Channel 26 META-SCIENCE : GROWTH. FORM AND STRUCTURE

Geometric Extensions of Consciousness

excerpts from Anne Griswold Tyng's article

This was first printed in Zodiac 19 - a review of con-temporary architecture .

The evolution of man's consciousness was built, atom byatom, into the configurations of matter and mind . Bothfor our understanding of its evolution and for its own ex-tensions of consciousness, the form of mind-matter findsclues in geometry .

The difficulty of tracing the history of man's consciousnessof space in a continuous sequence lies in the cyclic natureof the evolution of total spatial awareness-a repeatingcycle in which man's perception and understanding havebeen stretched asymmetrically in different shapes of ten-sion between the individual and the collective, and be-tween consciousness and unconsciousness . Thus the moreintroverted phases of the cycle tend to appear as a regres-sion (a "return to the unconscious" when vitality is re-newed through a reunion with primitive natural sources)instead of being seen as part of a continuous process ofexpanding spatial awareness .

The cycle itself proceeds from simplicity to complexity andfrom a balanced axial bilateral order to the movement ofrotation to the serpentine flow of the helix to the animat-ed form of the spiral - a building up of form and energywhich are integrated in a new cycle . (The synthesis of bi-lateral symmetry, a new simplicity of order which includesand integrates the previous complexity, begins a newcycle of spatial awareness from bilateral (synthesis) to ro-tational (space) to helical (time) to spiral (space-time) .)As "generative molecular elements" inherent in man'sown evolution, these principles may provide geometriclinks in the extension of man's consciousness .

today when we know that the relationships of form ex-pressed in these five Platonic Solids are involved in theway in which 'fundamental' particles-protons and neu-trons-are built up into atoms of about a hundred dif-ferent elements (according to Pauling's Close-Packed-Spheron Theory and Fuller's proposals of atomic close-packing)' and are involved in the way in which differentarrangements of these atoms form the building blocks of amillion or so different forms of matter . both natural andsynthetic .

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These five Platonic Solids-the only regular forms possiblein three dimensional space, each with all of its faces thesame and with the angles at which the faces meet eachother the same-are involved, not only in the spatial or-ganization of forms at the level of nuclei of atoms andmolecules, but also in cells, organs, plants, animals, thehuman embryo, the psychic structure of man, the worksof man and in the astronomical forms of the universewhich pre-existed man. Previously invisible ordering ofthe primordial atoms within us, revealed by the electronmicroscope, gives proof of internal geometry in naturalforms .

The four stages of symmetric form in this geometric pro-gression, as in the cyclic extensions of human conscious-ness, I have called bilateral, rotational, helical, and spiral,with each stage seen as the motion of simpler forms defin-ing the outline of more complex shapes.

In 450 B.C ., in his search for an 'atomic' order of spatialconcepts, Empedocles proposed as the building blocks ofeverything fire, air, earth and water . On mathemeticalgrounds Plato, in his Timaeus, determined the 'exact'forms of the smallest parts of these elements as the fiveshapes we now call the Platonic Solids ; fire the tetrahe-dron, earth the cube, air the octahedron, water the icosa-hedron, and as the symbol of the cosmos, the dodecahe-dron. This intuitive concept is given a measure of validity

Anne Tyng was one of the first women to receive an architecturaldegree (M. Arch.) from the Harvard Graduate School of Design .She has worked for a number of years for and with the architectLouis Kahn, associating with him on the "Project for a CityTower", featured in the Museum of Modern Art exhibit "Vision-ary Architecture". Based on a triangulated three dimensional sys-tem which had been used previously only as structure separatefrom usable space, as in Bucky Fuller's "octet truss", the undulat-ing geometry of this tower, which appears to have a life of itsown, is probably the first to be conceived as occupiable space asin a bee's honeycomb. Her independent research in formingprinciples, for which she received a Graham foundation grant in'65, has been oriented primarily toward principles of asymmetry,proportion, and hierarchical ordering of form . As Maria Bottero,editor of Zodiac, states, "It is geometry, with its oscillations be-tween symmetry and asymmetry, which, according to Anne Tyng,offers the key to the reading of the processes and phases of or-ganic and cognitive becoming." Inspired by Louis Kahn, BuckyFuller, Lancelot Law Whyte (Accent on Form & Aspects of Form),the zoologist Adolf Portmann (Animal Forms and Patterns), andthe work of the psychologist Carl G. J ung (Man and His Symbols& Memories, Dreams and Reflections), as well as by recent devel-opments in molecular biology, her work is one of synthesis . Inher articles, Urban Space Systems as Living Form (published in theR.A .I .C . Journal Architecture Canada) and Geometric Extensionsof Consciousness (in Zodiac 19), she has found links betweenbasic geometric principles (which Bucky Fuller has called "herdiscovery of Golden Mean relationships between the whole fam-ily of Platonic Solids not previously known by man .") and atomicstructure, molecular configurations, biological forms, psychic,structure and human creativity .

The polarity of a tetrahedron can be expressed in the po-larization of two of its four edges (as Fuller has suggest-ed) . One tetrahedron in two positions, which have a

point to face polarity, can establish the corners of a cube .Two other positions of a tetrahedron, also in polarity,define the corners of an octahedron . These three simplerPlatonic Solids-the tetrahedron, the cube, and octahe-dron-represent the bilateral forms of the geometric pro-gression . The cube in five positions, in rotation, definesthe twenty corners of the dodecahedron, and five posi-tions of the octahedron, again in rotation, establish thetwelve corners of the icosahedron . The tetrahedron infour positions, with rotational ordering, also defines thetwelve corners of the icosahedron and, in addition, onecorner of each of the four positions extend beyond theicosahedron to form the corners of a larger tetrahedron,disclosing a 'vestigial' polarity in this arrangement . Thesemore complex of the Platonic Solids, the dodecahedronand icosahedron, represent the stage of rotational formsin the geometric progression and, in the way they areformed, express Divine Proportion ratios (1 :1 .618) in theirrelation to the simpler solids, the dodecahedron to thecube and the icosahedron to the octahedron .

The 'fourth dimensional' extension of these rotationalforms along an axis perpendicular to the radius of rota-tion, expressing again the tension of polarity, defines thehelical forms of the geometric progression. Since both ofthe rotational forms have pentagonal symmetry around acenter, the plan of their helical extensions is based on thedecagon with its side in Divine Proportion to its 'radius'(of the circumscribed circle) . The vertical extension ofeach turn is in Divine Proportion ratio to the side of thedecagon, making a Divine Proportion-vertical turn =ø,horizontal turn = ø2, and radius of turn = ø 3 .

A proportional increase in the radius of rotation of thehelical forms, expressing rotational tensions, results inspiral forms, the fourth stage of complexity in the cycle .The only ratio which satisfies the condition of a logarith-mic spiral in which width of turn' increases at a fixed ratioto length is again the ratio of the Divine Proportion. Theshifting order of these forms between polarity and rota-tion includes the previous order within the new order, sothat rotational includes the polarity of bilateral, helicalwith its own polarity includes rotation plus polarity, andspiral with its own dominance of rotation includes polarityplus rotation plus polarity, with the new bilateral phaseincluding all the ordering of form of the previous cycle .

While it clearly appears to be a special achievement of liv-ing forms, the repeating cycle of bilateral, rotational, heli-cal and spiral apparently is not valid for non-living or 'in-organic' forms . The energies and configurationsprogressively built up in the rhythmic interplay of rotationand polarity result in the gradual intensification of struc-ture and the flexible vitality which is a special achieve-ment of 'higher' living forms . 'Inorganic' form is based ona generally more rigid bilateral symmetry, as in such atom-ic structures as graphite, salt, peronskite, copper, dia-mond, carbon dioxide, and cristabalite .

An example which does indicate evolution of formthrough a complete cycle is the structure of hemoglobin,which took the 22 years work of Perutz and his associatesto uncover . This extraordinary configuration of 10,000atoms includes the bilateral tetrahedral bonding of car-bon atoms in the glycine molecules, the rotational cluster-ing in the heme molecules, the intricately helical alphaand beta chains which in turn are folded into irregularspirals, and finally, each of the four spiralling myoglobin--type parts nestled and interlocked in a symmetrical tetra-hedral arrangement to form an overall bilateral symmetry.This bilateral symmetry reaffirms a basic simplicity of or-ganization over the complexity of differentiated parts tostart a new cycle-a hierarchy of form . With all the inter-nal complexity of this structure, we can barely conceive ofthe fantastic number of hierarchies within hierarchieswhich include and give meaningful organization to the280 million such hemoglobin molecules contained in asingle red blood cell - which in itself takes the rotationalform of a disc. Not only does there appear to be aprogress in the life forms corresponding to the geometricprogression toward complexity and increase in scale, hutthis progression can be seen as a repeating one with each

new cycle building hierarchy upon hierarchy which in-dicate at each stage of development the record of its earli-er evolution, the hierarchies of form and the hierarchiesof energy evolving from the interplay of polarity and rota-tion .

Form thus finds its own form, extending feelers, gills andtentacles to the world around it, in its rotational tension-ing, expanding its magic circle to new concepts of space

- from the first articulation of fin or finger to the spiritualdimensions of human creativity .

Form finds new helical dimensions, elongating to dif-ferentiate intake and output, strengthening backbone be-tween tusk and tail, head and anal poles, articulating thetensions between spirituality and sexuality, stretching tonew concepts of time between awareness of darkest ori-gin and highest aspiration, between the depths of the un-conscious mind and conscious thought .

Form stretches to elaborate both length and breadth inspiralling shells and branches, antennae and antlers, divid-ing and subdividing into the intricate filigree of blood ves-sels and delicate nerve ends, tensioning in space and timetoward an infinity of matter .

In the fleeting moments of balance between the tensionof rotation and polarity, the tensions of space andtime areresolved in bilateral living form, the transformation of theend of complexity to a new beginning of simplicity - ahigher order-the discovery of the cycle .

The life cycle of the butterfly is clearly defined in fourphases: the rotational symmetry of its eggs, the helicalsymmetry in its form as a caterpillar or larva, the spiralsymmetry of the pupa of chrysalis form and its dramaticrebirth in a magnificent form of bilateral symmetry . Thefrog follows a cycle from the rotational symmetry of thezygote, to helical embryonic bodystalk, to spiral form ofthe tapering tadpole to bilateral symmetry of the maturefrog . The bilateral human being evolved from numberlesshierarchies of cycles of form, from the primordial order-ing of atoms and molecules, goes through the cycle againin the early stages of embryonic development from the bi-lateral, then rotational cleavages of the ovum, to thehelical bodystalk of 18 or 19 days, to the spiral embryo ofabout 4 weeks to the miniature complexity integrated intoultimate bilateral form as a 10 week 2 inch embryo of po-tential human being .

The psychic synthesis of 'rebirth' is far removed from thestructure of hemoglobin, but in each case the simplifiedrelationship of complex internal structure creates a newunity, and, in the process of psychic individuation, anew involution of structure creates from all the complex-ity of a collective and primitive origin an uniquely individ-ual form . So a relationship to the principles of space, time,causality and synchronicity is valid for the psychic cycle,space expressing the tension of individual man with col-lective consciousness of external environment, time thetension between conscious thought and unconsciousmemory, causality the interrelated and combined ten-sions of both space and time, between the individuatingconscious psyche and the vast reaches of primordialmemory in the collective unconscious, and synchronicitythe balancing of tensions, the synthesis of space and timein concepts such as immortality . Jung wrote " ... Thefeeling of immortality, it seems to me, has its origin in apeculiar feeling of extension in space and time ."'

Generally in cycles of human creativity the periods of ro-tational tension relate to periods of external ordering ofthe psyche, periods of expansion, of materialism andpracticality, of openness, space, of concern with life, light,sun, of physical comfort and pleasure, of rationalism, ofbelief in the essential goodness and creativity of man ; thephases of helical tension relate to periods of internal or-dering of the psyche, of subjectivity and instinct, of con-tainment and verticality (polarity), of concern with ori-gins, with past and future and the element of time, withdeath and the principle of evil, of darkness, of emphasison irrationality, emotion and inner spirit; the phases ofspiral tension relate to periods of increased tension andcontainment of opposites, of complexity, of bizarre andexotic styles, of fascination with the occult, of exaggeratedmotion and energising of form, forms with complexcurves, pointed arches, ovals, winding processionals andlabyrinths, tapering towers and spires, forms dema-terialized by light, forms with weightlessness, with pro-gressively diminishing horizontal and vertical dimensions,of the combined tensions of space and time; and thephases of bilateral synthesis of tensions relate to periodsof serenity and balance, of unity, of cubic forms with em-phasis on horizontality and planar surfaces, forms withrectilinearity, simplicity, axiality and solidity, forms ex-pressing the integration of tensions in equilibrium, formsembodying the principle of abstraction free of space andtime .

From countless levels of such hierarchies the brain of manwas formed, the evolution of human consciousness andthe psychic potentials of 'individuation' and rebirth,man's search for the secret of creation, for concepts of im-mortality free of time, space, causality - for synchronicityfor the immortal 'static' synthesis of mortal 'kinetic' con-cepts .

1 . Buckminister Fuller lectured on and illustrated relationships ofclose-packed spheres as proposals of atomic configurations in 1949 .printed as "Item 0" prepared by North Carolina State School ofDesign students in 1955 . Linus Pauling's Clos e-Packed-SpheronTheory of the Nucleus appeared in Science October 1965 .

2 . P. 142, C. G . Jung, The Archetypes and the Collective Uncon-scious,

3 . See C. G. Jung, "Synchronicity - An Acasual Collection Prin-ciple," in The Interpretation of Nature and the Psyche, with W .Pauli, Bollingen 1955 Pantheon .

Videotapes of lecture by Anne Griswold

Tyng at Parson's School of Design avail-

able . Contact Beryl Korot, P .O .B . 135,

Ruby N .Y . 12475 for further information .Edited by Joan HennessyCharts by Anne Tyng

Helical

Spiral

Rotational