite death, his conception of parallel universe, that comes to be part of the Copernican theory which marked the birth of modern astronomy. A momentous time, if we consider that the concept of infinite abandons the philosophy for enter in science. And in fact, in such a revolution, the thought of the philosopher Spinoza has been stating, who conceived the idea of infinite as an immanent attribute of reason, fulcrum on which he will argue the current of the German idealism, promoter of the great debate about the absolute, to which, in recent times, reflections of Auguste Blanqui and Borges will be joined. And it is the latter, Borges, who in the Metamorphosis of the turtle, in Otras Inquisiciones (1952) writes about the concept of infinite from which was totally fascinated.
"... I'm speaking about Infinite. Sometimes I whished to fill its moving history. ... The great Hydra (swamp monster that is like a foreshadowing or emblem of geometric progressions) ... that should crown the sordid nightmares of Kafka, and his central chapters not ignore the guesswork of that remote German Cardinal Nikolaus Krebs (Nicola Cusano), who saw a polygon with an infinite number of angles in the circumference, and he wrote that an infinite line would be a straight line, it would be a triangle, it would be a circle, it would be a sphere... Five, seven years of apprenticeship metaphysical, theological, mathematical, could put me in a position to plan decently perhaps this book. Needless to say, life prohibits to me the above hope and also this adverb. ... " But Borges did not disdain even the enumeration, that exercised a great interest over him, to the point that he wrote: "Theoretically, the number of numeration systems has no limits. The more complex, in use to the gods and the angels would record an infinite number of symbols, one for each integer. The easier requires only two of these: zero is written 0, one 1, two 10, three 11, four 100, five 101, six 110, seven 111, eight in 1000 ... " (Leibniz's invention that apparently had been stimulated by the celestial hexagrams of I CHING). It is impossible to enumerate all the logical-mathematical paradoxes about the infinite: we would like to remember the Paradox of Albertus de Saxonia (1316-1390), which speaks about of beams of infinite length, typology often taken as pattern in the texts of Structural Mechanics [7] where, of course, there is no reference to the paradox. It is: the volume of a beam of infinite length is equal to the volume of the whole Space. Undoubtedly related to the concept of infinity are both the representation of the number 8 and a particular algebraic curve in the shape of eight inverted, the Lemniscate of Bernoulli that is, as it is known, a particular case of the oval of Cassini (1680). 2.1 The enigma of the number eight The graphic representation of the number eight (as well as the infinite symbol) is a curve that the human hand can repeat indefinitely, without ever disconnect the pencil out of the paper. The cases in which the number eight achieves a particular importance (not only symbolically) are various and heterogeneous: from the Esoterism to the Metaphysical Sciences, to the Mathematic, to the Classic Physic etc. Some examples of what, are listed below, between science and no-science:
the number 8 is universally regarded as the cosmic number
888 is the number of the portal toward the infinite, the eighth dimension!
8 Hertz is the Schuman's frequency, the heartbeat, or the pulsation of the Earth
8 hertz is the frequency of the mystics healers
Fig. 3: Evolution of the number 8 from the Indians to the Europeans [8]. Apparently, among other things, the symbol has taken inspiration from the shape of the Via Lactea, since in some ancient texts it was considered a huge serpent of light residing in the sky and that circled the Earth. Then, it was widely used by mathematicians from all over Europe, but in a rather random. In fact, until the mid-nineteenth century, the dominant philosophical idea was derived from Aristotle, so that the actual concept of infinite, not only was not included, but there was neither in the empirical sciences, nor in those math. The sign of infinite was only designed to express a numerical continuity, endless, inaccessible and unattainable, function and domain of a single Almighty Creator. Additionally, the use of rigorous symbols to express concepts, was equivalent to the alchemical invention of complex codes, through the which use it was possible to achieve solutions, decipherments that in a sense revealed the secrets of God. 2.2 The Lemniscate In mathematic the Lemniscate is an algebraic curve approximately with the form of an 8 rotated of 90°. It has a node in the origin and two double nodal points in its cyclic parts [9]. In Cartesian coordinates it is described by a rational quadric
(2) The graph of this function produces a curve similar to the infinite symbol, which, in turn, is often called just Lemniscate.
Fig. 4: Graphic representation. a) Lemniscate; b) Watt's system A famous type of Lemniscate was described for the first time in 1694 by Jakob Bernoulli, as a modification of the ellipse, which is the locus of points for which the sum of the distances from two fixed points (called foci) is constant. A Lemniscate, vice versa, is the locus of points for which the product of these distances is constant. Bernoulli called it Lemniscus, which is the Latin equivalent of "pendant ribbon". The Leminiscate can be drawn with the system devised by James Watt (1736-1819). The Watt's parallelogram is a mechanical connection, in which the movable central point of the system is forced to move approximately in a straight line. It consists of a series of three bars, two longer and measurement equal between them, both connected to a smaller central bar. The outer ends of the long bars are pivoted in fixed positions between them, while the two joints between the three bars are free to rotate. Taking into account the fixed distance between the outer ends, the system is an example of a "connection to the four arms. A strange formation of crop circle appeared the August 8, 2008 (8-8-8) in Wiltshire (U. K.), and naturally this fact interested very much all the Ufologists, that spoke about a mysterious signal came from the sky. They though, perhaps, that the portal toward the infinite has been opening just that day [10].
Fig. 5: Formation of crop circle appeared the August 8, 2008 (8-8-8) in Wiltshire. a) Overview; b) preliminary diagram. 3 The spirals An Archimedean spiral is a curve which can be described in polar coordinates (r, θ) by the following equation
(3)
with a and b real numbers, and b strictly positive. The change of the parameter a rotates the spiral, while b controls the distance between the arms. The spiral of Archimedes [11] is distinguished from
the logarithmic spiral for the fact that the subsequent arms have a fixed distance, while in a logarithmic spiral the distances follow a geometric progression.
Fig. 6: a) Crop circle in the Wiltshire countryside; b) Spiral of Archimedes. A logarithmic spiral, equiangular spiral or growth spiral is a special type of spiral that is often found in nature. The logarithmic spiral was first described by Descartes (1596-1650) and later extensively investigated by Bernoulli, who called it spira mirabilis, "the marvelous spiral". In the logarithmic spiral, the radius grows while it turns. As it approaches the pole, the curve "wrap itself" around this without ever reaching it. In polar coordinates, the curve can be written as (4)
Fig. 7: a) Stonehenge, July 7, 1996; b) logarithmic spiral. The problem of rectification of the circle, which cost much effort to the ancient surveyors, was also solved by Archimedes, with the introducing of a new curve, over than those that can be generated using only a straightedge and a compass, that is his spiral. He was able to produce a result that, if one thinks of mathematical tools of the time, is incredible. Consider the first circle of Archimedes (figure 8). Draw the line s normal to the radius of the first circle AH and passing through the origin of the spiral A. Consider, then, the straight line tangent to the spiral in H that intersects the line s in a point called F. Archimedes shows that the segment AF is the rectification of the circumference of the circle of radius AH. So doing, Archimedes moves the problem of rectification of the circumference to tracing the tangent to the spiral, which with the only use of a ruler and compass it's impossible [12].
Fig. 8: Archimedean construction In geometry, a logarithmic spiral with factor of growth b and growth equal to φ, with
(5) that is the golden section, is called "golden spiral" [13, 14]. The graphical representation of this spiral is based on a series of squares that can be insert inside the golden rectangle (Fig. 9 a). The realized construction can be related to the Fibonacci sequence, because the squares have, as sides, just the values of the series: 1, 1, 2, 3, 5, 8 ... See Figure 9 b
Fig. 9: a) The construction of the golden spiral; b) relation with the Fibonacci sequence 4 A recent crop circle with an “energetic” meaning Consistent with what has been said in the preceding paragraphs, a singular crop circle has been examined. The crop circle appeared in Robella, Italy, on June 30, 2013 could be the confirm about the expression of energy represented by the signs [15]. It represents an eight-pointed star (in Par. 2.1 we have talked about the symbolism of number eight), but to correlate it with a mathematical algorithm is really complicated, and this needs of a team of scientists with much different competences. Taking into account only its geometrical image, it seems to suggest a solution to energy problems, also emphasizing that its nuclear fusion reaction could be used to generate electricity. This crop circle has been read as an example of nuclear reactions needing a low quantity of energy, so called “cold fusion”, and it was even related to the “LENR” or “low energy nuclear reactions” studied by a lot of important researchers in the world [16]. The cold fusion energy generators will not need to be connected to an electrical grid… In Robella’s crop circle, a binary code individuates four chemical elements, that are Hydrogen (H), Potassium (K), Sulfur (S) and Deuterium (D) (Fig. 10 a).
Fig. 10: a) June, 30, 2013. The crop circle appeared in Robella, Italy; b) exemplary specimen. According to the scientists, through the gradual addition of Hydrogen and Deuterium, it is possible to melt the Sulfur into Potassium, with the help of other six elements (protons or neurons?). The interpretations that we consider most appropriate are two: Hypothesis 1. The elements depicted in one of the areas of the drawing are 6 (Fig. 10 a), and 6 is the atomic number of carbon, chemical symbol C. It is known that the letter C in Latin alphabet had the function of G, and the union of the various elements would seem to draw just a Greek gamma (Fig. 10 b), which is also the third letter of the alphabet. In addition, training is in the field of binary code corresponding to H, and GH of the Atlantis' ancient alphabet has a very similar form, as an y lying with one leg shorter and one longer. This could imply the chemical symbol for carbon and hydrogen? Four carbon allotropes are known, including might have interest in this case the graphite that, because of the relocation of the π bond, conducts electricity. In additions, C combined with the hydrogen forms oil, coal, natural gas and other compounds jointly called hydrocarbons. Hypothesis 2. There is another metal that might have an interest in this case, the Gallium, that has isotopes 6. Its chemical symbol is Ga and it has semiconductor properties, especially in form of arsendite gallium arsenide (GaAs), that is able to convert electricity in light. Old studies, recently developed, have shown that an aluminum-gallium alloy can spontaneously produce hydrogen by water [18]. The hydrogen could then be used to run an internal combustion engine. 5 From the crop circles until the seismic prevention To talk about the importance to find new forms of energy is really superfluous, because it is evident in
every aspect of the modern life. In particular, for what concerns the field of the constructions, the strategy to design buildings really efficient from the energetic point of view, is the more actual theme. Moreover, in the filed of seismic control, it is known that some studies have remained in stand by precisely to the huge use of energy that the application of them would have required, as in the case of the active control processes on the seismic response of structures [19], the study about the use of which has stopped just for the cited motivations. For this motive, in the last anti-seismic projects, for the most part passive isolators were used. The idea of using the energy itself triggered by an earthquake and convert it into mechanisms control through a system of sensors and actuators positioned on the structure (closed loop) is very interesting, but it remains in stand by because of the great quantity of energy required by the electrical box: this has to read the entity of the earthquake, to convert it in energy able to stabilizing the oscillations of the building, and finally to transmit the signals to the sensors. The sensors have to transmit the command to the actuators, that finally set in motion the structure. It is clear that, being able to use for this purpose a source of alternative energy, taken directly from nature, it would be essential and it would imply a real development of the studies. 6 Conclusions So, the message contained in the crop circles remains misterious. A researcher that approaches the study of the crop circle formations, will find a lot of arguments about the energy. But it is enough impossible to find something that treats the question in real "scientific" terms. Also if looking at Fig. 11, it appears clear that the raffiguration is an “expression of energy”, but is it connected or not with the actual scientific knowledge about the matter? And in that case it would be possible to find some information involving the energy-saving built? Or is it only a legend by esoteric culture?
Fig. 11: June, 17, 1996. Alton Barnes (Wiltshire) At the time the studies are still in progress, and we cannot exclude that a link between the “graphics” of the formations, and the algorithms about the energetical expressions, could be found. Moreover, in this paper any hypothesis about the origin of crop circles are formulated, because this is not the aim, but it has to be the first step for studies about a new renewable form of energy applied to the constructions, emphasizing also the link that from ever joins men and Universe, that expresses itself by the Signs and tries to explain itself with the Mathematic, as already it has been done at other times in the past [20]. Bibliographical References
[1] KEPLER, Johannes. Harmonice Mundi. 1619 (available on line)
[2] PLANCK, Max. On the Law of Distribution of Energy in the Normal Spectrum. Annalen der Physik, vol. 4, p. 553, 1901 (available on line)
[3] DURAND, Gilbert. Le strutture antropologiche dell'immaginario, Bari, Dedalo, 2009, pp. 391-4.
[4] Referring Web Pages: http://en.wikipedia.org/wiki/Ley_line
[5] CENNAMO, Claudia, CHIAIA, Bernardino. Biomimetics: strategies and intuition from Nature to man-made structures. In: Research in Scientific Feng Shui and the Built Environment. Torino, 2008, Y. Mak & T. SO, ISBN: 978-962-937-172-2.
[6] BRUNO, Giordano. De l'Infinito universo et mondi. Epistola Proemiale, 1584 (available on line)
[7] FRANCIOSI, Vincenzo, Problemi di Scienza delle Costruzioni. Napoli: Liguori, 1982. Vol. I, ch. XII. [8] Referring Web Pages: http://it.wikipedia.org/wiki/8_(numero)
[10] Referring Web Pages: http://www.ufonetwork.it/cerchi_08.asp
[11] NAPOLITANI, Pier Daniele. Archimede: alle radici della scienza moderna. Collana I grandi della scienza, curata dalla rivista Le Scienze IV, n. 22, ottobre 2001.
[12] Referring Web Pages: http://it.wikipedia.org/wiki/Spirale_archimedea
[13] Referring Web Pages: http://www.archweb.it/geometrie/spirale_aurea.htm
[14] CENNAMO, Claudia, CHIAIA, Bernardino, FANTILLI, Alessandro. Defining the crack pattern of RC beams through the golden section. Journal of Physics, Conference Series, 2011. ISSN: 1742-6588
[15] CENNAMO, Claudia, DRAGO, Rosanna, Design of energy systems through the crop circles geometry. In: Proceedings of XI Forum Internazionale di Studi Le Vie dei Mercanti, La Scuola di Pitagora Ed, Napoli, 2013.
[16] B. KRIVIT, Steven, MARWANB, Jan, A new look at low-energy nuclear reaction research, Journal of Enviromental Monitoring, 2009, 11, 1731-1746.
[17] Referring Web Pages: http://www.cropcircleconnector.com/2013/Cavallo/articles.html
[18] ZIEBARTH, JT, WOODALL, JM, KRAMER, RA, CHOI, G, Liquid phase-enabled reaction of Al–Ga and Al–Ga–In–Sn alloys with water, Journal of Hydrogen Energy, 2011, V. 36, 9, pp 5271-5279.
[19] BARATTA, Alessandro, CENNAMO, Claudia, VOIELLO, Giuliana (2006). Design of linear active control algorithms for non-stationary unknown forcing function. Structural Control and Health Monitoring, ISSN: 1545-2263, doi: 10.1002/stc.4300050202
[20] BENVENUTO, Edoardo. Sui principi di filosofia naturale che orientarono la ricerca di Saint-Venant, in Omaggio a Giulio Ceradini, Dip. di Ingegneria Strutturale, Università “La Sapienza” di Roma, Roma 1988, pp. 125-138.
