Vortex-sphere drawing theory
A transdisciplinary comprehensive approach to nonlinear phenomena
in fluids induced by acoustic stimulation.
Consonant proto-languages and the hierarchy of
biomorphic standing waves
Key words: sound, vibration, fluid, sphere, vortex, standing wave, biomorphic, language, morphodynamic, computational
Sound visualization is one of the ancestral human dreams. Capturing the image of the ephemeral sound still concerns scientific environments. Almost everything in the universe is in a state of vibration. Clarification of hierarchies, of the origin and consubstantiality of form with the natural language algorithm, inextricably grafted on phenomenal synergies, places form as a result of standing waves interferences in conjunction with symbolic inferences induced by language. Thus the cosmos is reduced to expressing order through sphere, a real brick of the open immeasurable, infinite construct, that punctually resides in the geometry of spheres with toroidal, discoid or spiral morphodynamic vicinities. This geometric minimalism allows for the infinitesimal thinness of terrestrial biosphere surface of all morphological diversity, subject to permanent rhythmic structuring and destructuring.
The influence of acoustic waves fields on liquids reveals striking morpho-dynamic conformations as a response of the fluid permeated by sound waves that induce and maintain standing waves, invisibly shaping the carefully selected liquid corpus.
The metamorphoses of the liquid matter traversed by vibrant energy reveals as dynamic patterns symmetries generated by interferential overlaps of loops and nodes of standing waves periodically stabilized in the fluid mass. Thus, visualized sound results from the confluence of sound-recipient-fluid interactions. The dynamics of standing waves occurred in the undulatory pulses of liquid circular surfaces establishes unifying bridges equivalent to the laws of musical harmony with polyhedral geometry.
Under the influence of sound waves, rhythmic fluctuations propagated in liquid media induce and maintain geometrical symmetries revealing the natural approach of conversion of harmonic movement in polygonal-polyhedral symmetries which are inextricably linked in the phenomenology of standing waves induced in liquid. Thus the periodic fluctuation cycle induces in the raw material fluid (situated between the two states of aggregation – solid and gaseous) oscillating cyclic temporal harmonic symmetries.
Studies on the morphology of standing waves aspects in multistage liquids reveals incredible facets, namely the appearance of humanoid-like forms with unsuspected implications, which require the reformulation of the entire conceptual fabric in the clarification and understanding of the origin of the living and non-living natural form.
The transformation of the original concentric circles in regular polygons, along with the emergence of cyclo-spiraling centripetal and centrifugal currents accompanying the dynamics of symmetrical waves propagated in the fluid subjected to vibration demonstrates the close connection between the periodic (time division) and the geometric (space structuring). Fluid matter thus becomes standing wave and therefore animated geometry. Vibration as fluctuation in time and space, perpetually propagated stimulus in an elastic medium, becomes visible once with the penetration into the liquid corpus which takes it over, structuring itself in surprising standing geometries. Liquid matter, a permissive receptacle of spatial-temporal fluctuations of pressure-depression, constructively modulates the harmonic interferences in the fluid mass which thus became a symmetric structure animated by vibration.
The resulting geometries are formed on the scale of different ascending-descending frequencies, like a complex vibrational language that surprisingly interferes with ancestral symbolic sources. Therefore, the circle becomes the common source of planar polygonal successive series the triangular, square, pentagonal symmetry, which tends to n radial spokes, just as in the three-dimensional the fluid sphere becomes a primordial matrix giving rise to regular polyhedra that oscillate in the cube-octahedron, icosahedron-dodecahedron polyhedral complementarities, etc. Therefore standing concentric circles become polygons from increasing noise intensity, while the sphere delivers polyhedral alternating oscillations. Vibrationally stimulated fluid spheres unify the indissoluble link between polyhedral antipodes as intermediate morphological stages of polyhedral structures couples; more precisely, the liquid sphere becomes a cube, and the cube becomes an octahedron, the alternation fluctuating with the acoustic stimulus’ frequency.
Vibration is an open periodic spatial and temporal fluctuation (plastic wave) or a standing wave in enclosures supported by interferential compositions of the reflected waves type from whose multiple reflections maximums and minimums (loops-nodes) in the liquid substance are stabilized, phase coherence depending on the geometry and size of the container, the enclosure in which the standing wave occurs. Thus the standing wave becomes a function derived from the interdependence between vibration-cavitary enclosure and liquid, gas or powder.
Triple influence resultants in chronological order: vibration-fluid-form eliminates the boundaries between sound-vibration and the acoustic enclosure dependent on form and geometry. The series of elastic waves that cooperates in form becoming and the complexity of unsuspected pulsing morphologies transmit to the slightly elastic liquid medium outside loops and nodes an effective and differentiated displacement of branched circular currents that rotate generating frequent vortexes in the standing fluid mass being similar to marine hydrous currents or atmospheric currents convections, and, at a biological scale, to blood-lymphatic circuits.
Therefore we have:
vibration - enclosure - liquid - reflection - interference - standing wave -
pressure - depression (pulsation) - movement - waveform
Standing wave in viscous liquids comes as an independent volumetric form traversed by compressions-decompressions, vibration turning into cycloids, vortexes and rotations with complex dynamics.
The diversity of vibrant formations, pulsations waken in maximum ventral dynamism areas, lead pulses like a heart, displacing the liquid to the areas of a nodal minimum of the oscillating network that attract fluid like a void-vacuum. So the fluid located in the three-dimensional nodal mesh of nodal-ventral, void-full complex networks becomes a vibrant corpus with an own morphology, stable as long as the vibration maintains the persistence of form. The fluid form maintains its stability, supported by the ethereal network of the vibrant watermark like an invisible skeleton, a kind of vibratile network in the eyelets in which matter throbs accelerated in loops and is drawn to nodes under the form of alternative convexities and concavities generating spiraling vorticities in nodal points, simultaneously with eruptive outbursts in the area of loops generating spheres that erupt ephemerally, remerging into the liquid mass similarly to the movement of tectonic plates (volcanoes being equivalent to loops, and swirls, tectonic subductions being equivalent to nodes).
The vibrant, resulting networks that maintain their form even if they have remarkable biomorphic looking similarities (trilobites, stages of embryonic development, facial fragments, etc.) are still deprived of the linguistic modeling input by which the genetic code operates in the living world. Most likely, the symphony of standing waves that maintain biodiversity works closely with the unseen information envelope that synergistically directs form and function in various living organisms.
Performing a vibrant forms system that, while increasing-decreasing sound frequency starting from infrasound and ending with ultrasounds, embodies linguistic expressions with real bases in the physicality of standing waves’ morphemes at certain frequencies would become the starting point of establishing analogies with the natural language of biological forms on morpho-vibrant-genetic bases, reducing the entire spectacle of diversification of the living world to the consonant language code-vibration-form similarly to a biomorphic alphabet. For example, gradually increasing the frequency of sinusoidal type sound waves further produces cruciform-like formation, which oscillate around a center; at 33Hz, standing waves are formed revolving to the left or right that pulsate during the rotation of the two vortices formed in the liquid mass, resembling the figure eight or the yin-yang sign.
As the frequency increases, the liquid creates pulsatile triangular formations. Further, at frequencies of 46-51Hz, curious human-like formations are obtained, both facial fragments with orbits, nose, zygomatic, mouth, and a sort of dancing homunculi that resemble Buddha’s statuary representation in lotus position.
Fig.3 46-51Hz standing wave with axial symmetry looking like a humanoid mask, Photo by Gabriel Kelemen 2009
By increasing the number of cycles per second at a frequency of 53-60Hz, the centered pentagon emerges, static or rotating, sometimes with an incomplete revolution, thus pulsing. Further increasing the frequency, hexagonal, heptagonal and so on geometric configurations emerge so that the series of forms that are formed in the liquid mass after sound stimulation pass successively from a circle to a multitude of rheological-vibrating interconnected circles, crossing both morphological geometrical stages and stages strikingly reminiscent of biological fragments, combining in an ascending-descending sequence of frequencies both symmetric geometries and biomorphic appearance forms.
Correspondences between vibration-frequency-intensity and form are obvious; forms that emerge from the liquid successively traverse figurative elements that represent ancient symbols in human culture: the circle, the circle with a center point, the yin-yang sign, the triangle, the pentagram, the Ouroboros, the hexagram, the Flower of Life dynamically alternating with the Fruit of Life, consecutive series containing together elements with a biomorphic, humanoid or animal or leaf-like, well defined flower corolla appearance, the array thus becoming a real hierarchical construct as a sound spectrum gamut in whose frequency range the afore-mentioned bio-symbolic morphologies exist.
Returning to the series of forms that occur within the acoustic range 1-70Hz, as mentioned, there are a number of independent morphologies corresponding to certain frequencies. In addition to the stable form, sometimes forms variable in time occur either by alternatively transitioning from one form to another, or by rotation about the axis to the left or right.
Looking at a formation that appears in the 27-33Hz frequency range and analyzing it more closely, we see a figure 8 form that constantly revolves to the right and which has in the center two permanent vortices that circularly push viscous liquid spirals. Towards the edge of the form, there are subduction currents that return following cochlear trails to the two vortices, in fact corresponding to the depression nodes belonging to the unseen oscillating network of the three-dimensional vibrational skeleton that crosses the thus constrained and shaped fluid by the standing sonic field envelope.
The sonic standing wave forms the viscous liquid situated on the elastic surface of the membrane. The liquid undergoes a series of elastic compressions-decompressions which, accumulated in areas of maximum amplitude, produce convexities (loops) becoming areas that gather liquid at the base they dissipate radially at the convex upper surface level. Simultaneously, concavities emerge in depression areas (nodes), the liquid moving centrifugally towards the concave funnels of nodal vortexes. Liquid is modeled by the sonic field simultaneously accompanying the standing wave, according to the principles of sonicity  developed by the illustrious Romanian scientist George Constantinescu. Theoretical and practical similarities between sonicity and electricity are almost identical; therefore and according to the theory of sonicity, liquids and considered as compressible, so that the stationary wave induced in liquid, besides generating the already mentioned cycloid currents, zonally imprint compressions and decompressions of the sonic currents type that amplify the thixotropic rheological behavior of the form subject to the sound field.
The dynamic behavior of the fluid located in the influence field between fluid and vibrant spatial structure, type standing wave, induces sonic fields related to the elasticity of liquids subjected to compression-decompression and thus explaining the form and behavior over time becomes apparent. In conclusion, the whole range of formations that arise and are maintained in the liquid depends on the vibrant network structure type standing wave and on the instant induction of the sonic field that work together thus modeling the liquid waveform. Watching closely and chronologically the acoustic induction of vibration in the liquid matter, from the sound source generated by the electronic oscillator that propagates through the aerial elastic medium in the immediate vicinity of the speaker, the sine wave train with set frequency meets the circular surface of the elastic membrane and induces convex-concave deformations that alternate dividing by frequency, diameter and elasticity the surface of the disc penetrated by the sound flow that vibrates constantly, a consonant response between the acoustic stimulus, air column and elastic membrane being established at its level.
Therefore, the standing wave falls under elastic, dimensional parameters, and, depending on their characteristics, the membrane surface division in concentric structures occurs that later become radial geometries specific to the frequency given. The more frequency increases, the more (at a constant diameter of the membrane disc) the surface is divided into concentric circles, which multiply with the emergence, once sound intensity is increase, of radial spokes that divide the circles starting from the center.
In addition to these divisions, at certain frequencies there are in two, three, four, or more circular centers that oscillate simultaneously in phase (simultaneously upward and then simultaneously downward). Experiments conducted in liquid membrane layers reveal and clarify the loop-nodal profile of the standing wave that animates the liquid disc. They capture the emergence of rapid rotary motion to the left or right arising from vibration, confirming the dynamic relationship between vibration and rotation. Both the quantity and viscosity, the surface tension, respectively, lead to the emergence of interrelated changes between the standing wave, air column, elastic membrane, and fluid. These standing waves reciprocally conformationally having as common base the continuous sinusoidal frequency of the wave train that runs through the three elastic media, which become stationarily synchronous. The input of frequency to obtain the morphological and geometrical configuration of the vibrating fluid results as a direct consequence, and this morphology largely depends on frequency, and, not at all negligible, on the above mentioned parameters (diameters, elasticity, distances, viscosities, and so on), and which together with amplitude completes the vibrating contour of the liquid waveform.
Following this interdependence between three elastic media, the emergence of higher harmonics in proportions dependent on size, distances, forms leads to the emergence of frequency intervals where a certain standing fluid conformation manifests, and thus, at different frequencies, varying the physical parameters, a limited array of the same forms in the liquid mass are obtained. In conclusion, between a certain frequency and a certain form a relative dependency is set. For example, the pentagonal oscillation can be obtained at several frequencies between 6 and 25Hz by altering some dimensional-quantitative parameters; however they cannot be obtained at any frequency outside this range due to the limits of the elastic characteristics of the transmission media (air, membrane-liquid) .
Therefore, as the number of genes in living nature is not directly and quantitatively related to the form or symmetry of a specimen, the different allotropic states of the elements (e.g., for carbon, diamond and graphite, phosphorus, white, red or black phosphorus, and the examples may continue for liquids stimulated vibrationally), the geometrical form does not depend strictly on a certain frequency, but rather on a plurality of synergic results emerging from the interaction between frequency and the related elastic media. If for a vibrating string the number of loops and nodes induced by resonance by means of the aerial elastic propagation medium is characterized by a certain frequency, the number of cycles per second depending on the string length, tension and diameter, for the vibrant elastic disk we see a more complex dependence of harmonic reports between the given frequency and a standing wave map determined in the elastic disk, and in the case of elastic spherical surfaces or volumes the complexity of maximum-minimum zones distribution increases.
Thus the vibrant cycloid becomes the common dynamic element found at the basis of the entire manifest interval. This prime module is a true base motor unit whose dual multiplication makes the entire diversity of stationary wave symmetries, merging archetypal essences with unsuspected references to symbol-pattern. Thus, these fluid models build unseen bridges to the ancient symbols. The surprising interferences between symbolic languages and liquid patterns divined as if by ancient mythologies supplement the analogy of apparent coincidences between geometric symbols and wave surfaces, leading to finding and explaining the archaic symbolism primordiality of standing waves.
Following the approach geometrically from simple to complex, we note the successive transition from line-string to disk-membrane, and finally sphere-oscillating volume, correspondences with geometry being obvious, the series being established from simple to complex (line, plane, volume). We could say that vibration’s encounter with matter gives rise to the number, polygons and polyhedra, however not in a strict and rigid dependence, but in a relatively elastic manner, rather statistically quasi-determined, and yet not by whatever means, but by having complex interdependent rules. Summarizing, vibration as energy fluctuating in time develops when impacting matter geometrical antipodes either in plane (convex-concave polygons) or in space (complementary alternating polyhedra, for which the peaks become centers of surfaces and vice versa alternatively); the vibration is thus transformed through liquid in polygonal symmetry, and the fluid sphere in dual polyhedral antipodes. Platonic polyhedra, namely tetrahedron-inverted tetrahedron, cube-octahedron, and finally dodecahedron-icosahedron, starting from the fluid sphere thus become successive oscillating terms of three pairs of transient states, of polyhedral standing waves in a cyclic tandem.
With all these related contributions of acoustics, sonicity, geometry, rheology, standing waves physics and the striking resemblance with biomorphic conformations, we cannot reduce the mysterious complexity of terrestrial life that may be operating with these phenomena enclosing them, but by no means being limited to them.
Regarding the emergence, origin and maintenance of form, Ian Steward in his book Nature’s Numbers, describing Morphomatic, brings into question the delicate relationship between the limits of mathematical interpretation, genetic code and natural geometric diversity:
“We need an effective mathematical theory of form, which is why I call my dream Morphomatics. Unfortunately, many branches of science are currently facing the exact opposite direction. For example, DNA programming is considered solely responsible for the forms and patterns in the body. However, current theories of biological development do not explain adequately why the organic and the inorganic world have so many common mathematical forms. Perhaps DNA rather encodes the dynamic rules of development than contain the code of the form developed to the end. If this is the case, then current theories ignore the crucial part of the development process”.
Continuing, Steward makes clear distinctions:
„ For example, as D’Arcy Thompson noted , there are similarities between the forms of the various organisms and forms of fluid flow, but fluid dynamics, as it is currently understood, uses equations that are too simple to model organisms”.
The close cooperation between genotype and phenotype finally realizes morphology. Emergences in liquids of elements flagrantly similar to biological forms on the one hand and geometrical patterns on the other hand suggest the existence of unsuspected subsidiary relationships between vibration and the nature of form.
Gabriel Kelemen, PhD.
 Constantinesco, George: Teoria sonicităţii, Editura Academiei R.S.R., Bucureşti, 1985
 Meyl, Konstantin: Scalar Waves Editura Universităţii tehnice din Berlin, 2003.
 Hildebrand, Stefan. & Anthony, J, Tromba: Mathematics and Optimal Form Scientific American Library Scientific American Books, Inc. Distributed by W. H. Freeman. New York, 1985
 Kelemen, Gabriel: Universalitatea principiului sferă-spirală, Editura de Vest, Timişoara, 2009,
 Stewart, Ian: Numerele naturii, Editura Humanitas, Bucureşti, 2006. pp.168-169
Raw material is a principle of unity and order in substance that remains unchanged behind apparent diversity. Raw material is not matter, but the matter’s possibility of becoming, described in contradictory terms.
Sonicity is the science of mechanical energy transmission through vibrations. Based on the theory of musical arrangements, Gogu Constantinescu found laws similar to electricity for remote transmission of energy by oscillations which propagate in continuous environments (solid liquids) due to their elasticity, sonicity being a branch of mechanics of continuous elastic media (solid, liquid or gaseous) that deals with the study of energy transmission by vibration or by longitudinal elastic waves
George (Gogu) Constantinescu (b.1881, Craiova d.1965, London) was an illustrious Romanian scientist and engineer, a prolific inventor with over 130 patents, including the inventor of a new science – sonicity.
 Op.cit. pag.168.
D'Arcy Thompson (England), in On Growth and Form, proposes a significant synthesis of the dynamics of biological forms with illustrations claiming some type of evolutionary metamorphosis of the anamorphical type.
 Op.cit. pag.169.