Darcy-Thompson, and Goethe before him wondered aloud over the geometrical regularity of animal and plant forms, leading Goethe to postulate the famous Urform, from which all biological forms are derived. This essay is the discovery of a working Urform presumably the Goethean Urform.
The Urform is the cube of eight cells resulting from the first three divisions of the egg cell.
Differences in the rate of further subdivision of each of the eight cells results in the three forms of nature, the sphere of the radial animals, the spiral of the the bilateral animals and the the concentric cylinders of the plant kingdom.
Organs are shaped by the fluid dynamic forces that guide the expanding embryonic membrane
A set of blueprints for the human body and other complex life is printed here for the benefit of those interested in creating artificial life, and for those merely curious about the natural manufacturing process. This is the summary of the work of the past twenty-five years, initiated by Stephen Jay Gould, with the participation of dozens of scientists and science illustrators.
By blueprints we mean a set of drawings that can depict the mechanical stages that direct the development of a complex structure from simpler elements. The stages depict the mechanical consequences of the endless serial subdivisions of a membrane-bound sphere of protoplasm.
The grand message is that function follows form. We walk because that is what happens when limbs alternately extend and flex. Birds fly for the same reason. Over billions of years the repetition of the same sequence of cell subdivisions, gradually extended, makes ever more complex configurations before the individual dies. Meanwhile the parts move according to how they were fabricated, with no regard for use or purpose. New configurations that lead to systemic failure go extinct. This is called natural selection.
It is no coincidence that nature looks organic. This tautology reflects that the organs of plant and animal seem to expand following the outlines of fluid dynamics—the shaping of river deltas, ships’ wakes, clouds, mushrooms. We demonstrate here that the body form of plant and animal is the result of the fluid dynamic patterns that self-organize in the expanding embryonic membrane.
Goethe famously said: “How is it I am able to immediately recognize life against the rocky, inorganic background, were there not an Urform?”
ORIGIN OF LIFE (The EVOLUTION Problem)
From Leonardo Da Vinci and Vesalius to Gray’s Anatomy, anatomists have provided meticulous descriptions of the bones, muscles, and organs of the body but have never offered a clue as to how these shapes came about. Darwin, and recent scholars, suggested that form arose out of the chaos of chemical errors. The changes in the beaks of finches and the path from fish to man have been plausibly described, but none account for the origin of the beak, not to mention the finch or the fish. Indeed, On the Origin of Species doesn’t actually address the rise of new species. Rather, it confronts the question of how selection might work once myriad variations have been established. Recent work highlighting the importance of genetic and epigenetic interactions in the evolution of Darwin’s finches suggests that neither one nor the other should be considered to have primacy in the emergence of new species. The mechanism underlying the origin of radically new forms thus remains an open question
The editors of scientific journals in anatomy and the chairpersons of university departments of biology are aware of their ignorance of the origin of the chief object of their study—the human body. Evolutionary biologists, under pressure to solve the problem, have come forth with ever new black box theories that, in common, offer no graphical causative descriptions of the steps from one cell to man. Biology, unlike physics and chemistry, remains a descriptive science, as was cosmology in biblical times.
Nineteenth century scientists observed the origin of life in the egg and the embryo but were finally literally unable to make head or tail of what they saw. Molecular biologists in the 20th century thought there was a code for the body in the DNA. They now know there is none. Despite its ambitious scope, the molecular revolution that began with the characterization of the double helix and is now culminating with the complete sequencing and manipulation of a variety of animal genomes, has not provided a satisfactory explanation for the emergence of animal form.
Zoologists know that all bilateral animal life—vertebrates, insects, crustaceans, etc.—share the same body form, i.e., a head with eyes and jaws with a tubular segmented body with pairs of pointed jointed limbs attached beneath. Based on the plausible theory that the origin of living form can be seen as a problem in geometry, a solution to this problem in the tradition of Euclid is presented herewith.
Embryo Geometry
The mechanobiological solution to the evolution problem:
The spherical blastula resulting from equal proliferation of the primal octet, deforms over time, as unequal proliferation causes evolutionary phasal changes of symmetry.
The proliferation of the first eight cells shapes the embryo.
Equal proliferation forms the spherical embryo of the radial animal.
Axial proliferation forms the cylindrical embryo of the plant.
Laterally asymmetrical proliferation forms the spiral embryo of the bilateral animal.
It follows that evolution is guided by embryo geometry
The body plans of complex organisms are predominantly radially or bilaterally symmetric. Animals with radial symmetry have vase-like bodies. Animals with bilateral symmetry comprise segmented tubes with anterior heads, dorsal eyes, and pairs of jointed and pointed limbs. All complex organisms initially develop from an egg that cleaves alternately along the three spatial axes, yielding eight cells arranged as the corners of a cuboid form. Further divisions create the blastula, a ball of hundreds of cells of fairly regular geometry derived from the earlier cuboid form. The blastula resembles an earth-like sphere, its populations of cells sequestered in north-south, east-west hemispheres, floating, continent-like, on the liquid core of the blastocoel. The model speculates that, like tectonic plates, these cell populations have ‘drifted’ over eons, resulting in perturbation and deformation of the original geometry of incipient animal forms. As will be demonstrated later, the major bauplane could plausibly have emerged from these cleavage patterns. ‘Embryo geometry,’ as we refer to it here, makes certain predictions about the morphology of animal forms that arise from global geometric constraints and mechanical forces acting—in conjunction with local mechanochemical and cellular mechanisms—on the shapes that characterize the organization of cell collectives in the early phases of morphogenesis. The speculative model offered here characterizes embryogenesis as a series of mechanically driven shifts in topology constrained by the physical properties of size and shape.
The single living cell undergoes serial binary fission to make a ball of cells, that upon proliferation produce one of only two possible configurations: the radial and the bilateral; the jellyfish and the fish; the flower and the bee. Biodiversity is the variations that occur in these fixed body plans by the altering of the proportions of the parts by varying the rates of growth.
Embryo geometry is the study of configurations that emerge in the mass of cells generated by the subdivision of the egg upon fertilization. The result is a coherent roadmap that directs evolution and guides the generation of the complex body from a single cell.
the mechanobiological events of vertebrate Embryology
Mechanobiology works by the strange behavior of ultra-viscous fluids called non-Newtonian, or thixatropic, a property shared by both the embryonic membrane and Turkish taffy. Illustrations of the following thumb-nail on development appear throughout the book.
“Turkish Taffy was originally sold in large sheets to Woolworth's stores, where pieces were broken off with a ball-peen hammer at the counter and sold by weight. In the late 1940s the company released a version in candy-bar size which the purchaser could whack against a hard surface to break into bite-sized pieces. This property of being shattered or broken by sudden shock, but still pliable and soft when chewed is possible because the candy is a non-Newtonian fluid. Since the pieces were both chewy and slow-melting in the mouth, it was a favorite for the frugal customer.[3] A bar still cost 5¢ in the 1960s. By that time, it was marketed by Gold Medal Candy Corporation of Brooklyn, New York”
The serial binary fission of the egg cell produces eight cells disposed as a cube within the spherical extracellular membrane. These are the foundation for the body axes, the upper four cells to form the head.
The proliferating cells form a patterned bi-layer sphere called the blastula. Rows and columns of cells form polar and circumferential bands. The latter will form articulated limbs as pressure differentials cause the blastula to enter its own interior, turning the surface pattern inside out, in an event called gastrulation.
Flat sheets of tissue roll up, forming hollow, tubular bones.
The limb girdle bands split apart at a polar midline forming articulated limbs as stress causes the limbs to fold and displace.
The origin of the regular coloration patterns in snakes and birds can be accounted for by this event, as banded patterns of the blastula enter the blastopore.
The embryo is formed as the blastula bi-layer flows as a viscous fluid within the spherical containing sphere.
The advancing embryonic membrane is a hyper-viscous liquid with the peculiar property called thixatropy. You can stretch a slab of thixatropic Turkish taffy slowly, but if you slap it sharply it will shatter like glass. Stress causes crystallization along planes that can part, leaving a surface with a rough texture compared to the smooth taffy surface.
The bones of the vertebrate skeleton are similarly differentiated. Every bone includes both surface types. The rough surfaces of crests and condyles provide the roadmap of the formation of the limb by flowing and shattering.
The exterior muscle layer acts like a tight body stocking for the skeleton. The viscous stretching of the muscle membrane sticks to bony landmarks.
The muscle membrane possesses the property called orientation, common in synthetic polymers. The sudden rapid stretching of the membrane causes a phenomenon called necking, where the tense sheet assumes a linear orientation of the molecules as they slide axially, making thinner, stronger segments: the tendons.
The fluid dynamics of the expanding embryonic membrane causes the edges to curl under, piercing the membrane, and emerging as claws in the limbs, and as the snout and jaws in the head.
THE GEOMETRY OF PALINGENESIS: THE SUBDIVISION OF SPHERES
In modern biology (e.g. Haeckel and Fritz Müller), palingenesis has been used for the exact reproduction of ancestral features by inheritance, as opposed to kenogenesis, in which the inherited characteristics are modified by environment. It was also applied to the quite different process supposed by Karl Beurlen to be the mechanism for his orthogenetic theory of evolution.
The subdivision of a sphere by sequential cleavages is described in the strange geometry called palintomy. Upon fertilization the hugely oversized egg cell is reduced to a same-size ball of hundreds of normal-size cells by ten or twelve rounds of binary fission. From the complex configurations resulting from the stacking of the subdividing cells, symmetrical patterns emerge, beginning with the first three subdivisions of the spherical egg, which produces eight hydrodynamically interconnected cells. Geometry and fluid dynamics can predict the configurations these eight cells will assume, based on simple parametric variables. These hypothetical constructions predict the forms assumed by plants and animals.
The idealized, hypothetical model of the last universal common ancestor is of eight cells arranged as the corners of a cube, each connected by a tube of membrane to the adjacent cells, and to a spherical plenum in the center of the configuration. These hypothetical constructions predict the forms assumed by plant and animal life. Radial body forms result in cases where the central plenum persists in development. When the central plenum does not develop, the four left and right pairs of connected adjacent cells are the progenitors of the bilateral body form.
Besides providing the emergent patterns of life, palintomy confers a remarkable topological property on the mass of cells produced by this interesting process—that of self-recording history—likened to the annual rings in trees. Sequential binary fission creates a generationally subdivided map recapitulating the steps of its creation, much as a map of a city shows the original historical subdivisions superimposed on the later subdivisions. Cells know their address and zip code. The dividing cells maintain their lineage spatially in the dividing cell mass. By the phenomenon of induction any past stage may be recalled, although stages may be skipped in embryogenesis as cells take shortcuts. These strange properties can account for two important otherwise unexplained phenomena: the inheritance of form and regeneration. The general architecture of complex plant and animal bodies can be deduced from the initial eight-cell figure, accounting for a wide variety of otherwise imponderable proclivities of nature for certain forms.
The inheritance of the fundamental cubical geometry of the first eight cells can explain the universal architectural styles and features of the animal body plan including Bilateral symmetry Head and body anterior-posterior symmetry, Dorsal–ventral symmetry.
The trilobites and crustaceans can roll up, as do the larvae and juveniles of insects, and vertebrate young. The vertebrate embryo is an unfurling spiral coil with the ear as the central axis, which easily rationalizes its complex spiral morphology.
Palintomy is the reduction of a cell by sequential binary fission alternately through the three axes of space. The first round of three cleavages produces the primordial cube inscribed in a sphere. The subsequent subdivision of each of these generates the bilateral animal body plan, alike for vertebrate, insect, crustacean or arachnid. Biodiversity is the continuation of patterns of expansion of form, predictable by the geometry of palintomy, or embryo geometry.
The configurations predicted by the fourth, fifth, and sixth cleavage upon the first three, in the absence of any constraint creates a sixty-four-cell figure of a form easy to imagine. But the egg subdivides with a fixed outer spherical shell that is a prime factor in the packing of the spheres that will ensue. The phenomenon of palintomy will endow the cell with the understanding of its ancestry in terms of a theoretical position in a cubical lattice however dislocated from its real place.
EUCLID AND EMBRYO GEOMETRY
The fundamental imponderables of biology derive from the chemistry and physics of the lipid molecule, which has the property of forming sheets that duplicate as a bilayer, and readily curve on themselves forming bubbles that subdivide and to regain their size in an eternal cycle.
The fundamental influence that is responsible for the diversity of the complexity of biological forms, in comparison to the inflexible rules for the accretion of inorganic crystals, is variants in in the lipid molecules—much in the way that brick structures differ from cinder-block structures or wood structures.
One ancient strain of cells came to form an outer, inflexible, spherical membrane that serves as a mold for the proliferating cells which quickly fill the space within. Thus, in the sequence of 1 cell, 2 cells, 4 cells, 8 cells, 16 cells, 32 cells, 64 cells, 128 cells and so on, the growing mass assumes a spherical form, concealing the underlying genealogical geometry that predicts complex plant and animal form.
The observed stages in early development are:
1. Fertilization
2. Binary fission
3. Compaction: the spherical constraint of the first eight cells
4. Blastulation: the formation of a multi-cell sphere
5. Gastrulation: the topological inversion of the blastula sphere
6. Organogenesis: the generative rule of fluid dynamics in late embryogenesis
The blastula is a multi-layer spherical membrane that replicates sequentially as a new interior spherical membrane, ultimately creating a multi-layer laminate within a hard, outer spherical shell, destined to form the skeleton. The interior layers form the muscles, which are oriented alternately latitudinally and longitudinally, neatly consistent with the famous striptease [MS1] of Vesalius. Inorganic geodes are formed likewise.
Plants
The outer constraining sphere bursts early on, leaving concentric cylinders free to root and shoot in opposite axial directions, the inner tubes squeezing through spiraling intersections as leaves, and fluorescing geometrically at the apex.
Animals
Embryo geometry is the topology of sphere inversion. In vertebrates, the skeleton layer separates in rectangular plates that curl to form tubular bones. Gastrulation brings the skeleton to the inside. The muscle layers stretch over the skeleton like a multi-layer body-stocking, attaching and splitting.
Insects invert twice in gastrulation, making an exoskeleton, classifying the insect as an inside-out vertebrate.
Jellyfish are an inside out blastula. You can turn an adult medusoid inside out.
Vertebrate organogenesis
The upper hemisphere of the blastula is the skull. The lower hemisphere is the torso, superficially banded by two limb girdles. The anterior surface of the scapula is contiguous with the skull. In development the scapula descends posteriorly, separating completely from the skull, but not without leaving evidence of the schism in the rough surface of the against-the-grain separations.
The innermost muscle layer is polar. The separation in hemispheres leaves two massive bundles that accumulate at the ventral and dorsal midlines, as the Rectus abdominus, and the Longitudinus dorsi.
The counterintuitive orthogonal disposition of the intercostals inside as well as outside the ribcage can possibly be deduced by the dynamics of gastrulation.
The diverse forms of hands and feet of tetrapods have managed to survive in spite of being, in most cases, poorly designed for the terrain, ever since crawling out of the water. These result from the early embryonic bone fractures that occur in the catastrophic mechanics of gastrulation.