Geometrical anatometry, with its inherent  geometrical-mathematical base, holds  potentiality as a particular tool of biological  science and as a general tool for  metapsychology, the relationship between the   psyche and physical, biological processes.  The ability to compare structures within a  common geometrical matrix is of particular  usefulness for the osteologist, the  morphologist, the physiologist, the anatomist,  the comparative biologist, the osteopath, the  physical anthropologist, the paleontologist. . .  For comparative studies it provides a matrix  within which features of comparison may be  described and delineated with geometrical  order. The intellect both apprehends and  comprehends structure more quickly when the  structure can be shown to embrace order. Pd  anatometry provides a rigorous geometrical  paradigm for the descriptive delineation of that  order even though such a defined rigor is not  so precisely expressed by a Nature that is  regularly irregular, rigidly flexuous, statically  dynamic..

A small portion of the work is reproduced here by permission of the author. For more information on Geometrical Anatometry the author may be contacted by snail-mail at:

Page 1 

Over two thousand years ago Pythagorean doctrine remarked the significance of the pentad and the decad: Nicomachus subsequently expressed the import of the tenfold, the decad, thus: "But as the Whole was an illimited multitude... an order was necessary... and it was in the Decad that a natural balance between the Whole and its elements was found to pre-exist.... That is why the All-Ordering God, acting in accordance with his Reasons, made use of the Decad as a canon for the Whole... and this is why the things from Earth to Heaven have for the whole as well as for their parts their ratios of concord based upon the Decad and are ordered accordingly." And: "Because it (the Decad) was used as a measure for the Whole, as a set square and a rule in the hand of the One who regulates all things."

It was also Pythagorean thought, doctrine, that declared that the dynamics of physical structures depend on the interaction of contraries or pairs of opposites. And that all physical things can be ultimately reduced to numerical relationships.

Initially this inquiry into a particular aspect of human morphology (osteology) was undertaken in order to demonstrate and establish that man's skeletal structure is disposed to an ordered geometrical configuration. Concomitant with the objective, graphic, delineative aspect of the work was an investigation of the mathematics "virtual" to the configuring geometry. This study, in turn, engendered an awareness of a threefold, reflective, integrative, relationship obtaining between the geometry, the mathematics, and the physical manifestation of human form and function with modes, aspects, of consciousness. The material is presented in three parts. Part I illustrates the conformation of the human skeletal system to a geometrical matrix, and is descriptive of the methodology employed in the configuration. The delineation of man's skeletal structure, as developed in this work, conformal to a particular geometry, is both comprehensive and detailed. It is considered that the presentation is of an order sufficient to demonstrate validity for the disposition.

Part II examines the geometry basic to the delineated configurations, and investigates the mathematics, the role of numbers and concepts of symmetry connate to that geometry. The math-geometrical matrix generating the configured structural forms derives from a five-tenfold (pentagon-decagon) expression of shape. The word geometry conjoined with a term combining anatomy (analysis, division, dissection), and metric (measurement) determined the title of the material. The subject is sometimes referred to as pectagonal-decagonal anatometry, or for the sake of brevity, pd anatometry. A multiplicity of animate entities are configured to this pentamerous-decamerous structure from viruses to flowers to man.

The pd structure is a form linked to the organic. It does not occur in the inanimate crystals. Crystallographers have shown that this pd disposition, in fact, cannot exist in inorganic crystalline structure. The pd form is thus consonantal with the animate, the organic, and with endogenous growth (additive growth from within as a consequence of cell division and metabolism), distinct from exogenous growth (growth from without by accretion, the inorganic crystals).

A germane analogy of the physical "within and without" growth processes is a psychological one: cognitive growth by the "within" of intuition distinct from cognizance by physical, reasoning, observation of the "without".

Part III assays an exegetical commentary directed to the expansion and integration of the material addressed in Parts I and 11. It endeavors a unifying exposition of the relationship extant between man's psyche, and his physical form as disclosed by geometrical anatometry, by geometry, number, and music (harmony).

Page 2          Page 1      Top

The full-figure representation of man, as delineated in these pages, displays a most agreeable harmony of proportion. This harmony of proportion derives from a remarkable mathematical, geometrical entity, a proportion variously known to mathematicians, ancient and modern, as the "sacred proportion," the "divine proportion," the "golden section," the "magic Fibonacci number," and "division into extreme and mean ratio." All of which testifies to the import this "divine" entity has had on the mathematical mind. This proportion is an intrinsic property of the math-geometric complex that serves as the matrix for the configurative delineations of the human skeletal structure, and in that structure dictates a harmonic coadunation of the parts with thee whole.

Because of its remarkable mathematical properties, numerical, algebraical, and geometrical, and its ubiquitous role in natural structures of existence, this "jewel of geometry," this "divine proportion," may well be considered a constant, a fundamental invariant immanent to Nature. And it may well be a link, a mathematical nexus, that connects the rational, the discrete, the sequential, causality, quantum, with the arational, the multi-dimensional, the simultaneous, the acausal, the continuum. Its role, its presence, both explicit and implicit, is paramount to this work. As will be demonstrated, the delineated human figure springs from a math-geometric complex that embodies cannons of harmony and proportion that is consubstantial with Nature's dynamic structures.

The ordered complexity of the geometric matrix is such that it accommodates the infinitude of variations consonant with human form while uniting them by a harmonic proportion linking parts to parts and parts to a whole. Related to this statement are two observations by J. Bronowski (Science and Human Values, Pelican, 1964) as they relate, in turn, to the nature of beauty, the discoveries of science, and the creativity of art:

"When Coleridge tried to define beauty, he returned always to one deep thought: beauty, he said, is unity in variety! Science is nothing else than the search to discover unity in the wild variety of nature, - or, more exactly, in the variety of our experience. Poetry, painting, the arts are the same search, in Coleridge's phrase, for Unity in Variety.

"The discoveries of science, the works of art are explorations - more, are explosions, of hidden likeness. The discoverer or the artist presents in them two aspects of nature and fuses them into one. This is the act of creation in which an original thought is born, and it is the same act in original science and original art. . ."

Figures 1.30, 31 and 32 illustrate variations of skull shapes, culled from the fossil record, that are related to particular relatives of modern man, and demonstrate their distinctive configuration within a common geometrical complex or matrix. Figure 1.2 and 1.1(g) further delineate flowers, a marine sea urchin, a cactus plant, and a virus, all configured to the same geometrical matrix, which remarks the spectrum of diversity, the variety of structures that are conformal to the same unifying geometry. Remarked, too, is the innate conation of man to harmonize, to integrate, to unify his perceptions, his intuitions, his feelings, his interpretations of the wholeness of himself and the connatural wholeness of Nature's structures of existence. This conatus for unity in variety, diversity, is basic to his endeavored sciences, his expressed arts, and his visions of beauty and spirituality.

It is with the analogous, the analogic, that we endeavor to express, to unify, these likenesses, these correlative interrelationships that exist between Nature's superabundant forms, processes, and ourselves, our physical-biological, psychological, creative-spiritual selves. It is of some relevancy to note the meaning of the word analogous as it derives from the Greek word analogos (according to a due ratio, proportionate, fr ana + logos ratio). This identification of the modern meaning of analogy (likeness, correspondence, resemblance of attributes, effects) with that of ratio-proportion links the modern meaning to the above noted "divine proportion," "division into extreme and mean ratio," and to the math-geometric complex of geometrical anatometry.

The pd geometrical matrix is a threefold study in symmetry: (1) a symmetry of division, reflection, mirror image (bilateral symmetry), (2) a symmetry of sequential transformation, mappings (translations, rotations,, dilations), that express and reflect processes of growth, differentiation, and variations of form, and (3) a symmetry of

Page 3            Page 2      Top

Proportion that defines a signal harmonic coadunation of the whole and its parts. These three notions, concepts, of symmetry have mathematical analogs in three expressions of proportional means: arithmetic, geometric, and harmonic. These symmetries, these themes of proportion, are conjoined in, and are immanent in the human skeletal schema, a dynamic conjugative symmetry of innate wholeness physically manifesting a harmonic synergism of facets of human evolution. These symmetries may be characterized as geometrical analogs of physical structures, processes, connatural with reasoning consciousness, as the intellect seeks to understand and describe perceptions of the natural world or comprehend more fully, new ideas, new concepts perceived intuitonally or by analogy.

It is of particular relevancy to note that bilateral symmetry is expressive of a static, equilibrated state, and that transformational symmetry defines dynamic, sequential movement in time that is expressed in precise quantum steps. These discrete, explicit steps being imposed on the pd matrix, by all implicit proportional symmetry that dictates a harmonic concordance of the parts and the whole. Vitruvius, lst century B.C. Roman architect and engineer, defined symmetry thus: "Symmetry results from proportion. . . Proportion is the commensuration of the various constituent parts with the whole." These expressions of the discrete, the sequential, and tile proportional, have an analog in musical rhythm: The dynamic interplay of proportion and static, discrete, periodic beat.

Discrete world and omnidimensional existence: The math-geometrical complex delineating human skeletal structure images features of noted contraries, dualities, dyads of complemental inequivalance, expressing a dynamic, harmonic, conjugative alternation of the rational - arational, reason-intuition, the symmetrical and the asymmetrical ¹,division -integration, the discrete and the extreme, actuality-potentiality, explicit-implicit, the sequential (time) and the simultaneous (timeless), the physical and tile psychological, contraction-expansion, the objective and the subjective, concrete-abstract, the discrete and the continuous ², and the rectilinear and the circular ³, discrete sided polygon and infinite sided circle, discrete integers and continuous, nonrational numbers, the static and the dynamic, the rigid and the flexible ⁴. . . . and science-art, prose-poetry, nonfiction-fiction, judgmental-nonjudgmental, particle-wave.... All are psychological and physical pairings reflecting the phylogenesis of human consciousness, human biology, and in their reciprocal action tile harmonic dynamics of the structures of being, psychological, biological, and physical.

In The Identity of Man, Natural History Press, Garden City, New York, J. Bronowski addressed a "dyad of inequivalence," science and art, and by examining complementary processes of science and art remarked a unifying, integrative, philosophy of the nature of man, identifying the "whole range of the mind as one: the single identity of man." And, discussing science, states: "We do not know whether nature is made of waves oi- particles. The fact is, we do not know what basic stuff she is made of, and if we knew, we could not understand it otherwise than by analogy."

Language promulgates and affirms our conscious recognition of, and concordance with, these fundamental dualities intrinsic to our projective-perceptive, our objective subjective stance in physical existence. The analogic, the metaphoric, and the allegoric express reason's cognizance, through language, of the innate Unifying interrelationships that

¹ Ihe introduction of asymmetry into a static, equilibrated, symmetrical system is a condition for the dynamics of phenomenon: a form of Currie's principle.

² Ilcrman Weyl, Symmetry, Princcton University Press. "The splitting into something discrete and something continuous seems to be a basic issue in all morphology."

³ In the Philebus, Plato defined beauty of shape as: "...not like the beauty of living beings or of paintings representing them, but something alternately rectilinear and circular, and the surfaces and solids which one can produce from the rectilinear and the circular, with compass, set square, and rule. Because these things are not, like the others, conditionally beautiful, but are beautiful in themselves"

⁴ The rigid and the flexible: rigid ego-flexible imagination, rigid skeletal material-flexible flesh, rigid harp-flexible strings; the psychological, the biological, and the physical.

Page 4           Page 3      Top

are conjugate to physical and Psychical existence. They elaborate and enrich our understanding, and stimulate our imagination. It is with the analogic that we particularly give definition to the likenesses, the unities inherent to all manifestations of existence.The pd configuration of the manifest form of the larynx is illustrated in Fig. 1.37 as it is detailed within a constituent segment of the geometrical matrix that also details the vertebral column demonstrated in Fig. 1.34 and Fig. 1.38. The configurations example the three definitive symmetries, previously remarked, that are intrinsic to the pentagonal-decagonal matrix: (1) Bilateral (the symmetry of left-right, reflection, mirror image), (2) Translational (movement, the shape is directionally, sequentially, repeated), and (3) Proportional (that symmetry that links, catenates, the parts to the whole as represented in Fig. 1.7 and Fig. 1.8). The laryngeal structure also illustrates the synergistic conjugation of dyads of inequivalence; the rigid and the flexible, the static and the dynamic, essential properties of any instrument of sound. Conjugate to the semi-rigid manifest structure of the larynx are the constituent, flexible vocal folds, and immanent to the manifest form is the rigid-elastic nature of the cartilaginous material that composes that form, a distinctive example of the synergistic coadunation of dyadic inequivalence; the static and the dynamic, the rigid and the flexible.

Inequivalence, the asymmetrical, is a requisite of action, and essential to the dynamics, the phenomena intrinsic to animate structure. In order to accommodate action, mobility, flexibility, lungs must be able to expand and contract, trees must bend in the wind, rigid skeletal forms must incorporate flexile, elastic cartilage and/or curvilinear surfaces in their articulating structures, and the lens of the eye must accommodate the near and the far. An initial description of human morphology might well begin with the analysis, the anatomy (division, dissection into constituent parts in order to investigate them separately or their relationship to a whole) of the rigid and the flexible, the static and the dynamic.

Evident in the human body is a basically rigid skeletal form supporting and protecting a dynamic physical complex of sentient, sensorial features, and interrelated systems of life processes which, when activated by the psyche, are able to move about in, manipulate in, and experience the physical world with varying degrees of joy and understanding. We are all familiar with the facet of reasoning consciousness that is rigid, dogmatic, bullheaded, both in ourselves and in others, especially in others. Contrasted are functional virtues of mental flexibility, broad mindedness, compassion, and dynamic imagination expressing a potential for mental synthesis, assimilation of new ideas, new concepts, derived from sources of experience, intuition, dreams, and creative instincts.

This synthesis of the static and the dynamic, the rigid and the flexible, examples the synergism of the remarked dualities, and the cooperative interaction of contraries that generate a total effect that is greater than the sum of the parts.

T'he rigid and the flexible, the discrete and the continuous, the rational and the nonrational... If an even strip of paper, embodying, rigidity and discreteness, in one plane, and flexibility and the continuous in the other is, under the direction of the psyche, turned in on itself (tied into a simple overhand knot) as illustrated, a precise geometric figure, a polygon of five equal sides, the pentagon, is generated.

Page 5           Page 4      Top

This particular expression of the conjugation of the rigid and the flexible, the discrete and the continuous, physically yields, sans drawing instruments, the pentagonal shape, the geometrical matrix, that is the basis for the delineation of the human skeletal construct detailed in Part 1, and the mathematical complex, virtual to that matrix, examined in Part II. Expository pd delineations attendant to Part I demonstrate how the pd geometric structure, and its symmetrical transformations, accommodate the multitudinous structural variations and differentiae that mark human kind. The pd disposition thus provides for a multiplicity of form correlative with the myriad of possibilities constitutional to the genetic code.

The composite structure of man describes a conjugation of morphological dualities evidenced as a physiological, rigid skeletal construction supporting and sheltering a dynamic, flexible biological complex that functions under the direction and guidance of a sentient psychological system that embodies the dualities as properties immanent to modes of consciousness. These dyads of complemental inequivalence, the static and the dynamic, discrete-continuous, rigid-flexible, reason-intuition... extend from man's psyche to his gross physical-biological structure, his molecular makeup, and to the pristine, beginningless-beginning state manifesting the duality of light as discrete particle and continuous wave.

Geometrical anatometry, with its inherent geometrical-mathematical base, holds potentiality as a particular tool of biological science and as a general tool for metapsychology, the relationship between the psyche and physical, biological processes. The ability to compare structures within a common geometrical matrix is of particular usefulness for the osteologist, the morphologist, the physiologist, the anatomist, the comparative biologist, the osteopath, the physical anthropologist, the paleontologist. . . For comparative studies it provides a matrix within which features of comparison may be described and delineated with geometrical order. The intellect both apprehends and comprehends structure more quickly when the structure can be shown to embrace order. Pd anatometry provides a rigorous geometrical paradigm for the descriptive delineation of that order even though such a defined rigor is not so precisely expressed by a Nature that is regularly irregular, rigidly flexuous, statically dynamic...

Nature's use of the explicit pentagonal-decagonal construct and its implicit, virtual structures holds relevant implication for the graphic artist, the designer, architect, sculptor, and engineer. ¹ An architecture for the environmental use of man, practically and aesthetically, surely needs to incorporate and express forms and proportions that are in harmonic consonance with man's own structure.

This work, Geometrical Anatometry, addresses themes, aspects of mathematics, music, anatomy, geometry, morphology, language, symmetry, physics, biology, and psychology. The work endeavors a demonstration of the consubstaniality of these multiple subjects as they exist within the psyche and reflect human form, physically, biologically, and psychologically-spiritually. It is a work expressive of the analogic, remarking and demonstrating the multiplicity of unities. that are embodied in the gestalt of our physical and biological structure, and our gestalt of being.

The human gestalt is awesomely complex and wondrously composed. Perhaps the present work will, in some measure, render it less complex, while quickening the wonder, and furthering our understanding of the concordant relationship that exists for the psyche and the physical body, and the psyche and matter. Plainly they are not unrelated, and any inquiry directed to the elucidation of the nature of one also serves to expose the nature of the other, and their inherent consubstantiality.


¹ Matila Ghyka, "7he Geometry of Art and Life," Dover Publications, New York NY, has provided an historical account of the use, by artists, architects, and engineers, ancient and modern, of aspects of penatagonal-decagonal geometry and the "divine proportion." It is hoped that the present work will stimulate renewed interest in this remarkable math-geometric matrix of form and life.

Page 6           Page 5     Top

FIGURES

1.1 Fundamental features of the geometrical matrix, complex, intrinsic to pentagonal-decagonal anatometry. They provide an introduction to the geometry that is described more fully in Part II.

(a) The basic regular polygon, the pentagons with a discrete number of sides, circumscribed by a circle, a planar shape considered as having an infinite or continuous number of sides. The discrete and the continuous, the angular and the curvilinear, the rational and the arational. Shown too, are a vertical axis and a horizontal axis, essential to any physical description of man because of his vertical stance, his orientation to an earthly "cube of space," and his bilateral construct. It may be noted, as exampled in Figs. 1.17, 1.25, and 1.26 that the horizontal axis bisects the head circle of "Decagonal Man" by a line, a plane, through the center of the eyes, a feature typical of modern Homo sapiens. This is a particularly distinctive lineament when compared with pd delineations of some of contemporary man's predecessors as configured in Figs. 1.30, 1.31, and 1.32.

(b) The pentagon and the star pentagram generated by lines connecting the vertices. Also shown is an inverted pentagram that is virtual to the primary pentagram. The "golden triangle," ABC, is a prominent structural feature of the thorax, and is described in this work as the thoracic deltoid. See Figs. 1.46, 1.48 and 1.51.

(c) The decagon, two pentagons inverted one to another, and its basic or primary grid matrix. The matrix is generated by lines, "virtuals," connecting the ten vertices of the decagon. In addition to these primary virtuals there are secondary virtuals that are implicit to the matrix which connect coordinate points of intersection, but are not indicated here. See Fig. 1.16 for an example of the use of these multiple virtuals in pd delineation. Circles are virtuals of extent, both full figure and component parts.

(d) The six-pointed hexagram, dashed lines, and the ten-pointed decagram inscribed in a circle show virtuals with congruence AB and CD. The geometrical and mathematical relationship existing between the decagon and the equilateral triangle, hexagon, is of particular relevancy with respect to the subject of geometrical anatometry. See Fig. 1.9, Fig. 1.11, Fig. 1.50, and Fig. 1.58 for anatomical and mathematical pertinence.

(e) A geometrical dilation (expansion-contraction) of the pentagoti. A multiplicity of such dilations provide linked, proportionate, and size variations of shapes intrinsic to the base pentagonal- decagonal matrix. See Fig. 1.42 for an example of its expression in pd delineation.

(f) The three linked circles within the decagonal matrix express a construction compresent to the matrix that is most important with respect to the pd delineation of the skull, and via the translations, the delineation of the spinal column. Refer to Figs. 1.15, 1.16, and 1.34.

(g) These two expressions of the pd matrix demonstrate the delineation of a dodecahedron, three-dimensional analog of the pentagons within that matrix. The dodecahedron is the form of the poliomyelitis virus, the virus that causes infantile paralysis in humans. It is remarked here in order to demonstrate the extremes in size and kind that are configured to a common geometrical matrix.

(h) The linear translation of the head matrix shown in (f). Two types of translations are illustrated; the contiguous (touching), and the concatenate (linked). See Figs. 1.34, 1.38, and 1.40.

The fundamentals of orthographic projection,. and the biological nomenclature conventional to the various views, planes, and axes. A schematized vertebra is exampled. It is from such drawings that automobiles are manufactured and skyscrapers are constructured. They represent an object as it is, physically positioned in the cube of space. Unlike perspective drawings, measurements can be made directly from the projected delineations.

The technique of pentagonal-decagonal delineation, configuration, is demonstrated in this figure. An object, in this case, the second cervical vertebra of man, the axis, is shown positioned in the cube of space, and its projection to the six planes or faces of the cube. Five of these views are shown in their conventional planar relationship, and the manner in which each view projects precisely to another. The several views illustrate the pd configuration of the vertebra as determined by connecting lines, "virtuals," of the base pentagonal-decagonal shape.

The virtuals delineate linearly the edges resultant to the juncture of planes, outlines of extent, and features intrinsic to growth processes. In this manner it thus becomes possible to delineate each separate bone, and through the linked geometric mappings, the dilations and translations of the pd matrix, the complete skeletal complex.

Five three-dimensional renderings of the vertebra, in the several views, are shown at the bottom of the figure. The horizontal Iining of certain surfaces, a sometime convention in this work, indicates the articulating surfaces of the vertebra with linked vertebrae above and below.

The human skeletal system and its relationship to the equilateral triangle. An equilateral triangulation of the full-figure circle divides the vertical axis of the circle into eight equal parts or subsidiary circles (primary or head circles). As is shown in the illustration, lines of the triangulation intersect joints. Although the skeletal form is configured to the pd matrix, the joints are "positioned" by a triangular geometry. This feature, as revealed by pd anatometry, relates to an essential rigidity basic to the equilateral triangle. Nature, in the skeletal system, compensates for the rigidity by providing flexible joints at functional locations. See Fig. 1.50.

1.11 The movement of the leg from the pd configuration to the standing configuration. This movement results in a standing height of the full figure equal to seven and three-quarters head lengths or primary circles. In this movement the knee joint continues to be conformal to the equilateral triangulation. It will be seen that this height is a result of the leg pivoting from the hip joint, an off-center position from the vertical axis of the full figure. Since the leg pivots from an off-center point, the foot would not describe an arc of a circle centered on the vertical axis as represented in the drawings by Albrecht Durer and Leonardo da Vinci. It appears that the angular movement of 27° is the natural extreme that the hip joints allow laterally. A compensating angular shift of the pelvis is required in more extreme lateral movements of the leg. The pd delineation determines a mid-point of standing height between pelvic pubes, and the inferior extremity of the pelvis which lies on the horizontal axis of the full-figure circle. This half-height is consistent with observed adult heights. The hip-pelvis structure is detailed in Fig. 1.45. The corresponding movement of the arm and shoulder girdle, from the pd configuration to the standing position, is delineated in Fig. 1.50.