home page (page 1)
home page (page 2)
E-Book

THE FRACTAL ORGANIZATION OF NATURE

                                                                                                                             JOHN A. GOWAN
                                                                                                                             (revised Jan., 2011)

Part 3. The Astrophysical Realm (4 Third-Order Equations)

Introduction
(See: The Fractal Organization of Nature (Table))

Just as atomic quantum numbers, charges, and their associated forces are the underlying organizing field of the Microphysical Realm, and DNA with its genetic code orders the whole of the biological realm, so gravitation is the dominant regulatory force of the astrophysical realm. Here, too, we find a 4x3 pattern in the fundamental field. In Einstein's own words, gravitation is a force described by 4 third-order equations: the changing rate of acceleration in four dimensions, x,y,z,t. The nucleosynthetic pathway, controlled by the alpha particle, is the other dominant force of the astrophysical realm, determining the life history and developmental sequence of stars. As we saw in the Microphysical Realm, this too is a force characterized by a 4x3 pattern, the 3 quarks and 4 nucleons of the alpha particle.

Because the hierarchy is ordered in terms of size, the astrophysical realm is placed last. In terms of time, or evolutionary sequence, it should perhaps be inserted between level 1 and 2 (since level 1 is the story of the Big Bang and level 2 deals with the production of the elements in stars). Nevertheless, presenting the hierarchy in terms of size is less confusing, and suits our illustrative purposes better than the very complexly intertwined actual evolutionary sequence.

The "Multiverse" Theory

Recently our conception of the Universe has become larger - increased, in fact, by one full level - with the general acceptance of the notion of the Multiverse. The Multiverse theory asserts that our Universe is but one of infinitely many which exist simultaneously, sequentially, or cyclically in an N-dimensional (or a-dimensional) hyper-space, brought into being by some type of inflationary explosions in the vacuum or void, like bubbles in champagne, perhaps requiring no net energy to produce.

The rationale for this seemingly crazy idea is that it may account for the peculiar values of the fundamental physical constants which are for some otherwise unexplained reason extremely delicately and precisely balanced to favor the existence and evolution of our universe and life form. While this is an essentially circular argument, like wondering why our skins fit our bodies so perfectly, the religious interpretation of this circumstance is that God created the Universe specifically for humankind. The "Multiverse" is the "rational" alternative to this religious assertion, but the theory is of such an extreme nature that the religious view is hardly less credible or more speculative.

The Multiverse theory, in the service of the "Anthropic Principle", states that humans of course find themselves living in just that Universe, of infinitely many possible Universes, in which the physical constants and the interactions of Natural Laws have such values and strengths - by chance alone - that the evolution of our life form is favored. Now while this is a perfectly "rational" explanation in a statistical sense, the more I think about it the more it seems to be indistinguishable (by any possible experiment) from the religious notion of a unique Universe purposefully created by God for humanity. We get the same special Universe in either case by a purely speculative, mysterious, and ultimately faith-based process which neither side can demonstrate or otherwise prove. Moreover, the religious side can always claim the Multiverse for its own: of course God creates many Universes. God can, by definition, do anything.

I personally like and endorse the Multiverse theory. I think it leaves the door wide open to either the religious (intuitive) or the scientific (rational) view of creation - it becomes a matter of taste and personal preference - which is certainly in the spirit of a General Systems solution to this enduring mystery and controversy (which is not likely to go away anytime soon). (For more on the Multiverse, see: In Search of the Multiverse by John Gribbin, 2009, John Wily and Sons, Inc.)

It is quite possible to accommodate the Multiverse vision within the 4x4 framework of the astrophysical realm. We have to "telescope" the first two levels of the original version - Planets and Stars - into one level, the Stellar level, such that level 8 - Gaia - remains as the true planetary level, and level 9 becomes the level of the Solar System and Star, in any case a very natural progression. The previous Galactic and Universe levels remain largely unchanged, and a new Level 12 is added, the Multiverse Level. I will outline the new arrangement below:

"Multiverse" Formulation of the Astrophysical Realm

Level 9: Star (Solar System)

(Kepler's laws of planetary motion - 3 laws in 4 dimensions)

The first cell of level 9 contains the mass particles and aggregates that will eventually condense to form the planets and stars of this first row of the Astrophysical Realm. These mass aggregates range from interstellar gas and dust particles to meteoroids, comets, asteroids, moons, planetismals, and full size planets up to and including "brown dwarfs" ("failed stars" of less than ~100 Jupiter masses). This cell recapitulates and carries forward the planetary size range of Gaia in the last biophysical level (row 8). Hence in this first cell we can place all the particles of the primordial solar nebula, including our own planet Earth and similar bodies, provided only that they are not so massive that they become stars.

The pairing function of this level refers to gravitationally bound orbital pairs on the astrophysical scale, including gas, dust, and debris fields, asteroids, comets, planetismals, moons, planets, etc., anything less than an actual star. The Earth-Moon is a typical example of the pairing function in the second stage of level 9. We do not know exactly what role our (very large) Moon played in the evolution of life on Earth, but we can be sure it has been very significant, including its influence on the ocean's tides, the length of the day, and the regulation of Earth's orbital tilt, and thereby the stabilization of Earth's seasonal climate. The Moon was once much closer to Earth than it is now, raised much larger ocean tides, and may also have significantly influenced the Earth's interior, continental drift, and the Earth's magnetic field (to say nothing of the effect upon Earth of the Moon's birth in an apparent collision between Earth and a Mars-sized proto-planet early in the Solar System's history). The Moon will also have had some effect upon the bombardment of Earth by asteroids and comets, since from a distance its presence increases the gravitational attraction of Earth's locus, while nearby it may serve to deflect or absorb incoming "missiles".

A particularly appropriate expression of the 4x3 organizing force in the group category of level 9 are Kepler's 3 laws of planetary motion, ordering the Sun's family of planets in the 4 spacetime dimensions. It seems quite likely that many if not most stars likewise shepherd a family of planets that is born with them as they condense out of the gas and dust clouds and debris fields of their stellar nebulas and nurseries. Our technology is just now advancing to the point where we can detect planetary companions of stars.

The group function of the Solar System is the gravitational interplay between the various members of the system as they form out of the primordial solar nebula, determining the major parameters of their orbits, spins, axial inclinations, etc., and including their sizes, composition, history of bombardment, and many other significant evolutionary features. Many of these details are still being worked out for our own system through computer simulations, and the dynamics of other such systems are still in the earliest phases of discovery and theoretical investigation.

The Sun, our glorious star, provides us with heat, light, and energy from its fusion reactions, converting hydrogen to helium. The sun is far and away the principle source of energy for life on earth, although recently a few forms of life have been discovered in the deep oceans utilizing the chemical and thermal energy of the earth's interior (around hydrothermal vents on the ocean floor. These findings are chiefly of interest for theories concerning the origin of life on the early earth, alien planets, or in otherwise unfavorable or unlikely environments). The Sun's gravitational field orders the Solar System and keeps us comfortably near the solar hearth. Earth's heavy elements, however, are not produced in the Sun, but in the galaxy of level 10.

Level 10: Galaxy

(the nucleosynthetic pathway - alpha particle - 3 quarks x 4 nucleons)

I identify a galaxy as the next complete level of natural organization above the star, which itself is the natural unit of the galactic level.

The pairing function at the galactic level is expressed through binary stars - as many as 50% of the stars in our galaxy are estimated to be binary. What is of greatest relevance for us, however, is that "failed" binaries produce planetary systems. In the case of our own solar system, if a little more debris and gas had escaped the central disk of the primordial Sun, Jupiter would have become a second Sun in a typical binary system. As it is, the most significant pairing in our own "failed" binary system is between Mother Earth and Father Sun, whose fertile embrace has produced the biophysical realm.

A second expression of the 4x3 pattern we have already visited, in level 2 of the microphysical realm, the nucleosynthetic pathway of element production. This pathway is a 4x3 process, controlled by the alpha particle, or helium nucleus: four nucleons each composed of three quarks. The nucleosynthetic pathway in stars proceeds mainly through the incremental addition of alpha particles such that atoms with even atomic numbers (numbers of protons) are more commonly produced than atoms with odd atomic numbers. Our Sun produces mostly helium, as it is not large enough to produce the heavier elements. Larger stars produce oxygen and carbon as the next major elemental product after helium. Carbon, as we have seen, is another perfect 4x3 resonance, 3 alpha particles each containing 4 nucleons. Oxygen is a 4x4 expression of the fundamental algorithm (a commonly seen extension or "filling out" of the 4x3 pattern). A final resonance of the 4x3 pattern, chromium 48, fails because of the electrical repulsion between protons, and settles at iron 56, where another alpha particle and 4 additional neutrons dampen the repulsive effect. Iron is the final exothermic (yielding energy) reaction product of the nucleosynthetic pathway; all heavier nuclei are endothermic products (requiring energy inputs). Thus the energetically favorable pathway ends with the failure of the 4x3 resonance, as we should expect. The nucleosynthetic pathway determines not only the relative abundance of the elements produced, but the life history of the star as well, in terms of its activity, longevity, and final state.

Stars are the workhorses of the Universe, the principle sites where the symmetry-conserving forces of matter and spacetime begin the process of reconverting particles of matter to photons of light. Stars can only begin this conversion, however, since all they can do is extract the excess binding energy from baryons, forming them into compound nuclei with lower total energy states. Stars cannot actually convert baryons directly to free energy, but only reduce their total mass, which is mostly composed of the binding energy of their constituent quarks.

Although we designate a star as one of the bound states of the hierarchy, what we typically think of as a star (our Sun, for example) is hardly a bound state at all, but rather a work in progress, whose principle activity is shedding as much energy as it can in its journey toward a true final bound state. Three final states of stars are known, all composed of exotic, superdense states of matter. One is the "white dwarf" (which is the final state for ordinary stars such as our Sun), in which the electron shells of atoms are crushed by gravity into an electron "sea" surrounding the atomic nuclei. A second, even denser state is the end point of heavier stars, produced in the death throes of a supernova, the "neutron star". In neutron stars the electrons have been forced to merge with protons, producing a giant atomic nucleus (of stellar mass) composed entirely of neutrons, all bound together by the force of gravity. The third final state of the heaviest stars is the black hole, in which gravity becomes so strong that it equals the electromagnetic constant c. No light can escape from such a region of spacetime. It seems likely that black holes are filled with gravitationally bound light, a sort of "light crystal", or light transformed to rest, a state Einstein thought impossible.

Whereas we can think of the ordinary rock as the free energy of light "transformed to rest" in an asymmetric state of mass and atomic information, we can similarly think of the black hole as the entropy drive of light (light's "intrinsic motion") "transformed to rest" in an asymmetric state of time and dimensional information. Time stands still and becomes visible as the surface area of a black hole's "event horizon" (the Bekenstein-Hawking theorem relating the entropy of a black hole to the surface area of its event horizon).

What happens to matter in a black hole is unknown, but baryon decay is a strong possibility. If baryon decay does occur in black holes, then the black hole would consist of nothing but bound light, a condition not unlike the state of ordinary gluons bound inside baryons. The black hole would therefore resemble an enormous single baryon, rather than an enormous compound nucleus, as in the neutron star. In such a process, gravitation would effectively reverse the procedure of the "Creation Event", imploding matter into light rather than exploding light into matter.

Stars have a birth, a developmental history, and a death, often spectacular and providing a legacy to future stellar generations which they both initiate with shock waves and endow with heavy elements. In black holes they may even find an "afterlife" and their true "salvation". Stephen Hawking, in perhaps his most remarkable achievement, showed that black holes, despite all expectations, succeed in producing radiation, and in fact will eventually convert their entire mass to light. We see in this stunning result the achievement of the ultimate goal of gravity and the other conservation forces, the complete conversion of mass to free energy. Given the absence of antimatter, proton decay is the only other process which has been proposed that can accomplish this task.

While there are definite limits to the size of white dwarfs and neutron stars, there are no limits to the size of black holes, and black holes of millions and billions of solar masses are believed to occupy the centers of most (possibly even all) galaxies, including our own. These central supermassive galactic black holes are the Astrophysical fractal analog of the atomic and cellular nuclei of the Microphysical and Biophysical realms. All galaxies apparently have a supermassive black hole at their center; the current thinking is that the black hole comes first (from the primordial generation of supernovas), and slowly gathers a galaxy around itself.

The production of heavy elements in stars and the distribution of these elements through space is of course of vital importance to biological life, which is based upon carbon (organic) chemistry. Life has also found uses for many of the other heavy elements. Biological life as we know it could not exist without stars to manufacture the heavy elements or supernovas to disperse them, form planetary systems, and provide a steady source of energy for the billions of years life requires to achieve the complexity of its higher forms. Stars, from the point of view of biology, are the indispensable focal point and generator of the biological pathway, just as they are the focus and initiator of the purely physical process of symmetry restoration. They are the most important objects in the Universe from any point of view, and it is small wonder that our own star is so often in human culture equated with the Deity.

The "Group" function of the galaxy is to provide this "social environment", especially in the arms of spirals, where multiple generations of stars are born together in giant molecular clouds and share their precious product of heavy elements with each other. Our Sun is presumed to be at least the "3rd generation" star of such a stellar "nursery" (and in a galaxy with four spiral arms). Galaxies have well defined structures, shapes, and even "personalities" (active, quiescent, young, old, bright, dull, blue, red, etc.) dependent on their age, size, and interactions with other galaxies.

Level 11: Universe

(gravitation: four 3rd-order equations)

The galaxy is the natural organizational unit at the scale of the Universe. The pairing function is therefore between galaxies. In our own case we have the Milky Way - Andromeda pair. Our giant neighbor is the furthest object visible to the naked eye (about 2.2 million light years distant). Galactic pairs are thought to be significantly involved in the formation of spiral galaxies, the gravitational interaction between them being deemed necessary to draw out the arms of the spirals. As we saw at level 10, the spiral structure is believed necessary for continuous star formation and the phenomenon of supernova generations. Hence the heavy elements upon which Earth-life is so dependent may well depend for their existence upon a ponderous dance between galactic partners.

As in the biological realm, the astrophysical realm soon becomes so complex that we lose the visible thread of the physical 4x3 process underlying the system, and we have to once again look for dynamic processes that follow the Land-Jaccaci sequence and thereby reveal its underlying organizational activity. Although we know that gravitation, a 4x3 force, Kepler's 3 laws, and the nucleosynthetic pathway, a 4x3 process, all underlie the dynamics of stellar and galactic formation, how are they expressed in galactic structure?

One way to see the workings of the general system is to note the curious similarity of galactic structures to those of a cell or organism in the 3rd column of the biological realm. While there are many types of galaxies, spiral galaxies such as our own may be the most likely to harbor life. Like the cell, our galaxy has a nucleus, a central dense object or region (apparently a supermassive black hole), which through its gravitational field dominates the organizational structure of the galaxy, apparently including its growth, much as the cell nucleus of biology (or the atomic nucleus of the Microphysical Realm) dominates its sphere of influence. (See: Sky and Telescope, April 2005, pages 43-47.) The galaxy contains discreet parts, such as globular star clusters, which seem to mimic the organelles of the cell; it contains giant molecular clouds, great regions of unconsolidated gas and dust which are its energy reserves, much as cytoplasmic sugars and starch molecules are for the cell; stars themselves act like mitochondria and chloroplasts, metabolizing these gas reserves and gravitational energy into the energetic and elemental output of the galaxy. Galaxies like our own contain discreet regions, a central bulge and halo, and great spiral arms which function as stellar nurseries, the birthplace of stars, and a reasonably well defined gravitational boundary and recognizable overall shape.

Galaxies have, moreover, a dynamic and developmental history, a youth, maturity, and old age. Like cells, they occur in many sizes and shapes. The function of a galaxy, in purely physical terms, is to convert as much mass as possible to light, reducing the bound energy symmetry debt of the Universe. It is far more efficient to do this in the gravitational "social context" of a galaxy than through proton decay in a disorganized cloud of gas scattered throughout spacetime. Gravitation, through its unrelenting and unidirectional force, can bring these great clouds together and through the nucleosynthetic pathway change at least some gravitational and mass energy back into its original form, light. From the viewpoint of biology, it might seem that the "purpose" of a galaxy is to manufacture and disperse the heavy elements of the periodic table. From the viewpoint of physics, and the conservation of energetic symmetry, however, the heavy elements are just byproducts of the debt-repayment pathway, whose end goal is the gravitational conversion of bound to free energy, in stars and via Hawking's "quantum radiance" of black holes.

When a giant star (much larger than our Sun) reaches the end of the nucleosynthetic pathway, which as we have seen, is the end of the 4x3 nuclear resonance series, its final product is iron. No further energy can be extracted from nuclear fusion, and it collapses upon itself and explodes as a supernova, throwing large amounts of heavy elements into the galaxy where they become incorporated into other cloud (gas and dust) systems. It is thought that the shock waves from supernova explosions trigger new waves of star formation by initiating the collapse of adjacent gas and dust clouds, so the death of one star triggers the birth of a new stellar generation. It is believed that at least 3 such supernova generations have occurred in the spiral arms of our galaxy; it is these arms which are the disturbed, active regions of star formation, death, and rebirth.

From the perspective of biology supernovas are dispersing heavy elements into situations where small, ordinary stars like our Sun can form and burn quietly for a long period of time, warming a family of planets (also composed of heavy elements) at the solar hearth. This sequence of events can obviously only take place in a galaxy, where there is a sufficient accumulation of gas and dust to support the supernova generations and statistically insure that at least some low-mass, quietly burning, "well behaved", long lived stars like our Sun will emerge with a precious load of heavy elements they themselves could not possibly produce.

The dispersal of heavy elements from one stellar generation to another is again rather analogous to the exchange of genetic information between generations of biological cells. So from the biological, or "anthropic" perspective, we see the galaxy as the astrophysical system necessary for producing the elemental requirements of life, and for nurturing its development. In doing so, it is not itself unlike a biological system with its specialized parts, dynamic, energy flow, organization, stellar generations, and developmental history.

The 4x3 gravitational drive toward symmetry repayment, which passes though the 4x3 nucleosynthetic pathway, produces within certain appropriate galactic systems, a 4x3 expression of electron-shell chemistry, DNA based life. Because life is so deeply embedded in the 4x3 algorithm, from the nucleosynthetic pathway of stars and the structure of galaxies to the electron shells of atoms and the DNA of cells, life cannot be construed as merely an accidental output of the system, anymore than is carbon. Life is rather a natural and inevitable consequence of the 4x3 fractal algorithm and system operation. (See: "Newton, Darwin, and the Origin and Abundance of Life in the Universe".)

In summary, in a spiral galaxy, such as our own Milky Way, we have the phenomenon of stellar generations and inheritance as well as stellar birth, development and death. There appears to have been 3 supernova generations since the formation of our galaxy; therefore, the heavy elements upon which Earth life is so dependent are evidently the legacy of 3 stellar generations within the four spiral arms of our galaxy.

The group property of the Universe lies in its great galactic clusters, where galaxies interact with each other on a time scale of eons, keeping each other "young" through these "social" interactions which are necessary to stir their nuclear hearths to life, producing new energy and elemental resources for life.

In the final cell of level 11, the Universe itself supplies us with certain essentials that the galaxy cannot; these include the four spacetime dimensions, three forms of electromagnetic energy (bosons, leptons, and fermions), and the interactive set of 4 physical forces that characterizes the physics of the Unified Field Theory, the regulating "metric" of spacetime and matter.

Level 12: Multiverse

(4 conservation laws connected in triplets; physical constants)

We begin with the unit Universe, inherited from the final stage of level 11. At the pairing stage we have the Universe-Anti-universe pair which we encountered briefly in level 1 during the Big Bang. The Big Bang was essentially the annihilation reaction between this matter-antimatter Universe pair, resulting in the residue of matter which forms our present Universe. In the group stage we have the concept of the Multiverse, the set of all possible (electromagnetic) Universes. In our model, the electromagnetic subset of the Multiverse would appear to have some sort of inherent 4x3 metric, which, if our own Universe is any guide, may have to do with conservation laws governing energy expressions - the simplest sufficient set, in all cases.

At the group level of organization (all electromagnetic universes), we identify 4 conservation laws connected in triplets as the primary 4x3 expression of the fractal algorithm governing all such Universes in the multiverse: the conservation of raw energy, the conservation of the symmetry of energy, and their associated principles, entropy and causality (see: "The Energy Tetrahedron"). But here we have only a sample of 1 Universe (or 2, if we include our own antiuniverse, which already displays some very small difference (asymmetry) from our own).

Our Universe has a shape, a center, and nucleus, the fiery "Big Bang", which establishes the initial conditions and controls all future development. (See: "A Spacetime Map of the Universe".) It has a surface membrane, the Universal Present Moment, of which we are a part. Between the surface and center stretches a sort of transparent cytoplasm consisting of a record of the Universe's evolutionary past (historic spacetime, the conservation domain of Information and matter's "causal matrix"), a portion of which we can observe. We see embedded in this past (presumably they also exist in the present) the "cellular foam" of galactic clusters. Within these cell walls are dominant galaxies and dominant galactic clusters, a further level of organization. Within the galaxies are central nuclei of supermassive black holes, and stellar "mitochondria and chloroplasts", energy producing "organelles", feeding upon the gas and dust gathered by gravitation, and converting gravitational energy and simple elements (rather than sunlight and simple molecules) to free energy and heavy elements (rather than to oxygen and sugar molecules), restoring the symmetry of the Universe while simultaneously creating an environment for life. The Universe even has a "body temperature" - the 2.7 degree Kelvin background radiation remaining from the Big Bang.

The Universe has a birth, developmental phase, and a death (either a "heat death" or a "Big Crunch"). It is a conserved and regulated structure, engendered by symmetry-breaking and with a metabolism driven by entropy (the intrinsic motions of light, time, and gravity), conserving total energy and symmetry. It is in constant communication with all its parts, its nervous system composed of the metrical web of spacetime which transmits the vibrations of free energy and information simultaneously (in their own reference frame) everywhere. It "knows", for example, via the neutrino and graviton, the location, mass, and identity of every elementary particle within its domain. A complete record of its past (matter's "causal matrix") is also contained within the historic domain of its onion-like structure. Its light-order has reproduced itself on many levels in both structure and process. It is a cosmic creature of energy and spacetime, itself the largest iteration of the 4x3 fractal set, achieving self-awareness through its biological information component, life. The basic relationships of physical law are succinctly represented in the Tetrahedron Model. (See: "Symmetry Principles of the Unified Field Theory".)

The final position of the Multiverse level (12th row) is assigned to "First Cause". The 4 levels of the Multiverse astrophysical model are thus: Star, Galaxy, Universe, Multiverse, an entirely natural sequence. As we have seen earlier, the function of the Multiverse, in either biological, physical, or metaphysical terms, is stated by the "Anthropic Principle": to produce so many varieties of Universe and in such numbers that by statistical chance alone at least one Universe friendly to life forms such as our own will be created.

At the group position of the Multiverse level, we found the subset of electromagnetic universes with differing physical constants, insofar as this is physically possible and still remain within the electromagnetic energy type (they can at least differ in size). These are all characterized by the conversion of free energy into mass (since otherwise they just remain as light), and all have a 4x3 "metric" or fractal algorithm like our own. Perhaps many of these are only virtual, and immediately fail to materialize; perhaps many others give rise to very different life forms. Of these realms we can only speculate. But it seems reasonable to suppose that if our Universe exists, so too do others, and if our life form exists, so too do others. It now seems far less reasonable to suppose that our life form, and even our Universe, are singular, isolated occurrences.

Do universes interact in some significant way at the group level? We know of one annihilation reaction between a universe of matter and a universe of antimatter which produced the material residue of which we are a part. This interaction was at the pairing level. Interactions at some "group" level would appear to be anterior to the annihilation event of our "Big Bang", but might be involved in its mysterious asymmetric character and the origin of its 4x3 fractal algorithm. Would such an interaction involve only the subset of electromagnetic universes? It seems likely, as what would be the basis for interaction with any other energy type of universe? Gravitation? (We do not know, for example, if other universes are founded upon electromagnetic energy; or if they are dimensional, or if so, how many dimensional. We do not know what their laws and constants might be, assuming (as I think we must) that they have such things.)

The emergent new unit in the final column position of level 12, First Cause, is by definition the All-in-All. In religious thought, this principle of unity is expressed as monotheism, the absolute sovereignty of a single omnipotent Divinity. Scientifically, this unity is expressed in the conception of a single all-embracing Multiverse, the complete set of all possible universes of all possible energy types. This is the level of pure natural law, as given for (inherent in) each energy type.

However we wish to characterize it, the First Cause of level 12 provides the basis of energy, natural law, and physical constants from which our manifest Universe, as one electromagnetic production of the Multiverse, bursts into a conserved dimensional expression of light, spacetime, gravity, and particles. In biblical terms, this is the realm of the "Tetragrammaton" and the "Word", the Logos or Nous which preceded light in the genesis of the Cosmos.

Links:

home page (page 1)
home page (page 2)
E-Book