8 Domes---Their Long History and Recent Developments

Chapter 8
Domes---Their Long History and Recent Developments

2Domes made by men have probably been used by man for hundreds of millenniums. Domes made by nature, as caves or as vaulted branches of trees, have probably been used by man throughout the million years we now, tentatively, concede that men may have been present somewhere around the earth.

3 Little, exquisite, bubble domes, too small for man-occupancy, are made by nature at possibly the highest mass production velocity anywhere manifest to man. Some are of split-second longevity. Some are of great longevity. Nature combines these miniscule domical structures in myriad varieties of complex structural arrangements occurring as both organic and inorganic compounds and as cellular agglomerates. Most of these complex domical structuring accomplishments by nature are realized at modular frequency magnitudes infra to man’s sensorial tunability and apprehension.

4 In consequence of this invisibility man has ignorantly classified the whole gamut of non-sensorial module structuring by nature under the illusion-evoked designation of ‘‘solid’’ matter. Consistent with this illusion man traffics in various solid substances known as ‘‘materials.’’

5 Men do not, however, in reality build structures out of materials. Men build visible module structures out of sub-visible module structures. There are generalized laws governing structure. The generalized laws of structure are oblivious to the special threshold existing between the man-tuned sensorial spectrum and the vast ranges of universe structuring infra and ultra sensorial to man’s narrowly tuned conscious reception faculties.

6 Man has, however, through instrumentation, increased the spectrum ranges of his modularly tunable scanning of universal structuring. Wherever he succeeds in tuning in he discovers either orderly arrays or provoking indications of orderliness only mildly clouded by as-yet-non-tuned frequency phenomena. All these instrumental explorations into the infra-ultra sensorial spectrum ranges of Nature’s universal structuring discover dynamic or potential events in mathematically regularized, complex interactions of omni-accommodative principles.

7 The mathematical patterning and inter-transformability of Nature’s geometrical structurings are the only reality of universe. The infinitely regenerative dynamism, always potential in the fundamental relationships of the principles, in itself constitutes the intellectually tunable and ever inescapable reality.

8 The relative abundances of reciprocally patterning principles everywhere constitute the so far discovered inventory of minimally complex, ergo fundamentally differentiable structurally regenerative universal governance. The complex of interactively accommodating principles and their relative abundance accommodation are reality—the only reality. What man, in his sensorially preoccupied misapprehending, has termed ‘‘abstract,’’ in contra-distinction to sensorial, as well as that which man has designated as metaphysical in contra-distinction to physical, are altogether one reality. The fact of meagerness of the experience-generated knowledge of man in respect to the omni-regenerative structure of reality and the observational facts taken in the twilight zones between the meager known and the as-yet-unexperienced, in no way alters the unitary integrity of the utter inter-accommodation of complex structural interaction of the principles as so far sum totally inventoried by the faithfully reported experiences of man.

9 In view of the foregoing, it is the working assumption of my present soliloquy that the principles of structure are universal constants, utterly independent of the phenomena ‘‘relative size’’ in any man-elected arbitrary modular terms.

10 So important have domes been throughout man’s total experience that the roots of the word for God, home and dome are the same—domus, domicile and dome. In the language of the sailors of Denmark and Iceland, the word is dom. Because of the age-long interactions of mysticism, with religions of hope and fear, in the daily lives of men, always centering in the home, the dome, ages ago, became symbolic of all the cosmic thoughts, hopes, supplications and glorious conceptions. From its comprehensive pre-eminence, the dome conception gave root to the words dominate and dominion. As a result of the slow process of communication casualties in the hearing and mimicking of sounds, prior to the written word, a great interchangeability of the consonants prefixing the syllable om took place. The D was interchanged with the T in designation of the dome as a mortuary shrine and with a W as the gestation or pre-nativity shrine. Thus man went from W-OM-B to T-OM-B via the H-OM-E. Even the B-OM-B is a derivative of dome as the super accelerated explosive nativity container. The Bikini bomb was dome-like in shape.

11 In ecological patterning, early man was the hunter and fisher, operating at extreme radius from his domicile center. His mate operated at the domicile. She became the dome-man, the homeman, the w-om-man; also, she was the man with the dome inside, the w-om-b man. Greater and lesser ecological circles, characterizing male and female peregrinations respectively, are still the ecological domains of the swift running wild mammalian life. Later the thought-hunting and recollection-researching male, hibernating in his domicile center, became H-OM-O sapiens, domo sapiens.

12 Some of my hypotheses regarding the evolutionary ramifications of the life-dominating word dome have been suggested by others, such as Baldwin Smith of Princeton University and Prof. Barry Bierman of the University of Natal, Durban, South Africa, but I will accept all negative reactions for the guessed-out significance of the OM root relationships in their widest historical-geographical patternings.

13 Early domes, made by men, fall into two important historical classes. One has a very long history, the other a very short history by comparison.

14 Class One of all history’s domes is comprised of the hundreds of millenniums-old upside-down baskets which include the later evolution of baskets into boats and the re-upside-downing, once more, of boats to form the roofs of community meeting places and its later derivative cathedral. The nave (navy; Naga, the sea serpent god of the sea) is the upside-down boat. In Japan, the word for ‘‘ceiling’’ or ‘‘roof’’ means also ‘‘the bottom of the boat.’’ In this oldest category of upside-down basket domes, there exists a word identification linking the earth’s extreme territories which seems to tell us that men in their wooden basketry boats, rafts, catamarans and canoes had indeed conquered the whole dominion of earth. The once only oral words of South Africa and the Eskimo, designating their domical enclosures, are now spelled respectively INDLU and IGLOO. Only man’s later invention of phonetic spelling and its interpretive application by geographically remote, different men, in the documentation of these extreme Northern and Southern Hemisphere sound-words, must alone have occasioned the difference in spelling.

15 Class Two of all history’s domes consists only of the last four millenniums during which occurred the hard stone, crystalline ‘‘permanentizing’’ by man of the organic fibre domes. Here man sought for inventions which might bring the lasting quality of the non-man-made stone caves into the man-invented association of non-man-invented organic structures of Nature in the pattern of domical stability.

16 It is a good guess that Nature suggested this permanentizing to man when snow and ice amalgamated accidently, so far as man is concerned, with his domical basketry of bent saplings, reeds, thatching grass and skins. Realizing and testing the ice-augmented strength of this early version of fibrously reinforced crystalline aggregates (as prototypes of modern steel reinforced concrete or glass-fibre reinforced plastic) men ascribed the predominant improvement in structuring to the hard compressive ice component. They considered the preliminary basketry dome, upon which the heavy ice dome had been later formed, as now constituting a preliminary and removable scaffolding die, or form, which they later employed as centerings. Thus, a man’s ice-cake igloos may have come into logical evolutionary development. From the ice igloo blocks, fashioned at the readily cutable snow crust block phase, may have been derived the successively experimental information leading to the discovery of the structural strategy geometry consisting of horizontal rings of wedge-shaped stone blocks, rising in graduated concentricity, to an apex. Experiments with ice and snow would have been less dangerous than with blocks of stone.

17 The earliest domical structures of stone fashioned by men are those of the early sea-dominating people of the eastern Mediterranean Sea. The Greek historian Herodotus speaks of the Egyptians going through the Red Sea and into the Indian Ocean to the land of Punt. The mural records in the tomb of Queen Hat-Shep-Sut in the Valley of the Kings, inscribed four millenniums ago, report clearly on her Egyptian commercial fleet expeditions to what is now Somaliland on the northeast African shore of the Indian Ocean. I learned in southeast Africa that the name Punt in ancient language means the color red. The letter T is a feminine suffix. The Red Sea was once the Pun Sea. This is the same Punt root as the punt boat of the East Indian Ocean, the Malay Straits, Thailand, and Burma. The punt boat was the pontoon of the pontoon bridge. The people who bridged oceans by boat relays—first across the Indian Ocean—were the Puns, the redskinned peoples. These are the Punicians later alliterated into Phoenicians, and later alliterated into Venetians, Vekings (Vikings)—the Punicians of the Punic Wars, the Pun settlers of both the north and south shores of the Mediterranean Sea and, reaching out into the Atlantic,—the Punta La Garda in the Azores, of the Portuguese (probably Puntagese). These are the sailormen to whom the sea was the normal life and whose harbored, canalled and pontooned cities constituted their regenerative birthings at the sea’s landed margin, as do the seals and sea lions go annually upon the land for their regenerative whelping and new launchings. These were pundits of India: the wiseman, navigators, and astronomers.

18 Professor Thornton White of Cape Town, South Africa, architect of the town planning of the island of Mauritius, in the bosom of the Indian Ocean, was commissioned by the British Government, because of his great Indian Ocean knowledge, to conduct a survey of the total smuggling operations between the Orient and the East African countries. Thornton White met with his friends, the fleet captains of the myriad swift-sailing Indian dhows at Mombassa on the east coast of Africa. He was deeply impressed with the scientific elegance of the dhow captain’s navigational art and with the profound simplicity of their reckoning and the integration of their total navigational experience. This lent credence to the dhow captain’s confidence that their trans-Indian Ocean navigational success had been operative for no less than ten thousand years.

19 In appraising the statement, it is to be noted that Prof. Thornton White’s status with the British Government is such that his opinion regarding the Indian Ocean smuggling governed the British Government’s decision to discount, rather than to oppose, the smuggling. Professor White, an Oxford scholar and world traveler, a frequent visitor to America, stated, in dramatic contradiction to the legendary concept held in the western world of an uncouth, savage Indian Ocean sea pirate, that the captains of the dhow fleet, as a class, are the most cultured, intellectually gentle people he has ever known.

20 I have been studying the evolution of ship design throughout the ages and have traveled around the world to discover and strengthen my theoretical conviction that men have been successfully circumnavigating the earth, not just the Indian Ocean, for at least ten thousand years. The navigators, coming upon the land, became priests, and the priests effectively counselled, but did not teach, the kings. The priest-navigator conspiracy methodically buried in the ocean’s depths the visible evidence gained through their mathematically calculated astronomical competence in navigation. Above all, the priest-navigator conspiracy kept, even from its sailors, their astronomically and geographically realized knowledge that they had circumnavigated the earth. The non-navigating sailors never knew where they were on the high sea. Though they returned to the same familiar places, they did not know by what routes they had come. All they saw was water.

21 In the comprehensive illiteracy of man and its preoccupation with the sensorially obvious, the conspiracy of a few scientifically operating men was easily accomplished. The conspiracy knew the earth was a sphere, a finite or closed system. Approximately all other men, including the kings, thought the earth to be a local plane surrounded by infinity, and each thought of his native region as the center of this infinite plane. World men lived in infinitely remote groups, each in its unique local pattern of relative abundance and scarcity of a priori resources. Only the conspiracy knew of the comprehensive and complimentary associability of the remotely distributed but finitely integratible world resources. The fabulous wealth-generating secrets of commerce-controllable resource integratibility lay wrapped in the invisible mathematical webs of spherical geometry and trigonometry, as applied both to the spheroidal ships and their off soundings, trans-ocean, trans-night, trans-fog, trans-fear, astronomical navigation and mathematical reckoning.

22 The early Aegean sailor’s high-sea peregrinations inevitably took them into Arctic island hibernations, in man-made caves, under their ice and snow-covered boats. Closing in their boat-roofed caves with rocks and ice cakes taught them of the ice fashioning of domes with horizontal rings of concentrically graduating wedge blocks. Nature’s annealing of the ice to form a monolithic ice-dome provided a cohering hydraulic crystalline advantage countering the tendency of outward thrust disintegration occurring at the haunches of nontensionally cohered, domically piled up, compressional components without comprehensively finite tensional bonds.

23 Wedge-shaped blocks, like corks, have a polarized asymmetry in which one pole’s shoulders have greater dimension than the other pole’s shoulders. Wedges and corks may be piled up in oriented regularity so that the greater shoulders are always on one side and the smaller shoulders on the other. Horizontal extension of such piles eventually describes a circle which closes back upon itself at the base of the pile. Vertical extension of the pile curves upwardly and inwardly from the base circle, eventually closing at a domical apex. The outer shoulders of the individual components prevent any one component from slipping or sliding inwardly through its allotted radial domain space segments, toward the center of diminishing cordal dimensioning, within the finitely closed system of hemispheroidal pile-up.

24 Save for the friction between the dome’s base and the earth’s crust below it and between the progressively lessening frictions of the upwardly and inwardly tiered wedge layers, there is naught to restrain any of the wedge blocks from slipping or oozing outwardly from the finite aggregate. In the ice domes, the annealed hydraulic tension cohered the mass. To understand man’s subsequently-arrived-at dome-building strategies, we must consider a plurality of factors which he differentiated intuitively, out of his hard-won sailor’s experiences of masts, booms, long oars and rigging. These experiences generated a gradually generalized structural wisdom in professional sea masters.

25 A man may support a long column, say a fifteen-foot wooden pole, in a vertical position with the palm of his hand outstretched from his downwardly oriented arm. Small and continuous motions of the hand will keep the pole in vertical balance. In vertical balance, earth’s gravity operates upon the pole only in a downward force, concentrated entirely within the neutral vertical axis of the pole, focusing all of gravity’s force into the palm of the man’s hand at the comfortably vertical arm-tension axis. An attempt by the same man to hold the same fifteen-foot pole horizontally with outstretched arm is frustrated by gravity operating perpendicularly—that is, at ninety degrees all along the pole as well as the length of his outstretched arm. The center of the gravitational force must be at center of gravity of the combined length of the pole and his arm—that is, at approximately eight feet outward horizontally from his shoulder. This means that gravity combines the total weight of the pole and the arm at the end of a horizontal lever eight feet (ninety-six inches) long from his shoulder fulcrum. This provides a force of twenty-five pounds at the lever’s ninety-six-inch end, operating against a two-inch muscle tissue lever increment at the other—a forty-eight to one advantage, resulting in a twelve-hundred-pound loading of the muscle tissue.

26 The vertical and horizontal pole experience of man leads to his sensitive understanding of the effects of gravity on domes.

27 Weather, quakes, and fire are secondary to gravity as structural challenges, for these secondary challenges are variable factors which may only accelerate structural yielding to the constantly operative gravity by eroding the minimal structural section dimensions. Both the vertical and horizontal limits of the gravitational effects upon pole behavior are present in the dome and all the intermediary gradation effects between those limits.

28 It is in evidence that the horizontal vulnerability of structural components to gravitational effects requires the strongest answering stratagems. Columns are easy; beams are difficult. Ships’ masts support their booms, not vice versa. The walls and columns of history’s ruins stand the longest. Rarely are the horizontal beams, elevated floorings or roofs to be found intact, if at all, with the exception of domes which combine both horizontal and vertical behaviors progressively translated into mutual synergetical aid and integrated success. Domes of Istanbul, the Near East and India, which respectively comprehended these combined behaviors, have been standing for centuries—the Pantheon of Rome for two millenniums, and the tomb domes of Greece for three and one half millenniums—in better condition than their contemporary and much later column-and-wall neighbors of antiquity. Not so antique, but of sad significance, the Japanese have made an historical monument of the dimensionally unaltered dome framing of the only structure that remained at target zero of the first atomic bomb dropped upon Hiroshima in 1945.

29 The top half of a dome’s circular arc is approximately horizontal and the bottom half’s circle of the dome’s total area is approximately vertical. The top half tends to operate as unit mass, yielding most readily to gravity, and the bottom half of the dome as a vertical ring reaching up to the dome’s haunches, tends to resist gravity with greatest ease for the same reasons that govern man’s experience with the vertical and horizontal pole-supporting. It is surprising news to the layman that the greatest compressive forces in wedge block domes occurs within the top twenty-degree arc of the dome. The pressures are far greater than those caused only by the dome’s dead weight around its bottom.

30 Domical pile-ups of wedge-shaped blocks, without any finitely cohering tensional bands, tend to yield to gravity in their top parts and oppose it in their bottoms by spewing the unopposed haunch height ring tiers outwardly—thus uncrowding the remainder of the top-tier wedges which now have ample room to fall individually inward and downward. If the spheroidal domical pile of wedge blocks is graduated upwardly and inwardly in almost vertical conformation in the form of an upended half agar, the gravitational forces will induce such friction as to occasion structural stability and approximate elimination of the critical haunch area.

31 I have visited the earliest known stone-wedge navigator-priest designed domical structure of man, the tomb of Atreus, approximate contemporary of Agamemnon in the latter’s kingdom of Mycenae in Greece. The Mycenaean Greeks are of the Cretean sea-master pre-Ionian people. This t-om-b d-om-e of Atreus is of elegantly modulated vertical cigar shape. The Greek name for the shape is tholus, which is the same name as the sailorman’s vertical cigar shaped wooden thole pin, two of which, mounted in the boat’s gunwale, formed an oar lock and one of which, mounted in the stern, formed a pivot for a sculling oar or rudder. Thole pins thrust through collar flanges around masts formed cleats on which were wound halyard ropes whose fastenings could be quickly loosed by pulling the thole pin not only out of the wound or belayed rope but also out of the collar hole to let the rope run free—(thole pin, tompion, belaying thomplong, belaying pin). To keep the thole pins from sliding downward through the holes and to render them most graspable by the hand, their top thirds have a hand-filling bulbousness immediately below which a shoulder rim flares out.

32 This thole pin shoulder and cigar head gives its nautical conformation name to man’s first stone dome. The Mycenae tholus dome is not only the oldest stone dome, but it is the most perfect in its preservation and dimensional elegance of any dome I have seen anywhere around the world. To insure its structural stability against outward thrusting of its wedge blocks by later vandals, ramming outward from inside the dome, the Greeks concealed this dome under a naturally appearing mountain of earth. The gravitational effect of the mountain of earth continually thrust the wedged blocks into the tightest possible self-crowding and successfully hid the tomb for three and one half millenniums.

33 The Roman arch, developing a thousand years later, was, of course, a far less intellectual borrowing of a simple curvature section taken through this Greek sailorman’s stone adaptation of his arctic ice-dome prototype. The Roman arch employed this domical plane section in parallel series, holding down the outwardly slippable wedge by superimposed masonry and earth. Because parallel planes do not aid one another, nor cohere, the whole parallel set had to be held together like vertically standing books by ‘‘book-ends’’ in the form of buttresses impinging perpendicularly upon the cylindrical arch’s terminal faces.

34 All stone, ceramic or concrete block dome buildings are covered by the principles outlined above. When men developed the capability to produce iron chain in quantities sufficient to make wrought iron anchor chains for ships, as a practical, feasible and economical commodity, closed circles of the chain were employed to gird the dome bases and haunches as safety factors against their outward thrusting. Because the domes were laid in vertically diminishing diameters of horizontal rings, if the chain were lifted upwardly it would be too large for congruence with the lessening circles, and its binding effectiveness eliminated. The vertical wedge-contoured section of the domes themselves tended to thrust such chains upward and their safety-factor function was rendered so dubious that many chain-bottomed stone domes fell in, throughout domical history. Stone, wedge, block componented domes, which have survived through time, have been characterized by one form or another of haunch height buttressing of inertial restraints in a variety of ingenious and sometimes visually illusive structural inventions. Some of these involve a plurality of domes around domes, as in the great dome complexes of Istanbul in the whole Byzantine cathedral complex.

35 I am not at all suggesting that all dome-builders consciously followed the Mycenaean prototype or were even aware of it and its possible, if not probable, arctic laboratory prototyping. However, both geometry and gravity are universal and their interactive behaviors are so fundamental that no matter how experiment begins, if its lessons are heeded, they must lead inevitably to the universal relationships, only mildly modified by the local environmental arrays of pneumatics, hydraulics, and crystalline constellations of nature’s constructional inventory. Little boys, wherever there have been snow falls, have learned to heap and roll the snow into plastically cohering masses. Thereafter, with their relentless experimental appetite, many children learn, through intuitive initiative, that they can dig a hole into the side of their miniature snow mountain, which can be entered by them and progressively enlarged until they have hollowed out the mountain—thus more or less inadvertently arriving at a dome. The dome’s end result of a coalesced crystalline aggregate is characterized by high tensional coherence re-employing the molecular surface tensions in highest geometrical advantage.

36 All this was inadvertent and entirely fortunate—as without engineering training, the child remembered, and probably later reconsidered the fact, that the structural integrity of his snow mountain was provided entirely by the outer zone of the snow mountain because he had removed its innards. Six millenniums ago, the Egyptians, with an even earlier boat-implemented background, wherein the ship’s hull provided the same force-patterning structural stability as a tensionally banded dome structure, demonstrated their knowledge that they could tunnel and hollow out both their man-made pyramidal mountains, as well as nature’s formulated mountains, when they riddled the Theban ranges with their tomb chambers.

37 The Egyptian mountain-chambering was an historical structural invention in marked contrast to the entirely solid formalized mountain-building of the Oriental temples, ghats and wats. However the Egyptian mountain-chambering, as well as the Sicilian domical tomb-chambering of mountains, were all in gross contradistinction to the performance per pound of involved physical resources employed by world sailormen in the designing and crafting of their ships. Ships represented a stabilized structural invention, thrusting their hollowness (hull) by displacement into the far more difficult hydraulic instability of the sea’s water mass.

38 Inspired by nature’s nut, mollusk, egg and vegetable shells, men learned, hundreds of millenniums ago, how to hollow out or build up and bind together hydraulic containers or vessels. Learning gradually how to expand the vessel-making art, men finally hollowed out large logs and then later shaped and tied together tree planks, later bent around stout frames. They pitch-sealed the seams between planks to exclude, rather than include, the hydraulic element, as had their earlier water-holding vessel, the vase—ves, short for vessel.

39 Vessels represent man’s discovery, not only of the generalized principles governing stabilized separations of liquids and gases anywhere locally in the universe, but also of the generalized solution of local environmental pattern-controlling by man within the otherwise frequently hostile, if not lethal, environmental event patterning of total energetic universe. Vessels embrace the fundamental principle of finite local system mathematics. Vessels, as systems, divide the universe into two main parts—the withinness and the withoutness.

40 The concavity within tends to regenerative reconcentrations of the withinness by angular convergence of vectors impinging or echoing from its surface. The convex outer surface separates, diffuses and thins the energies impinging upon it or echoing from it by angular divergence. The convex-concave surfaces of systems are the prime energy pattern differentializers of universe. They occasion energy patternings either as local regenerative atomic masses or as universally divergent radiation.

41 Vessels are specialized realizations by man of finite local system theory, itself the realest of the real. The theoretically most effective vessels are the symmetrical and spherical (combined positivenegative hemisphere domes). The inseparably occurring positive (convex) and negative (concave) surfaces represent the most effective angular valving or shunting by man of the operative energy events of his environment, purposefully including this, excluding that. Vessels tentatively divide energetic universe into two fundamental domains, the macrocosm without and the microcosm within. Vessels constitute omni-directional spherical dams (domes). Physical experiments discovered and identified entropy as demonstrating the law of the increase of the random element which shows that locally confined energies will ultimately escape their local restraints.

42 Local perpetual motion systems are not permitted by a universe which, however, is itself the minimal, arid always most efficiently and perpetually integrating system of comprehensive motions. Anti-entropy shows that local systems also continually inhibit energies, but at rates inherently slower than their entropic rates. Both entropies therefore demonstrate that all spherical dams are inherently porous. Any spherical dam dome, described in the terms of atoms, consists only of a swarm of points contoured as the spherical zone occurring between concentric spheres. The spherical zone swarm of points are the non-intercontacting crowds of individual atoms and their respective individual nuclear components, all occurring only as different magnitudes of separate energy-event constellations.

43 Entropic and anti-entropic energy traffic between macrocosm and microcosm is only retarded by the frequency of angular pattern modulations occasioned by the successive, precessive shoulderings experienced by inward or outward bound energies passing through the primary atomic and secondary nuclear constellation crowds of the spherical dam.

44 Vessel design is concerned with the rate-valving and angular direction-shunting of the undeniable energy transactions between the dammed-apart macrocosm and the microcosm. As the traffic pattern of inward and outward energies are altered, inevitable energy pile-ups as differentials of potential energy-event directions are concentrated on either the inside or the outside of the micro-macro dam. These pile-up differentials result because vessel structuring and valving design inherently occasion frequency and angle modulations which can only decelerate the otherwise non-impeded energy traffic velocities of non-domedammed universe.

45 Vessels as local environment controlling structures are only challenged by four categories of force patterning:

1.

Internal energy pile-up differential, beyond the critical limit of the dome dam structures’ containment capability.

2.

External energy pile-up differential beyond the critical limit of the dome dam structures’ exclusion capability.

3.

(Either #1 or #2) Differentials articulated as exquisitely concentrated energy actions.

4.

(Either #1 or #2) Differentials articulated as approximately evenly distributed energy actions.

46 In the earliest known record of the land dome vessel, upside-downed and launched into the water, we find the probably earlier round pitch sealed basket vessel of the rivers, ovalled and axially elongated to form a round bottom, keel and rib-stiffened, sea-capable vessel with directionally favored low-resistance shape (ship) with (hollowed—holed) hulls which had already reached a scientifically sophisticated maturity of differentiated tension and compression functions in a multiplicity of highly specialized energy-structure tasks as spars, halyards, sails, rudders, oars and anchors. This earliest known picture of a ship, circa 2500 b.c., is inscribed in color upon the walls of that period’s most advanced dry land structural innovating. This most advanced land building art of that time, consisting of the Egyptians’ chiseled out cave tombs, and of their gargantuanly cumbersome stone columns (improvised in a treeless land) and their enormous, yet short spanning, horizontal stone block lintels, or beams. These historically novel and then most advanced land structures were in marked physical contrast to the exquisite performing efficiency, per unit of weight to function ratio, of the upside-down dome vessel ship-designing.

47 The Egyptians used their structurally innovated stone post and beam courts above grade for their living kings’ pageantry and their below grade, man or nature-made, mountain-chambering for the ‘‘bon voyage’’ advantage of dead kings. The bon voyage equipment was frequently implemented with either drawings or full-scale, fully equipped real ships. The ships and their equipment were intended to take them into cosmic dominion. They were symbolized, in the mural drawings, as around a great sphere. The sphere has been interpreted as meaning ‘around the sun’ or ‘traveling around the earth,’ as the sun appeared to do. These priest-authorized symbols disclosed both the priests’ fundamental knowledge of earth’s sphericity and their purposefully deceptive legend duplicity.

48 Some of the so-called sun ships of the Egyptian kings have been so well preserved in their tomb vaults that it is now possible to witness that their hull lines, overall dimensions, structural techniques, floorings, thwartings, plank lacings, prow and stem rudder-designing were approximately the same as the Viking (pronounced Veeking) ships of the North Atlantic—one of which also was in recent years unearthed in Norway from its burial ceremony grave. The Norseman’s (North man’s) ship and ceremonial equipment of tools and foods corresponded so intimately, by either a ship-designer’s viewpoint, or a mechanic’s viewpoint, that, despite the centuries, miles and skin-pigmentation differences between the Norsemen and the Egyptians, it is clear that the same maritime engineering and its concomitant philosophies and superstitions and economic tasks governed their contriving and use—as with one mind in one time upon one sea. In Scandinavia there are still traces of the upside-downing of these boats upon the land and their elevation as a nave in a subsequently evolved religious meeting building. The keel and ribbed ships inaugurated ridge poles (rigid pole keel) and rafters.

49 In Egypt, the living kings, when not enjoying their royal quarters upon the decks or within the cabins of their Nile River traffic-ruling ships, dwelt upon the land, not in their stone-columned courts, nor in their mountain-chambering, but in mud hutments (hut, hat, hus, haus, house, head, helmet, another epistemological transformation of the dome concept).

50 The only virtue of the stone-structuring innovations as a permanentizing of the wood and other organic products of nature, employed in building, was the expediency of finding a substitute for diminishing supplies of trees brought about by deforestation and failure to stem erosion and wind-blown desert growth. Because the permanentizing substitution of stone for wood took place in the Eastern Mediterranean, Asia Minor and the Orient and in the Americas, we do have records of structural conceptioning of the ages as previously realized only in perishable woods and theretofore utterly obliterated from the historical scene. The quality of durability is extraordinarily useful to us as an historical information relay system, but the virtue of durability, as a cosmic, infinity-seeking, security of the ego, is of dubious validity.

51 The anonymous millions of constructors of the pyramids and the techniques of their labors pique our curiosity and often provoke our admiration in greater measure than that spontaneously accorded by us to the name-identified kings. Certainly, the conflagrations of history that have taken place within stone walls have been equally devastating to those occurring in stone-free wooden structures. Fires burn more hotly inside furnaces than in the open. Today we witness would-be permanent wave after would-be permanent wave of progressively obsolete skyscrapers and lesser masonry buildings being expensively demolished and once more replaced with theoretically satisfactory ‘‘permanent buildings,’’ emulating many of the permanentizing masonry techniques and sanitary theories inherited from the only-six-thousand-millenniums-long era of statical fallacy governing the misconception of building or aught else as desirably ‘‘permanent.’’

52 In diametric conceptioning, air and sea ships of steel and aluminum evolute economically as we progressively scrap, recover, reprocess, realloy and structurally reinvest the utterly indestructible elemental metal atoms in the swiftly evoluting series of accelerating safety and velocity capabilities of man’s universe, interlinking and probing and resource-transporting vessels.

53 Separation of the whole history of domes into the two main classes of technology provides enlightenment upon the swiftly evoluting contemporary domical structures scene. While there are some borderline exceptions, the two classes have remained essentially distinct due to economic factors prevailing to mid-twentieth century, which have been suddenly changed due to the sudden maturing of world industrialization. The two main classes persist. The age-old Class One has evolved from grass basketry and wooden boats, through steel and aluminum shipbuilding, into the aero-astro air-frame power plant instrumentation and its production technologies. Class One is now entering the astronautical structuring era of remotely controlled environments.

54 The only six-thousand-years-long secondary Class Two is still clinging, interiorily and exteriorily, to the crudely upended, crudely borrowed and translated, steel ship skeleton skyscrapers. Class Two embraces the whole of public and private building construction arts.

55 Class One’s original spherical symmetries have been, in many cases, spheroidally elongated to accommodate the swift directional functioning of their domical enclosures as transport environmental controls. Generally speaking, Class One embraces all curvilinear environment-controlling structures wherein the prime objectives are high performance per units of invested energies, weights, hours; successfully anticipated violent stressability; swiftly evolving, initially strong transitorial, specific longevity structuring; high priority access to intellectual, mathematical and industrial resource supremacies; extreme degree of functional differentiation with concomitant elimination of irrelevant fat and piggy-back alternate functions occasioning, for instance, fifty-thousand types of unique, non-fat parts and multimillions of pieces per each intercontinental bomber; world-around resources, world-around usage.

56 Class Two’s structures, from the very outset of their permanentizing attempt, demonstrated a fundamental geometrical deterioration of the original dome’s spherical competency of conceptioning and structuring. Curvilinear structures could be readily accomplished through bending of organic components without any important mathematical computation. The components, themselves, yielded in extraordinarily efficient e-curves. Catenaries draped themselves in mathematically elegant progression of changing inclination rates. When non-binding stone masonry was adopted, primarily for its fortification virtues, lack of mathematical capability imposed the straight line of sight technique and its simplest possible closure by a two-line parallel cross grid. The whole of Class Two’s six-thousand-year history has been characterized by retreat from the curvilinear dome to the square cut domes or rectilinearity built upon rectilinear plans of unnatural square grid town planning, subdividing an unbounded plane, vanishing tangentially from a circular horizon of reality. Class Two, in contemporary youthful vernacular, is a ‘‘square.’’

57 Class Two structures are characterized by non-consideration of weight performance ratios in their conceptioning; inertia as premium stability; hopeful permanency; non-functional differentiation; non-functional fat; local craft production; safety factors ratioed by politicians; evolution flaunting; increasingly deferred payment occasioning long-term debt, socially underwritten, whose increasing hundreds of billions of dollars represent the relative inefficiency of Class Two’s domical arts in competence, as ratioed to Class One’s domical science competence.

58 Despite its obvious advantage over Class Two, Class One’s domical science competence has much to learn and is possibly entering upon the most accelerated evolutionary phase in its whole history. This statement is predicated upon the following facts. Throughout all history, up to now, high scientific domical competence was so scarce a resource as to be made available only to the most exquisite high-seas and high-skies tasks of empire building and empire survival. This forced relinquishment of the everyday environment-controlling tasks to the residually ignorant expediencies. Between its invention in 1904, and 1954, the airplane’s long-distance-striking premiership brought it the combined subsidies of the nations of the earth, amounting to over two and a half trillion dollars of capital. As a result, the greatest industrial tooling and scientific competence of all history has been built into a gargantuan productive capacity for Class One’s environment-controlling domical science.

59 When the first Sputnik accomplished orbit, it knocked the airplane out of the sky, as a weapon, as the airplane had knocked the battleship off the seas.

60 This has left derelict approximately the whole of the historically unprecedented industrial capability operative at Class One level of high scientific competence.

61 Because airplane-designing teams know nothing whatsoever about man’s fundamental ecological evolution from a local cave man to an around-the-world air man and its inexorable requirement for competent living and working facility solution, the airplane design teams, charged by their business administration with tackling the problem of employing aircraft technology to everyday building arts, revert spontaneously to substituting, brick and shingle emulating, aluminum sheet stampings for wall components within the unchallenged, unconsidered over-all rectilinear inertness of Class Two’s building arts ignorance. A myriad of other kept-mistress hangovers of the aircraft industry, from its golden subsidy days, will, for long, frustrate aircraft industries’ organization man’s attempt to exploit the obvious potential replacement of the ignorant building arts of Class Two hangover by Class One’s now available capabilities and capacities.

62 Unfortunately for the American and Western economies in general, the Communist bloc countries are unhobbled by vast dollar-powerful lobbies concerned primarily with permanentizing all the inefficiencies of Class Two environment-controlling arts, and their economics and negative financing. Russia gives every indication of converting its Class One technologies into the direct solution of its own and its allied world’s environment controlling and standard of living synchronization, with world oriented air-astro age living-pattern requirements. Missile, satellite and jet transport production requirements of the combined world powers can and will use but a minor fraction of the total capacity and capability of the now-boiling-over Class One technology of the world. Russia and China may establish so swift a pace in application of the Class One technology directly to the living needs of the air-astro man that the western world will be forced by survival competition to junk its whole gamut of would-be-permanent, inert Class Two technological economics, and, sweeping the board clear, take advantage of its fortunately geared-up tool and instrument capability to accommodate the whole new philosophy of preoccupation with Class One conceptionings and realizations. A complete revolution in the educational system and business philosophy is obviously involved.

63 As a preview of surprising technical vistas already opening to early phases of the major reorientation of Class One priority occupation in weapons development to prior occupation with air-ocean-world high standard of living requirements, I am submitting unembarrassed explorations, already undertaken, under generalized Class One domical strategy postulations. Reducing the generalized postulates to special case solution and practice discovers diametric reversal of the age-old engineering concept, which held that the law of diminishing return characterized the logistics and economics of progressively larger dimensioned building enterprises in any one schematic class. We discover in our special case exploration of unembarrassed Class One reduction to practice that the larger the clear span structure, the more efficient it becomes both in resource investments and in industrial logistics economics.

64 Problem

65 With minimum overall investment of technical-advantageresources—that is, of controlled energy, either as chemical ‘‘materials’’ or as ‘‘work,’’ the problem is to develop a universal and integral structural system:

(a)

capable of sustained enclosure and controlled isolation of conditions favorable to man’s activities ranging in magnitude from single-family dwellings to major industries housed on the moon;

(b)

capable of inheriting all probable stresses’ receipts and providing all positional advantage requirements;

(c)

capable of supporting appropriate mechanisms for valving all locally impinging random or periodic energy environment receipts into preferred patterns complementary to man and machine processes;

(d)

capable of elective omnidirectional penetrations.

66 The Generalized Structural Logic Resource

67 The tetrahedron (four-faceted structure) is minimum—prime divisor of omnidirectional universe into two fundamental domains—the withinness and the withoutness, the included and excluded, the microcosm and macrocosm. Tetrahedron is therefore basic structural system; all structure is a transformative phase or complex of tetrahedronal transformations.

68 All polyhedra may be subdivided into component tetrahedra, but no tetrahedron may be subdivided into component polyhedra of less than four faces.

69 The regular six-chord-edged tetrahedron encloses (defines) the minimum volume with the most surface of all ‘‘geometric’’ polyhedra or structural systems, whereas the sphere encloses (defines) most volume with least surface.

70 As there may be no absolute division of energetic universe into ‘‘isolated’’ or non-communicable parts, there is no absolute enclosed surface or absolutely enclosed volume; therefore no true or absolutely defined simultaneous surface ‘‘sphere’’ integrity. Therefore, a sphere is a polyhedron of invisible plurality of trussed facets—trussed because all polygons are reduceable to triangles or trusses and are further irreduceable. (Trusses are, therefore, basic polygons.) An infinite polyhedron is infinitely faceted by basic trusses.

71 When stressed with high relative internal pressure all polyhedra tend to define the maximum volume with the minimum surface—that is, toward the spherical convex-arc edged tetrahedra (the basketball and baseball are tetra-structured).

72 When stressed by high relative external pressure, structures tend to enclose the minimum volume with the most surface— that is, toward the chordal or concave tetrahedron (the symmetric collapsed baseball). The ‘‘regular’’ planar-bound tetrahedron is the zero-phase between the convex tetrahedron (that is, the spherical tetrahedron) and the concave tetrahedron (that is, the four-webbed interaction between the six exterior edges of the tetrahedron and its center of gravity).

73 Great circle arcs represent the shortest lines between points on the surface of a sphere, and great circle segment chords represent the shortest distance between two surface points on the surface of a sphere. Therefore, the great circle arcs represent the limit structural transformative tendency of outward surface tensing by internal pressures; and great circle segment chords represent the limit structural optimum for axes of compression-resisting columns opposing external pressure by surface spreading.

74 We are seeking a structure impervious not only to extreme differential between internal and external pressure dominances, but also to highly concentrated internal or external loads or impact forces—yet permitting omni-directionally effective controlled penetrability.

75 Chordal-edged tetrahedronal structures best resist external forces and their vertexes best resist concentrated loads, while arc-edged tetrahedra best resist internal pressures, and their surface arc vertexes best resist concentrated internal pressures and impact forces, yet both permit omni-directionally valved penetrations. However, as the number of trussed faces of symmetric polyhedra are increased from the chordal and arc structural tetrahedra, through the hierarchy of great circle arc and chord trussed ‘‘solids’’—that is, the octahedron (eight) and the icosahedron (twenty), the number of vertexes and edges increase providing more and dispersed structural interactions for resisting concentrated loads from more directions and also more and shorter chords, thus providing increasingly favorable slenderness ratios for the component compression columns. As the number of external truss facets of polyhedra are increased, the more nearly does the chordal (or compression) structure approach congruence with the arc or tension-structured sphere. The icosahedron has the highest number of identical and symmetric surface truss facets of all great circle defined polyhedra, providing twenty faces, twelve vertexes and thirty edges.

76 It is seen that if a further approach to the congruence of all-trussed chordal polyhedra with arc-structured spheres can be accomplished, not only will the vertexes and trussed facets (or penetration points) multiply, providing increased advantage in more directions against concentrated loads and more directions of penetration, as well as greater numbers of shorter compression columns to share the load—to be realized progressively with more economical slenderness ratios and sections, but also will the condition be rapidly approached when both the chord and arc lengths and spherical surface angles and their chordal facet angles become ‘‘practically’’ indistinguishable, and the polyhedron’s surface becomes indistinguishable from the sphere.

77 It was discovered that a three-way great-circle grid may symmetrically subdivide the trussed facets of the icosahedron. This is what is designated as a three-way grid geodesic structure. Its frequency of modular subdivision of edge of the icosahedron’s facets may be multiplied at will, once the spherical trigonometry rates of change of central and surface angle subdivisions have been solved.

78 This is the essence of the geodesic structures. At an edge frequency of sixteen modules, the arc and chord tend to ‘‘indistinguishable’’ differences of dimension. However, as the number of truss facets increase, the convex vertexial interactions approach a zero altitude condition, which, though ideal for tension or internal pressure, tends to allow concentrated external loads to push the convex chordal vertexes inside out—that is, to a dimpled or concave condition, whereafter the continuing concentrated external pressure will be resisted by tension increase in omnisurface direction (as a rubber ball draws on its skin as it resists punching in and gains reaction and spring back, causing bounce).

79 If we make microscopic inspection of a pneumatic balloon, we will find that the balloon skin is full of holes between its molecular chains, with a secondary and far smaller space continuity of ‘‘all holes’’ or ‘‘continuous space’’ between the remotely islanded energetic components of each molecule’s respective atomic nuclear constellations. All these humanly invisible balloon ‘‘holes’’ are too small for molecules of gas to escape through. Because the balloon skin is full of holes it is really a sub-visible spherical netting rather than a ‘‘flexibly solid film’’ within which the gaseous element molecules are crowded into lesser volume than required by their respective energetic, ecological domains, as fish might be within a seiner’s net.

80 The resultants of forces of all these net-frustrated molecular actions are angularly outward of the balloon’s geometrical center. Each surface molecule of the interior group of pressured gas has a vectorial action and reaction pattern identical to a spherical chord. In such enclosure-pressured gas, random sizes of molecules, each and all too large for the spherical molecular netting’s holes, impinge randomly upon the interior webbing of the spherically tensioned net. There are, therefore, more outwardly pressing molecules and more inwardly restraining net components than are necessary to the structurally resultant balloon pattern integrity. However, in the geodesic, tensional integrity, spherical nets, the islands of interior compressional chordal struts impinge in discreet order at the exact vertices of the enclosing finite tensional network. My independent satellite or moon structures are then the most economical, frequency modulated, dynamic balances between outward-bound resultants of force and inward-bound resultants of force. The exterior tensional net is a finite system successfully binding the otherwise randomly entropic infinity of outbound, self-disassociative forces.

81 As already noted, a tensionally finite system has no inherent limit of length to section—that is, no inherent slenderness ratio, as do the explicitly limited slenderness ratios of compression members, wherefore we have in the Geodesic Tensegrity (my name for the discontinuous-compression, continuous-tension structures) the ability to assemble unprecedentedly large, clearspan structures, whose over-all diameter dimensions are limited only by the relative alloyed coherence of the associated metallic atoms therein involved, whose improving coherences are in swiftly multiplying metallurgical evolution augmentation. We can go, therefore, into the same magnitudes of clear-spanning dimensions as our largest suspension bridges. As suspension bridges demonstrate the continually improving tensile capabilities of constantly improving alloys, a suspension bridge could now be made twice the size of the Golden Gate Bridge. We may, therefore, consider clear-span, geodesic, tensegrity spheres in the magnitudes of two miles diameter as now realizable for use as satellite environmental controls or as hemispherical, or other spherical segment, earth-contacting enclosures (in which the earth completes the sphere)—for example, for arctic city environmental controls or as water float-able enclosures.

82 Whereas suspension bridge cables are parallel to one another and therefore give one another no more anti-rhombic structural stability aid than do the parallel tension wires of a barrel—in the case of the geodesic tensegrities, all the tension members cross one another in great-circle-chorded triangulations, thus providing the highest possible dimensional stability.

83 The progression of technical events above described alters all old engineering concepts regarding the relative increase in the over-all weights of structures. The now-obsoleted concepts heretofore provided early limits of practicality in the relative increase of weights of structures which occur when the linear dimensions are proportionally increased. In Egyptian pyramid buildings, doubling of the linear base dimensions brought about a four-fold increase in the surface and an eight-fold increase in the weight of the pyramid—that is, a linear dimension increase of two brings about a second power surface dimension increase of four and a concomitant third power volumetric dimension increase of eight.

84 When men later learned that the highest capability in strength of structures existed in their surfaces, due to the greater action-reaction, leverage distance that opposite sides of the systems provided, they hollowed out their buildings, which they had theretofore thought of only as formalized, solid compression warts on solid earth, such as the solid wats of Ankor Wat, or the ghats of Burma and India and the pyramids of Babylon and Egypt. But it is probable that men had first hollowed out their food and water vessels; later their log boats; centuries later, their buildings and, millenniums later, their sailing crafts’ racing spars.

85 Later, when men learned that the structural strength at the surface was not provided by the ‘‘solid’’ quality of the exterior shell, but by triangularly inter-stabilized lines of force operative within that shell, they perforated the shell with holes between the force lines. The minimum holes were triangular. The pattern of triangulated force lines peppered with triangular holes in the hollowed-out structural shell became what we call a truss. We can say then, firstly, that the hollowing-out automatically reduced the third power volumetric multiplication of relative weight increase of structures as they increase in respect to their primary linear dimension increases. We can also say, secondly, that the piercing of the shells with triangular holes reduced the solid or continuous surface of second power increase of the shells and brought the rate of structures’ growth into something nearer an over-all first power or linear rate of gain, for the force lines are only linear. When we introduce the tensegrity structure and its many surprises already accounted, we see that we have broken through to a structural knowledge and technique, which permits a progressively decreasing relative weight of structure as proportioned to the linear gain. This is to say, the gain of weight in structures, as ratioed to basic linear dimensions, is as one is to one minus 1 ⁄ x weight ratio, as the same structure is multiplied in relative size.

86 In the above progression, as geodesic tensegrity frequencies increase, the sizes of the islands of compression diminish. Islands of compression are the only residual ‘‘solids’’ and their diminishing size diminishes their relative weights at a cube root progression of advantage. Having the size of a solid spar reduces its relative weight by eight. Having the size of a hollow spar reduces weight by a factor of approximately four.

87 When we combine the ability to introduce the above orderly mathematics into the electronic computers, permitting practical calculation and engineering ‘‘feasibility’’ nonexistent prior to that fact, with the fact that the higher the frequency the smaller the tensional sectional area (yet the higher the tensional sectional capability and the smaller the local islands of compression), it is statable that the higher the frequency, the more ephemeral the tensegrity complex and lesser the total weight of structure per given level of performance, and the less vulnerable the whole structure, by total violations by any or many inwardly or outwardly originating impinging forces, will be.