Michael
M. Lieber, Ph.D. |

L
The Golden Ratio as a Dimensionless Biological Constant Enabling a Dynamic Uniformity in Specificity within Various Self-Similar Scales in Natural Phenomena: Its Operation within the Universal, Dimensional Physical Constants and in a Fractal Reality
Part I: The Dimensionless Biological Constant and Forces
In an earlier publication, this author demonstrated that many, if not all, of the universal, dimensional physical constants of physics have either as an underlying common component a dimensionless constant equal to the golden ratio or its reciprocal, or in some cases combinations of both or powers of both. Such a constant was referred to as a biological dimensionless constant as it was shown to represent biological properties or features, such as regeneration. This was also shown to be a dimensionless situation within the physical constants, independent of the system of units, but allowed to be detected and sustained by means of a constant, geometrical parameter, π, through which the metric system is based, and which system the physical dimensional constants use. (See, Lieber. 1998a. View PDF of Article.) The biological dimensionless constant, referred to as Q in this article, rather than as Φ, reflects or defines an inherent, regenerative-template feature within all natural phenomena, biological as well as physical. This regenerative, significant template feature, under and through the constraint of non-uniform physical forces, also geometrically structured or mediated, reflects a universal drive within all phenomena. This is the drive to maximize, utilizing that very non-uniform constraint, the uniformity of forces within the specificities or non-uniformities through all space within each of the phenomena or systems. (The physicist, Paul Lieber describes such a drive and its significance in a 1969 publication.) As the author illustrates as to
feasibility, this is enabled through geometries or geometrical
constraints only allowed by or enabled through this dimensionless
constant, Q, its template, (or its reciprocal Q Subsequently, A. Einstein, using Riemann's mathematics,
creatively and elegantly demonstrated in various publications that
gravitational forces and their patterns are due to or are marked by
curvatures within a space-time metric field influenced or structured by
means of matter-energy and its inherent geometry. And this could be a
geometry having Q as a constant and enabling parameter. In fact, it can
be shown through the author's analysis that Q Though not addressing this particular possibility regarding Q's relevance to matter-energy, Dr. Einstein's theories have nevertheless been supported experimentally. Dr. Clifford further pointed out that magnetic forces could be due to twists within space. If this were the situation, might these twists within space, or within a space-time metric field, be spirals whose parameters are defined and enabled by Q? Many developments in physics have attempted to unite the various forces of nature, such as gravity and electromagnetic forces, as unified distortions of a space-time metric at higher dimensions. Q could extend to and be involved in such dimensions, and thereby the specific forces allowed by or expressed through such dimensions. Even though each specific, dimensional physical constant refers to specific forces and/or energies, the common underlying dimensionless constant Q also defines a universal common behavior to all the various forces. Namely, irrespective of the particular force or energy, all forces would maximize their uniformity of magnitude, probably of the lowest degree also for maximum stability, within specificities or non-uniformities, with the consequence of such maximum uniformity achieving a maximum stability throughout all the space and time within the system or structural phenomenon, hence adaptation or its preservation, its constancy through regeneration by means of forces geometrically patterned through Q. Only through a regenerative dynamic shaped by a geometry modeled or determined or allowed or enabled through Q could such a uniformity of force be achieved or approached within or utilizing all space in a compact, though uniform manner. The existence or operation of the physical constants would be based upon and reflect this. In view of this, let us first
take a look at a physical dimensional constant and see how the
regenerative sub-constant Q or its reciprocal Q The geometry of propagation might
also indicate the operation of a constant endemic universally to various
geometries, and that constant is pi or
Π. And in
fact, this constant is readily evident in the expression of the velocity
of light constant, namely
5Q A important connection between Π and Q is thus indicated regarding the maintenance of the underlining dimensionless constancy within the dimensional constants due to the regeneration involving forces and their uniformity of manifestation. This connection is also seen in some mathematical equations: In various, complex trigonometric equations or expressions Q can be expressed in terms of Π, where in other trigonometric equations, Π is expressed in terms of Q. (See Posamentier, A. & Lehmann, 2012.) These expressions might also point to constant relationships within physical phenomena, perhaps involving self-similar waves, yet to be discovered. Trigonometric functions do in fact describe electromagnetic phenomena, such as waves, and the Fourier series shows how trigonometric functions combine into complex wave-functions, which describe actual physical wave behavior in fluid dynamics. In a well known physical-chemical situation, there is the operation of Avogadro's constant. This is a remarkable trans-scalar and trans-chemical constant, which points to quantifiable identities or uniformities of different physical entities, across different levels of organization and chemical variation, from electrons through atoms to molecules, within the scale of the gram-mole. This, thereby, links uniformly in terms of a constant number all of these different physical entities across different levels of organization and through chemical variation in a self-similar manner. That is, within the scale of the gram-mole of any chemical, the respective numbers of electrons, atoms, and molecules would be the same, irrespective of the scale of these different entities and irrespective of any chemical. For example, a gram-mole of any chemical such as sulfur would have respectively the same number of electrons, atoms, and molecules as any other chemical, such as gold. And this same or constant number would apply respectively to the number of electrons, atoms, and molecules in a gram-mole.
This constant number is denoted by Avogadro's
constant. This
dimensional constant in terms of Q and Π is Π As one
reviews the various dimensional physical constants, one sees how Q and Π
become repeatedly revealed. Though the relationships between Q and Π
(or between the various forms of Q
such as Q The
various relationships of these different forms of Q evident through the
various dimensional constants might be constructed into some type of
periodic table. Such a table might provide deeper meanings of these
relationships and more insight as to how the various constants
repeatedly relate more fully and completely to one another through
repeating patterns involving Q and Q This generation of self-similarity on different scales, defined/guided or allowed geometrically-physically by Q or its forms, could predictably be the generation of the maximum degree of force uniformity within specificity on different scales and between such scales. This would be geometrically represented by a fractal structure, stabilized uniformly by such forces. Such a fractal would represent a Principle of Universal Correspondence applied to the dynamic drive for maximum force uniformity on various, respective levels composing the different organizational scales of physical phenomena and the corresponding, similar behavior of these different scaled phenomena by means of such forces. In 1968 and 1969 publications, Paul Lieber described such a Principle operating in connection to a dynamic of maximum uniformity in non-uniformity and the universal, physical constants. Forces or their non-uniformity were seen as an aspect of the non-uniformity of a manifold that ontologically featured impenetrability. As Paul Lieber described, this non-uniform impenetrability is the essence of forces and their constraints. Such impenetrability would allow for their geometrical structuring and patterning, guided by the dimensionless components of the universal constants, towards their greater uniformity within non-uniformity through different self-similar scales. As Paul Lieber noted, the increase in dynamic uniformity is the reduction of non-uniform constraints within physical and biological systems independent of scale or level, with the consequent stabilization, and hence, the generation of the very constancy of the systems, independent of scale, from the quantum level to the cosmological level. This exemplifies the the Principle of Universal Correspondence of dynamical processes on different scales. In anticipation of this, in a 1961, unpublished article, Paul Lieber describes a variational principle for the hydrodynamics of viscous fluids. According to this principle "vortex and spiral motions are indeed preferred in non-uniform flows. " Such motions are a uniformity in non-uniformity, and "the measure of [their] uniformity is maximum for real flows." This means that hydodynamically a vortical or spiral pattern of motion represents a maximum uniformity of forces in a non-uniformity of forces within the fluid's motions. This coherent pattern, as an accommodation to non-uniform stress or force in a fluid, would apply to all hydrodynamical and phenomological levels, especially to coherent, cooperative, biological phenomena, as Paul Lieber noted.
Part II: The Biological Dimensionless Constant in Dynamics, Fractals and Reality
We have seen the golden ratio or Q as defining or representing a common regenerative, guiding, template component or factor within all the dimensional universal constants of physics. These respective constants deal with or address specific physical situations and the specific forces endemic to those situations. Yet, such constants also refer to and depend on a common regenerative factor or dynamic that is a-specific, but which is within all specific physical situations. Paul Lieber (1968, 1969) demonstrated that the dimensional physical constants mark or represent the accommodating, adaptive nature of forces in specific situations irrespective of the types of forces involved. This accommodation has geometrical-dynamical constraints in and between various physical and biological systems. That is, the systems display a drive to achieve a maximum degree of force uniformity within and between themselves under or within the constraints of non-uniform forces or stresses present within and between such systems. The physical constants would mark this constancy of the dynamic drive in various situations, including biological, to achieve a maximum uniformity of force within a non-uniformity of force or non-uniform constraints or stress, and consequently, a maximization of the degree of stability within and between the systems, enabling the accommodation and conservation of such. within a context of stress. The golden ratio, the Q template, or its reciprocal, common to all these dimensional constants, would mark or describe or guide geometrically-physically how this is achieved or enabled, the effective avenue of the "go of it." Biological processes or systems
clearly show this. In bud development of plants, the buds develop in
such a way spatially with respect to one another that there is likely
the lowest degree or magnitude of mechanical pressure or stress between
themselves. (See Posamentier and Lehmann, 2012.) This manifestation of a
low degree of stress between buds is due to the angle of divergence
between them. A component of this angle of divergence is the golden
ratio or Q, as can be seen: 360
One can interpret this inhibiting, high pressure as being non-uniform, thereby a non-uniform stress. In this regard, the golden ratio or golden angle manifests and enables an adaptive and stabilizing process that would otherwise be greatly constrained or inhibited through, non-uniform, high stress. Moreover, in terrestrial plants having this golden angle between specific developing buds also enables these buds to achieve the maximum exposure to light for essential photosynthesis. Hence, the maximum uniform display of a low degree of mechanical pressure or force throughout a system of developing buds as well as a maximum exposure to light by means of the golden angle between the buds allows for different types of accommodation or adaptation, and thereby a stabilization of the living process.
At a higher scale, the overall arrangement of emerging buds from a plant stem or the arrangement of seeds within a sunflower with respect to one another is itself a spiral. This is a specific type of generating logarithmic spiral whose inner ratio of growth dimensions is Q. This spiral, the golden spiral, is a logarithmic spiral whose growth factor or rate is Q per every 90 degrees of its turn or curve, irrespective of scale. This particular developing spiral arrangement or pattern of emerging buds and seeds at this higher scale is referred to as phyllotaxis, and it appears that the golden ratio at the lower scale enables this arrangement. In the case of the sunflower seeds, one group develops along a given number of counterclockwise logarithmic spirals, while another group develops along clockwise logarithmic golden spirals of another number. The ratio of these two numbers of clockwise and counter clockwise logarithmic spirals, irrespective of the developing sunflower plant, is always Q. The same situation can be seen in various plants with regard to location of repeating structures making up the spirals, such as in the surface morphology of the pineapple, and the seed-arrangements of a pinecone, where the ratios of the numbers making up the different classes of phyllotaxis-logrithmetic spirals (e.g., clockwise and counterclockwise directions) are always equal to Q.
These numbers occur within what is called a Fibonacci sequence, and adjacent numbers within that sequence have the ratio Q with respect to one another. This also demonstrates that at a still higher level of organization or scale, the golden ratio becomes again manifest in development, suggesting that the golden ratio is necessary for effective development. In fact, it is pointed out again that the involvement of the golden ratio within the generating spiral arrangement enables the most efficient use of space (Livio, 2002), and probably of minimal force manifested uniformly through that space. Again, this may indicate a dynamic cause of phyllotaxis and the critical role that Q plays in such, especially as it appears to dynamically link coherently and with stability the different levels of developing organization, which appear to be self-similar.
In this regard, a physical experiment strongly supports the involvement of Q in a physically organizing process: A dish containing silicone oil was placed in a magnetic field in such a way that the magnetic field was stronger at the dish's edge than at its center. "Drops of a magnetic fluid , which act like tiny bar magnets were dropped periodically at the center of dish. The tiny magnets repelled each other and were pushed radially by the magnetic field gradient. Doudy and Couder [the experimenters] found patterns that oscillated about, but generally converged to, a spiral on which the golden angle separated successive drops. Physical systems usually settle into states that [uniformly] minimize the energy [or magnitude of force used through space]. The suggestion is therefore that phyllotaxis simply represents a minimal energy [low magnitude force transferred uniformly through space] for a system of mutually repelling buds." (Livio, 2002.) A developmental patterning on different scales, guided or allowed by Q, would appear to only ensue in the maximization of a uniformly low magnitude of force through all limited space-time. In effect, the dimensionless constant Q appears to have enabled the most efficient, stabilizing, adaptive utilization of force in various biological and physical situations. Again, this is further illustrated. In other growth situations, the golden ratio or its reciprocal might also represent geometrically the best means or avenues to achieving a maximum degree of low uniform stress within a growing biological system, achieving maximal stability through space and time in the system. For example, terrestrial plants, such as the angiosperms, produce branches (and leaves), and these appear to be produced or displayed in a self-similar, reduction manner, taking the form of a fractal. Let us consider the following in this regard: To achieve a maximum degree of a generating, reducing self-similarity, whereby all space between developing branches (and between developing leaves) are achieved with minimal contact between branches and between leaves, and thereby minimal mechanical pressure between the structures, the reduction fraction or factor of each branch produced would be 0.618 (the reciprocal of the golden ratio) the length of the branch from which it began as a bud. This reduction fraction would enable the production of the maximum degree of uniformity of branching within the tree or bush with the minimal of touching. This geometrical uniformity of
branching, utilizing all space and generated through time and
mathematically defined or allowed by the reciprocal of Q, could
represent the maximum uniformity of low stress throughout the organism,
that is, the tree. This fractal process based on or requiring the
reciprocal of Q, which achieves through a reducing iteration such an
adaptive, branching geometry, can be demonstrated by means of a very
similar, fractal-generating procedure due to the reciprocal of Q, that
is Q Q We see how the generation of a fractal geometry, enabled or allowed through a constant reducing factor common to the dimensional physical constants, could represent a dynamical, adaptive situation for biological systems. This adaptive, fractal process, whose constant, determining parameter or constraint is the reciprocal of Q, or of the golden ratio, also generates different, self-similar specificities, between which, more and more low level, uniform stress is likely enabled. Hence, the very generation of this self-similar specificity could allow for or provide more and more avenues for the creation of greater force uniformity throughout the specificities on and between the different fractal levels or scales, hence would ensue greater dynamical uniformity through greater specificity or non-uniformity, thereby greater stability in space and time. The different physical,
dimensional constants represent different types of forces. But, due to
the common, underlining constant that defines a common behavior of such
forces, the various forces would also be governed by a fractal, dynamic
drive to achieve a maximum degree of uniformity of magnitude within
situations and periods wherein such forces operate or are prevalent.
This drive operates through either decreasing or increasing scales of
the physical and biological realms. With regard to lower scales, as we
have seen, this fractal, dynamic drive can have the constant of the
reciprocal of the golden ratio, as only in this dynamical reducing
factor, producing one type of self-similarity of increasing specificity
throughout smaller and smaller scales, could ensue in a situation
whereby a maximum uniformity of force is achieved throughout different
levels of self-similar, specificity based on the reciprocal of the
golden ratio. However, from the perspective of going from these lower
scales to the higher, Q becomes manifest. It would appear that Q Thus in development, the golden ratio itself, 1.618 or Q, can also enable-guide an increasing self-similarity on higher and higher scales. This in fact generates a fractal or morphology based on or determined by a dilation factor or ratio, which is the golden ratio itself. By virtue of this particular ratio, or dilation factor, a growth is achieved that displays geometrically a maximum degree of uniformity within an increasing specificity or non-uniformity on higher and higher scales. This morphology or growth is displayed or arranged as a logarithmic golden spiral, and it is displayed at various scales in the growth patterns and organization of various plants and animals, such as the shell of the Nautilus. And in fact, a 65 million-old fossil of a mollusk has a shape that closely resembles a logarithmic golden spiral, perhaps indicating that Q has been significantly relevant to life for millions of years, an effective constant through time. The logarithmic golden spiral and
its constant features would appear to be applicable to or define or
represent, from various perspectives and analyses, including
geometrical, the generative designs of existence, which scientists have
denoted as physical and biological. The constants or constancies through
the logarithmic golden spiral are clearly evident, and they become very
important in elucidating the nature and dynamic of those designs, which
we define as reality or existence. As noted, the logarithmic golden
spiral increases by a constant factor of Q over every 90 In other words, irrespective of the scale or total size of the golden spiral, it grows by the constant factor of Q for every turn it makes over 90 degrees at its scale or size relative to its origin and coordinates. As stated in Wikapedia: "In geometry, a golden spiral is a logarithmic spiral whose growth factor is phi [or Q], the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of phi for every quarter [90 degree] turn it makes." Also, in its constant rate of logarithmic generation, the golden spiral concurrently generates equal angles of 30 degrees (and 60 degrees) with vertices along the curve, and the respective cotangents of each of these equal angles are always equal to 1.718 at every scale within a given set of scales. In short, the spiral also generates a constant growth rate of 1.718 per unit rotation angle of the spiral radius per or within each equal angle, which rate is also referred to as the logarithmic rate (Clifford, 1899). In other words, this particular growth magnitude is the cotangent of each of the equal 30 degree angles (and equal 60 degree angles) generated along the spiral curve, where tangents and spiral radii intersect to form the basis of triangles with identical angles along the spiral curve. (Please note Figure S.) Triangles having identical angles with respect to one another are referred to as equiangular. Though mathematical, this reflects a generative, guiding process in existence creating constancy through change. More specifically, the unit angle rotation of the spiral radius with respect to the coordinates also generates the 90 degree angle of each of the equiangular or identical triangles, and thereby each of the equiangular triangles and their other angles, which happen to be 30 degrees and 60 degrees, and which are equiangular or equivalent in all the triangles generated along the spiral curve. The cotangents of these two angles, whose respective vertices occur on the curve, are 1.718. In effect, the generation of these equiangles along the curve, giving rise to the cotangent 1.718, depends on the generation of the 90 degree angle, and thereby the cotangent 1.718 also depends on such. The generation of the cotangent 1.718 of each equal angle of 30 degrees (or of 60 degrees) per unit angle of radius rotation, with respect to the coordinate, always results in the exponential growth constant, e, which is 2.718, per unit angle of radius rotation per each 30 degree (or 60 degree) equal angle generation and per each 90 degree angle of the triangles generated along and defining the spiral curve. (Clifford, 1899). Adding the two components of this cotangent ratio always give a scalar magnitude of 2.718. The generation of e through equal angles, along the spiral curve, is enabled through a 90 degree rotation of the growing radius section or triangle side within and generating each triangle. This is a 90 degree rotation or the operation i with respect to a side of each generating triangle. A rotation of 90 degrees is equivalent to the rotation or operation i within a complex number, coordinate system. The generation of e becomes contingent on this particular 90 degree rotation (or i ) over a much smaller scale relative to or associated with the generating curve, a scale involved in generating the equiangular triangles along and defining the curve (Clifford, 1899). This generation of e becomes another constant feature of the logarithmic spiral along with Q. Q would concurrently define the constant growth of the spiral over 90 degrees, and thereby also i, at much higher scales. These are scales much higher compared to the scales of 90 degrees, and thereby i, at which the constant logarithmic rate 1.718, underlining or ensuing in e, occurs. This occurrence or generation of 1.718 is through the generation of the equiangular triangles, whose generating hypotenuses become the generating, very small sections or arcs of the logarithmic spiral curve, thereby at the scale of the spiral's curve. In effect, constant generations of 1+1.718 (or e) through 90 degrees, or i, over lower scales, associated with the curve, integrate as a totality to compose generations of 1.618 (or Q) over 90 degrees, or i, at higher scales with respect to spiral generation at 90 degrees relative to or associated with its spiral coordinates and origin. This points to a critical relationship between Q, e, and i, and one defined by an important equation, whose deeper meaning will be addressed shortly. Also, e is an extremely important mathematical constant, and it shows up in calculations dealing with physical phenomena. It was noted that 1.718, the generative basis for e, might also occur as another generative, dimensionless component of some other dimensional physical constants (Lieber, 1998a), including those to be discovered. Note that 1.718 is almost identical to 1.618, that is Q, which is also significant. In that publication, 1.718 was denoted as Q1. Could Q and Q1 refer to features of the same, basic generative operation or situation within different sets of scales within existence, where each scale-set corresponds to a different infinity, such as those described by the mathematician, G. Cantor, at the end of the 19th century? Could Q and Q1 be applied interchangeably? The near identity of these two generative constants has an important meaning or significance, which as yet is not clear. However, there may be some clarification in this regard through considering the following paragraph. (Relevantly, also, see Table 1 pertaining to the generalized logarithmic equation and its representation of different constant growth rates in the turns of different logarithmetic spirals. These constant rates of generation may also correspond to other dimensionless, biological constants involved in other natural vortical generations not based on Q.) As noted, the logarithmic rate 1.718
results generatively in e, an example of constancy through generation.
And most significantly, there is a clear relationship between e and Q,
as This constant rate of curving generation over 90 These dimensionless or trans-dimensional constants, and collectively their relationships to one another, become manifest (from their seat in the plenum) through the consequent generation of the self-similar logarithmic spirals on and through all scales (and their sets), and correspondingly, in and through various phenomena. This points to the operation of such dimensionless constants explicitly and implicitly in the maintenance of the dimensional constants within various scales of phenomena, hence of the phenomena themselves in which the dimensional constants become manifest. Their operation also points to their involvement---through the dimensionless constants---in the generation of a geometry of force that displays through and is defined or enabled by a logarithmic spiral, a maximum uniformity of curving force within specificity through various scales, and hence an ongoing generation of stability through those various physical scales. In other words, it is thus likely that, structured via the dimensionless constants, this vortical geometry of maximum uniformity within specificity also represents, defines, and allows a maximum uniformity of forces within specificity on all scales or levels of organization, from the very smallest to the very largest. The golden ratio and its reciprocal would appear to be those dimensionless constants that, within a spiral configuration, enable or determine, or through which, a maximum uniformity of force of various types, such as mechanical and magnetic, can be achieved throughout different levels or scales of specificity, and in so doing, would give stability through those different scales, and thus be adaptive. The feasibility of this view or hypothesis can again be demonstrated experimentally with physical phenomena. In an experiment involving the changing spatial relationship of ferromagnetic balls being added in a magnetic field, a regular spiral pattern based on Q was generated within the changing geometrical arrangement of those balls. (See Posmentier &Lehman, 2012.) This might also suggest an arrangement of magnetic forces between all these balls that can only achieve a maximum degree of global uniformity or stability by means of a spiral or vortical geometry enabled by the golden ratio. As pointed out by Posamentier and Lehmann (2012),"this pattern building [based on the golden ratio] has also been observed in many other non-biological systems." Predictably, these could be systems where the dimensionless constants appear to become evident. In fact, quasi-crystals display a geometry of inter-connected and partially over-lapping decagons in which all space is completely used in such a manner that a maximum uniformity of pattern or symmetry is created compactly throughout all space on different scales of the crystal. The geometry is based on Q throughout the geometry and could not have arisen without it. (See Posamentier & Lehmann, 2012.) It is likely the geometry is sustained or manifests itself by forces in uniform magnitude throughout the crystal, indicating that the forces sustained or manifested the geometry by means of Q's influence on that very geometry. And it could have only done so if the forces emerging from the crystal themselves eventually displayed configurationally a maximally uniform force magnitude throughout the crystal due to the operation of Q on the crystal's geometry. In effect, within a natural substance, Q could have guided or enabled or determined---perhaps via a type of geometrically-mediated, global imprinting---the generation of a maximal, stabilizing force configuration throughout the crystal and through its geometry, a force configuration manifested by a Q-enabled geometry. In short, Q would define or guide the crystal's geometry of forces, and those forces retroactively and stably would maintain or enable that very geometry by means of Q's guidance. By means of Q, the geometry of the crystal would manifest the forces and their patterns, and the forces and their patterns would manifest the geometry. Through this, stability, and thereby constancy, would become manifest and enabled through the dynamic of regeneration represented and guided by the dimensionless biological constant. Significantly regarding the operation of a Q based geometry and force in genetics, a given cross-section of the DNA molecule making up the genetic material is also a decagon in which Q is represented. (See Hemenway, 2005.) Again, this might suggest a maximum degree of patterning, uniform force throughout the DNA molecule for its stabilization, a stabilization enabled by Q, and a stabilization that would be needed for a stable or constant inheritance of the gene. Perhaps, also contributing to this stabilization is the ratio of the angstrom distance of one complete turn of the DNA molecule to its diameter, which according to some reports is 1.618 or Q (34A/21A). (For example, see Hemenway, 2005.) Other reports have the diameter as being 20A. Even so, the ensuing ratio of 1.7 is not significantly different from Q. When DNA is in less than a stable state, as when it is under stress, it can undergo mutation. In this regard, when hypermutation
rates within genes in bacteria due to environmental stress are divided
by spontaneous mutation rates of such genes under no stress, the factor
Q (or its various forms such as Q From the preceding accounts, we see how or why these implicit and explicit dimensionless constants, such as i and Q, compose in different forms and ways the various, universal dimensional constants of physics, and, in so doing, playing a significant role in physical and biological phenomena on various scales, and hence the deep connection between force and its vortical generation and the dimensionless constants. Namely and more inclusively, the dimensionless constants, which include primarily and explicitly the various forms or powers of Q and Π, and, also implicitly, the constants i and e, the exponential constant, collectively represent, define and enable a constant dynamical drive to achieve a maximum degree of dynamical uniformity within and between all specific scales of physical and biological systems or phenomena. Such a drive would be through the fractal creation---primarily defined or enabled by the golden ratio and its reciprocal or powers---of those very specific systems on all levels of fractal organization. Through the explicit, primary and necessary presence and application of this underlining, biological dimensionless constant, Q, or its forms, could a maximum uniformity of force be achieved in various systems or phenomena, and hence their maximum stability or integrity. In this situation, the other dimensionless constants would appear to define and represent implicitly the constant, contributing, curvilinear or vortical generations involved, thereby behaving in a subsidiary though significant role in achieving a maximum, adaptive stability. The deeper implication is that
golden ratio or its reciprocal applies to all particular levels or
scales of reality, from below the quantum level, where six, hidden, new
curled dimensions might be present in addition to the four dimensions of
our reality, to the cosmological level. Physicists conjecture in various
publications that there may be ten dimensions to the universe or
physical reality. This might be denoted or indicated subtly by the
repeated occurrence of Π The Plank length or scale, which is
Q x (Π According to Penrose, it would be the incompatibility of these different gravity-fields, their geometries, with one another, and a dynamically driven accommodation to such, that would determine the resolution towards the reduced, stable quantum state, that is, the collapse of the unstable, superimposed, non-local system into a reduced, local, dynamically stable system. Might this stability be based on a uniformity and fusion of forces defined by a geometry structurally defined by Q, and whose stabilization through Q might eventually be made evident through experiment? Also, the persistence of the superimposed quantum states through evolution could involve a spiral regeneration of force through time, a regeneration defined by the imaginary constant 1i within an algebra of complex numbers As Penrose points out, such an evolution represents a determinism at the quantum or micro-level of existence, and this evolution of superimposition can only be described through the algebra of complex numbers. This process of stabilization, the ceasing of superimposition, might even occur in a self-similar manner through and by means of an increasing fractal whose geometry is based on Q and the imaginary number or constant, i. (There is a very close regeneration-correspondence between Q and i as pointed out by Lieber, 1998a.) Perhaps, such stability of states would also occur concurrently at an extremely small scale connected or opening to the six hidden, curled dimensions on and of even smaller scales within fractals, whose very generation could predictably also be defined by Q, i, and Π. Reality would appear to be a complex fractal composed of adjacent fractals within fractals---possibly vortical in configuration through self-similar, logarithmic golden spirals reflecting or defined by Q---whose underlining constant parameter is Q or its reciprocal. (See Figure 1 and Figure 2.) Perhaps, this fractal reality could also be described by an analytical geometry of complex numbers, as well as by Π, e and Q. This constant, dimensionless parameter Q could determine or enable the morphogenesis of this universal fractal and the self-similar, uniform configuration of forces that would sustain it stably through all various scales, by means of a projected universal template, whose facets would be universally featured, expressed, and templately defined through Q, and the the other dimensionless constants. This would be a projection via imprinting through all scales. It would be a morphogenesis, occurring through such projection, that enables and represents maximum inner, dynamic, stable accommodation within and between all scales of itself. Might some day will the likely inherent logarithmic, vortical structure and vortical dynamic of reality at all scales become experimentally demonstrated, even at the quantum level. It is predicted that this will be the situation. One way this vortical process
structering reality could be represented mathematically is with the
following equation similar to the one in
Table 1: F Regarding these vortices, the boundaries between the adjacent, unfolded vortical fractals could represent the discontinuities on the quantum scale, but, inclusively at a higher scale, these boundaries could have themselves the shape of intersecting logarithmic spirals of curved lines of force emerging from the non-unformities within a generative plenum. In effect, relevant to quantum mechanics, discontinuities of energy or force could emerge from and be defined by the intersecting, spirally curvilinear continuities of geometrically enabled or mediated force via Q. A unity is brought to bear through different structures of reality.
III. Epilogue
Many computer simulations of various fractal morphologies, visible on screens, have been generated using a basic iterative or feed-back loop algorithm (with exponents) involving the complex number plane based on the imaginary number, i. Many of these fractals take on the intricate and complex shapes of various integrated spirals or vortices. Others closely resemble complex, branching aquatic plants and small, aquatic invertebrate animals. (See Pickover, 1990.) This iteration is involved in the generation of the self-similarity on various scales of these vortical and biomorphic fractals. Such iteration or looping or twisting, producing, engaging and connecting various scales, would be a necessary, significant feature of regeneration, especially of biological regeneration. As pointed out earlier by this author (Lieber, 1998a), the biological constant, Q, the golden ratio, or powers thereof, defines or reflects a self-similar process of regeneration through forces on various scales by means of a repeated imprinting within and through the space-time geometry of physical and biological reality. The very computer generation of the various fractal morphologies use and depend on electric and magnetic forces and the electromagnetic energy of light for the display to be visible. Though computer programs enable the generation of the fractals, such programs must be designed in such a way whereby the dynamic process of regenerative-iteration through different scales, via those forces and energies, has to take place, in effect duplicating or representing on a much smaller scale what is occurring in nature. In the natural world, the regeneration of self-similarity through iteration is defined or enabled by Q. This regeneration, spirally manifested and defined by Q, would also become the manifestation of a projective template, a type of symmetrical imperfection within an underlining plenum or substratum, creating imprinted non-uniformities within this plenum. This imperfection would self-similarly modify and define a space-time-field geometry and would manifest itself transcalerly through its evolving imprinting on the underlining plenum, and, in so doing, would create and projectively evolve stabilizing forces on all scales within phenomena. This could be a template that might be considered an inherent, universal program and a source of a guiding mathematics, as geometry. This would be especially a force-involved geometry structured and mediated by and through the dimensionless constants, such as Q. Such dimensionless constants would be the guiding avenues, through forces, that would enable the universals of the inherent template to extend into our existence and give dynamic structure and form to it.
IV. Concluding Remarks
One may ask whether such a template Yet, what about the traditionally empirical or scientific method of testability that has been so successful? At this stage of scientific development, the existence of such a template may not be directly testable due to the current limitations of our scientific apparatus or technology. However, with the development of new scientific technologies and approaches via the application of mind, the hypothesis could very well become testable. In fact, there may be an approach to such an experimental test through the known organization of the microtubules composing the neurons of the brain. The organization of the protein components of the microtubules, which make up the functioning cytoskeleton of cells, is based on numbers having the golden ratio with respect to one another. (See Penrose, 1994.) As Professor Penrose points out, various scientists argue that this particular organization, which is based on the golden ratio, enables the microtubules to function more efficiently through changes in their conformations mediated by electrical forces, behaving as information processors. As Penrose further shows, these microtubules are very likely required for higher consciousness, which he says could manifest itself as a quantum coherence on the macro-scale throughout all the interconnected microtubules of all the neurons of the brain. As the golden ratio, the dimensionless biological constant, is essential for the organization within the microtubules, is it not possible that the golden ratio can imprint itself ---via the dynamics shaped by this organization---on this coherence, our minds? And that this imprinting could be demonstrated? And through this imprinting, could we not have grasped its deeper significance and universality, a universality which would also suggest an imprinting of a template represented by Q onto our consciousness, our minds? Perhaps, through a further investigation of such microtubules, and their enabling of consciousness, could such an imprinting-template be accessed or demonstrated, perhaps holographically projected outside our minds.
In view of this, such testability could be feasible at some time. At present, this may seem to be a difficult if not a nearly impossible task. Yet, the history of science demonstrates how the development of new technologies and approaches enabled the testing of important theories, which before seemed impossible to test. Thus, it should remain for the future to ascertain whether or not such a template exists, but it will be a determination that ultimately depends on mind creatively connecting to and using a deeper reality, perhaps one reflected through a multidimensional geometry based on and allowed through the eternal, transdimensional, universal constants Q, e, i and Π. This should be the source of our motivation and courage to find new truths.
V. Further Thoughts about Predictions
Does physical theory predict the existence of the very dimensional physical constants of nature? According to Paul Lieber (1968), the very generation of stability and constancy within nature through forces would indicate the necessary existence of the physical dimensional constants in physical phenomena and their representation in a mathematical description of such phenomena. As he argues, the Special Theory of Relativity demands that the speed of light be a constant. The deeper implication of this is that the Special Theory of Relativity would have predicted the constancy of the speed of light, even if such constancy had not been demonstrated before. In the same article, Paul Lieber shows that the stability, a type of constancy, of the states of matter within quantum mechanics demands the existence of a constant such as the quantum of action, Plank's constant. Again, this implicitly predicts the necessary existence of such a constant had it not been demonstrated before the stability of the quantum states of matter was discovered. In view of this, it is likely that the other dimensional constants, such as the charge of the electron and proton, could have also been predicted. The physical dimensional constants represent or define the deep connection between constancy or stability in nature and the forces, such constancy and stabilization depending on the forces and their patterns. Implicitly, the dimensionless biological constant or its forms should also be predicted if these compose the physical dimensional constants. As it appears that this biological constant defines or represents a vortical regeneration that gives enhanced stability, and thereby constancy, to physical and biological processes or phenomena at various levels of organizations, this dimensionless constant Q can also be predicted as being demonstrated through such phenomena, especially processes having a vortically structural dynamic. The very constant electrical charge of an electron (and implicitly of a proton) demands the predicted occurrence of Q as a necessary structural representation of this charge. Why would this be the case as we note the direct evidence of Q in this constant? As pointed out (Lieber, 1998a), there is evidence that “particles” such as electrons and protons have an asymmetrical vortical structure, where one charge, say negative, occurs when the vortex is asymmetrically left-handed and the opposite charge occurs if the asymmetrical vortex is right-handed, or vice versa. In other words, electrical charge is dependent on complementary, asymmetrical configurations, that is, mirror images of the respective vortices. As pointed out in that article, such vortices suggest an inherent hydrodynamic nature of reality on all scales, including the quantum mechanical. Based on the vortical morphology of matter on the quantum level, one can predict that a constant of matter on that level would be directly represented by Q, which also defines the vortex of the logarithmic spiral and its generation of increasing stability from the quantum level to cosmological level. It would seem that Q and its forms, as well as other critical constants such as π and i, represent and manifest a template within the plenum that through these constants projectively guides and enables, via imprinting, the self-similar generation of those force patterns allowing for a maximum uniformity of force within non-uniform force magnitudes throughout all physical scales and situations, achieving through such uniformity, dynamic completion. And thereby, such a projective template into space-time would enable, through those constants, a vortical, hydrodynamic drive or conserving regeneration towards stability and dynamic completion on all physical-temporal scales. Through such a transformation process, a conservation of physical integrity or physical system, and connected physical laws, would be achieved or enabled continually, and hence what physicists refer to as physical symmetry. This would be through all scales of reality by means of vortical regeneration. This symmetry as continuity of system integrity or its dynamic maintenance would reflect a constancy through all spatial-temporal scales via vortical regeneration enabled by Q and the other constants related to the universal template. Might those enabling constants in effect be extensions of such a universal template, by virtue of which, those enabling constants such as Q being templates themselves? (See Figure 3 and Figure 4, and also see Figure 14, which could represent a vortically-structured reality emerging biologically from a plenum-field through a template projection manifested by Q's guidance.) As this article has argued for the existence of an enabling, underlying dimensionless biological constant within all the dimensional physical constants, providing evidence for its existence, thereby linking the physical constants, this article has not only predicted the existence of such a biological constant and its forms, it has shown their existence is very likely, especially their role in achieving maximum stability. Also, had this dimensionless biological constant been demonstrated in physical phenomena before the discovery of the dimensional physical constants, the latter might very well have been predicted because of the former. Though other enabling, biological constants analogous to Q might also be found. These would be respective growth constants per 90 degree turns of respective logarithmic spirals other than the golden spiral. These biological, growth constants, analogous to Q, and their defining logarithmic spirals would geometrically represent and pattern further vortical, generative features of projecting realities on all levels, some yet to be discovered. And thereby could manifest or determine new physical constants yet to be elucidated. These would be realities or systems, guided by constants, vortically developing towards a maximum dynamic uniformity or completion in non-uniformity, and thereby increasing stability, integrity and connected accommodation of the systems on all scales to a dynamic milieu. It is predicted that if physicists and biologists seriously consider the existence and enabling operation of a biological constant such as Q, or analogous biological constants, incorporating it or others into their approaches, a truly unified view of reality may ensue, showing a drive towards a maximization of stability and dynamic completion on all levels of physical reality or physical scales, a unification through an enabling constant such as Q. This would be a universal, transdimensional constant of particular vortical regeneration through which the known universal, physical, dimensional constants appear to arise and endure. In 1969(b), Paul Lieber was thinking along similar lines with regard to the universal dimensional constants of physics when he wrote: "These constants, I believe, are the universal constants of nature and therefore in particular of the life process. It is in these constants and in what their existence manifests that the processes which we arbitrarily ascribe to to the inanimate and animate worlds merge, and where we must search for their synthesis and thus their comprehension." And through a concurrent investigation of the universal, dimensionless biological constant, and of similar constants, such a universal comprehension will be achieved. This would be a comprehension that would conceptually grasp in total, and hence enumerate completely, the drive for dynamical completion through out the universe. Michael M. Lieber February, 2015 November, 2015
Some Dimensional Physical Constants Exhibiting a Dimensionless Biological Constant, Q (and its forms), Equivalent to the Golden Ratio, 1.618 and The Dimensionless Geometrical Constant, Π or Pi. Two dimensionless, physical constants are also included.
The electric charge constant of the
electron: Q x (Π Proton/electron mass ratio:
3Q Mass of the hydrogen atom: Very close
to Q x (Π Kinetic energy of a V volt electron:
V(Q x (Π Gravitational constant: (Π The quantum of action (Plank's
constant): 4Q x (Π Plank's constant is also: (2Q + Q Velocity of light: 5Q Avogadro's constant: ∏ Boltzmann constant:
2Q x (∏ Bohr radius of the atom: 2Q Nuclear magneton: (Π Compton wavelength of the proton and
neutron: 6(Q + Q Fine
structure constant: 12Q
Plank length: Q x (∏ 8Π is a
component within a compact expression (shown in various publications) of
Einstein's field equations for General Relativity. 8Π = 4Π
In physics, the equation or
expression s = 1/2gt Most physicists conjecture
that the universe has 10 dimensions. In this expression,
Π As 4
Π
*This method of calculating volumes through the rotation of areas is described in all calculus textbooks.
Equation for the logarithmic golden
spiral whose growth factor is Q per 90 degrees of turning: r(ϴ)
= e The generalized equation
representing the family of logarithmic spirals, each with a different, constant
growth factor as it turns with respect to its origin, can also simply be As represented by or implicit in this generalized equation, r is the vector-radius of the growing spiral that increases by some constant factor over each 90 degrees of its rotation (which in the case of the golden spiral would be Q) with respect to the coordinates and origin, and by a rate of 1.718 per 90 degree rotation with respect to each small segment of the growing curve, where i represents an operation of a 90 degree twist or curving turn as described by Clifford. However, when the vector's length remains constant during rotation, the equation defines a circle in a complex number plane. These coordinate systems would appear to be suitable for illustrating the generation of various vortical designs in nature with constant features through space-time. Some of these yet to be discovered. The numeral 2 appearing in the equation for the golden spiral, and in some of the equations or expressions of the universal dimensional constants, might denote the duality inherent in the projective imprinting by a template upon the plenum, which generates non-uniformity within reality or space-time, a projected non-uniformity through which the further imprinting generation of vortical designs can be. Might the very exponents composed of 2 in the equations also denote the enhanced contribution of this dynamic property.
A group of adjacent, vortically shaped succulents. Might such a living group reflect as a generative structure the true, vortically dynamic, discontinuous morphology of space-time emerging on all scales from a pervasive plenum. The elongated, reddish structures could be symbolic of the projections of an underlying template within the plenum.
A Begonia flower displaying a fluid-like, vortical morphology. This morphology might reflect an inherent generative, dynamic pattern projected within all scales of reality and enabled through the biological dimensionless constant, Q.
Figure 3
Vortical morphology of a Gastropod shell. Might such a morphology also recapitulate the inherent, dynamically-patterned extension into space-time, through and from a fluid-like plenum, of a universal template manifested through all scales of reality via the constants.
The fluid-like, vortical morphology of a Ranunculus flower. Might such a morphology also recapitulate the inherent, dynamically-patterned extension of a universal template manifested through all scales of reality via the constants.
Figure 5
A Rose with a fluid-like, vortical morphology, possibly reflecting and recapitulating an inherent, enabled, vortically dynamic structure within the field-fabric of space-time on various scales.
Figure 6
A group of an integer number of vortical roses. Would this group represent the emergence of vortical morphogenesis with quantum discontinuities from a deep substratum of existence?
A section of the equiangular or logarithmic spiral. From Clifford, 1899.
The angles of 30 degrees and 60 degrees originate at the vertices in all the small, right triangles along the curve. These small triangles occur through the intersection of the growing spiral radii with the curve. Through the generation of these small, identical triangles, based on the increasing radii during rotations of 90 degrees of such, the growing spiral is defined. The respective cotangents of a 30 degree angle and of a 60 degree angle are the same, namely 1.718. The identical cotangents of these angles along the curve correspond to the constant logarithmic rate of differential growth of the spiral curve. The constant logarithmic rate occurs via the rotation or twist of the growing, end segment of each radius through 90 degrees, forming the 90 degree angle of each identical triangle. This constant growth rate ensues in the generation of the constant, e, which is also a parameter of logarithmic spiral growth. This geometrical analysis of the logarithmic spiral makes manifest the constant parameters or dimensionless constants involved in the generation of the spiral, and thereby, become explicit in or defined in phenomena.
References
Clifford, W. 1899, re-published 1946 &
1955. Hemenway, P. 2005. Lieber, M. 1998a. The Living Spiral. A Dimensionless Constant Gives a New Perspective to Physics.
Lieber, M. 1998b. Enviromentally
Responsive Mutator Systems: Towards a Unifying Perspective. Lieber, P. 1968. Constants of Nature:
Biological Theory and Natural Law. In Lieber, P. 1969. Aspects of Evolution
and a Principle of Maximum Uniformity In Lieber, P. 1969(b). Categories of
Information. In Livio, M. 2002. Penrose, R. 1994. Pickover, C. 1990. Posamentier,
A. & Lehmann, I., 2012. March, 2015
Statement by Michael Lieber, fall
1981:
SOME CONSIDERATIONS RELATING TO THEORETICAL BIOLOGY* by Professor Paul Lieber* University of California, April, 1961
Since 1943, I have been engaged in a study relating to the universal-physical constants leading to a number of inferences and conjectures, which may be relevant to furthering our capability in comprehending the life process. Concurrently, I was also engaged in studies on the foundations of classical mechanics and hydrodynamics in which the thinking was naturally influenced by the work done on the universal-physical constants, and conversely. The cross-fertilization and interplay of ideas springing forth from what may appear as completely separated areas of investigation has been constructive in producing a somewhat coherent picture relating these apparently disjointed studies. I have been especially concerned with the task of resolving a fundamental conceptual dilemma arising in Hertz's mechanics, and which refers specifically to the problem of materializing in a precise way the geometrical connections invoked by Hertz, and upon which, his theory rests in a crucial manner. An attribute of matter has been isolated, namely impermeability, which achieves this end conceptually at least, and which appears to have an equal significance in resolving a fundamental question arising in the study of the universal-physical constants. This correspondence in turn renders a concept which may appear significant toward evolving a way of thinking constructively and systematically about biological phenomena. My work in classical mechanics suggests with increasing force and conviction the operation in nature of a universal correspondence principle, which may be regarded as an extension of the correspondence principle formulated by Niels Bohr; and which may find its fullest and most perfect expression in the biological process. According to the extended correspondence principle, appropriately extended laws of classical mechanics would, for example, bear direct and complete rather than asymptotic correspondence to the principles of relativistic mechanics as they do in the sense of Bohr. This conjecture implies, of course, that the present statements of the principles of classical mechanics are in a sense incomplete and that a more complete, non-relativistic expression of them operates within the domain of classical mechanics; and that this form is in a certain sense incomplete rather than in asymptotic correspondence with the principles of relativistic and quantum mechanics. In this connection, it may be hard to imagine a sense in which the known principles of classical mechanics may be regarded as being incomplete. It is now conventional to regard them as deterministic, and it is accordingly tacitly assumed that in principle they render coordinate information with arbitrary accuracy. That is, that in principle there exists no upper bound upon the accuracy of coordinate information sought and obtainable in numerical representation. This, however, is but a conjecture which does not have its basis in proof. The number of solutions obtained by analytical representation for the coordinates of mechanical systems is indeed strikingly limited by their complexity; as reflected, for example, by the unsolved classical problem of three bodies. The fact that the fundamental connections between classical mechanics and thermodynamics remain to be grasped suggests a possible incompleteness in the principles of classical mechanics as they are known. There is indeed now reason to believe
that such an extended form of the principles of mechanics may find
expression as a variational principle in which space and time are
interconnected in a way which The theorem relating to the distribution of internal forces in a many-body system has also rendered information basic to the conception and formulation of what appears to be a fundamental variational principle for the hydrodynamics of viscous fluids. This variational principle seems among other things to resolve two long outstanding paradoxes of hydrodynamics, and implies a direction connection between asymmetric and time-dependent motions. What is equally interesting is that according to this variational principle, vortex and spiral motions are indeed preferred in non-uniform flows. Although this variational principle has been established as a theorem on the basis of the principle of conservation of matter, it renders much more information about flows than has been inferred from this conservation law directly. Its formulation may indeed be regarded as a synthetic technique for extracting the information cited above. Noting this variational principle is relevant here since it conditions cooperative phenomena in fluids and furnishes an example of how fundamental and new information about flows is obtained by such a formulation. This principle asserts that a measure of uniformity is a maximum for real flows. It is accordingly related in some sense to the extended variational principle mentioned above pertaining to many-body systems. A biological system may be regarded as a complex consisting of component systems which, according to our conventional way of thinking, are individually either predominately classical or non-classical in their operation. For example, the hydrodynamics of such a complex may be assumed to be governed by the laws of classical mechanics and thermodynamics, whereas the molecular chemistry is essentially quantum mechanical and quantum electrodynamical in nature. It is the strong and highly controlled interplay between these component systems which is indeed overwhelmingly characteristic of biological material. It is difficult to conceive how this interplay and control can find its expression in a correspondence principle which emphasizes their connections as being asymptotic rather than universal. Conversely, this observation concerning the function of biological material evokes compulsively the operation of a universal correspondence principle which emphasizes the strong rather than the asymptotic connections between the principles governing the function of their components. My research work pertaining to a study
of the structure of the universal-physical constants led to some
interesting and unusual conclusions. These require careful
re-examination and possibly both amplification and extension before they
can be submitted as results for publication. One conclusion reached
under this study, which may have fundamental implications to biology, is
that the universal-physical constants imply the existence of a class of
physical phenomena whose manifestations are detectable but nevertheless
in principle are not amenable to control. In this sense, this class of
phenomena are in principle inaccessible to direct observation and
manipulation by the application of impressed forces. The implication to
biology suggested here refers more specifically to the operation of
biological control systems, in which the control mechanisms are in a
sense inaccessible to disturbances emanating from the systems which they
control and dominate. The universal physical-constants thus appear to
provide The search for elementary particles in
physics appears motivated by and may be attributed to a conceptual
framework which implicitly adheres to the idea that all matter is
composed of elementary bodies and that any and all of the changes
occurring therein may be attributed to displacements in space and time
of those elementary particles. The identification with the
universal-physical constants of invariant space-time structures, which
propagate as a Biological control systems are striking in their precision of operation and power of organizing the function of macro-systems within an organism. What is an equally striking fact, and which must also be accommodated by any acceptable way of thinking about biological systems, is that these control systems are materialized in but a small fraction of the material constituting the organisms, which they control. These observations compulsively suggest that the material of which the control systems are made may be inaccessible to observation and thus to direct experiment. They may accordingly be identified with the class of physical phenomena, which according to the universal-physical constants, are in principle inaccessible to direct experiment. This inaccessibility would account for the immunity enjoyed by the biological control systems with respect to the functions of the macro-systems, which they control and dominate. The universal-physical constants imply a connection, as noted above, between a class of non-observable and observable phenomena, which are engaged irreversibly so to insure the invariance of elementary spatial temporal structures in the non-observable domain. It is here conjectured that these structures may be fundamental to biological control systems, and their invariance and inaccessibility may account for the immunity which they appear to enjoy with respect to observable changes.
*First published on this website. Copyright (c) 2016 by Michael Lieber
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photos and written content (c) Michael M. Lieber |

Michael M Lieber, GENADYNE
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Organogenesis Michael M. Lieber Ph.D.

M.M. Lieber