15 Continuity, Discreteness, and Resolution

3 As a student at Milton Academy, Buckminster Fuller would be puzzled by his teacher’s assertion that points have no dimension, but that placed side by side, such dimensionless points could constitute a line having no width. There, the teacher claimed, such insubstantial lines could be juxtaposed to form a plane having no thickness, and such planes could be stacked to form a solid. Young Buckminster felt that it would not be possible to assemble substance out of such ethereal objects having no finite dimensions.

4 Of course, Fuller was quite correct. Not only was he correct, but he put his finger on a flaw in the teaching of mathematics which has led to a great deal of muddled thinking. Fuller’s skepticism at this early stage of his development prepared him for many of his trail-blazing discoveries. The traditional approach to geometry asserts that a square generated out of line segments having unit length, if it itself has unit width, will have unit area. This claim assigns a fundamental importance to the square, which is, however, not necessarily correct, because it cannot be proved or disproved experimentally. It is an arbitrary convention; one might equally well, and frequently to greater advantage, assign unit area to an equilateral triangle having unit edge length.

5 The fact is that length, area, and volume are as different from one another as are force and electric charge, acceleration, electric current, voltage, and magnetic field. Experiments relating them measure changes in variables caused by changes in other variables, not absolute values; they establish proportionalities. Proportionality constants are introduced as a result of these experiments, such as dielectric constant, mass, resistance, etc. Analogously, relationships between length, area, and volume involve proportionalities: area and volume are proportional to the second and third power respectively of linear dimension.¹’^2,} The proportionality constants relating length, area, and volume depend on the shape of the object being measured: square, triangle, and tetrahedron each have their own shape constant; geometrically similar shapes will have the same shape constant. Geometric formulas relating different shapes are obtained by comparing geometrical shapes and transforming them, as explained in the three references given above.

6 Fuller noted that, in contradistinction with his teacher’s introduction of insubstantial points and lines, every point, line, and surface is made up of finite particles. A chalk line, he asserted, is made up of chalk particles, which in turn are made up of calcium, carbon, oxygen, and other ions, which in turn consist of protons, neutrons, and electrons.

7 Fuller espoused the view that matter is discrete. His line was a chalk line, made up of discrete particles. A circle to him was a polygon having a great

8 ’A. L. Loeb, ‘‘Buckminster Fuller versus the Irrational, A Double Entendre, in Morphology and Architecture,’’ a special issue of the International Journal of Space Structures, ed. Haresh Lalvani, 11,141-154 (1996).

9 ²A. L. Loeb, ‘‘Buckminster Fuller and the Relevant Pattern,’’ in Beyond the Cube, ed. J. F. Gabriel (New York: John Wiley, 1997).

10 ³ A. L. Loeb, ‘‘Deconstruction of the Cube,’’ in Beyond the Cube, ed. J. F. Gabriel.

11 many, but a finite number of, sides. The number k, which is irrational, may be approximated by the rational number 22/7. However, the fact that Fuller finds significance in the numbers 22 and 7 makes me uncomfortable.

12 In his tetrahelix, tetrahedra are located directly above and below each other after five rotations, because the dihedral angle of the tetrahedron equals essentially 72°, he believed. Fuller eschewed irrational numbers: just as a circle is really a polygon, he asserted, the dihedral angle of the regular tetrahedron, which is actually arc cos (1/3), should be rounded off to the rational one fifth of a complete revolution.

13 Here I believe that Fuller failed to realize that he was violating one of his own fundamental rules. Although the angles 72° and arc cos (X) differ in magnitude by only a little more than 1°, the former occurs in icosahedra, dome structures, etc., in planes perpendicular to axes of fivefold rotational symmetry, the latter between axes of threefold rotational symmetry. By ignoring these different spatial orientations, Fuller was in point of fact guilty of linear thinking. This is too bad, because in nature helical structures are significandy mismatched: if leaves were juxtaposed exactly above each other, they would intercept sunlight, hindering growth! We have pointed out (ref. 3) that most angles significant to Fuller, although not rational fractions of a complete revolution, do have rational trigonometric functions.

14 Whether our world is discrete or continuous has been a point of contention for thousands of years. In our century Buckminster Fuller was definitely on the side of discreteness. Fractal theory is on the side of continuity: coastal and cloud formations display geometrically similar structures at vasdy different levels of scale. Of course, the clouds eventually consist of drops of water, etc., but a continuous model is significandy successful at many levels. Thus it behooves us to compare Fuller’s polygonal model of the circle and other shapes with the mathematician’s definition of such forms, and, if possible, to bring them into some sort of conformance.

15 A circle, in mathematics, is defined as the locus of all points equidistant from a given point. This definition as I state it here is in fact redundant, because a locus by definition already encompasses all points that satisfy a given condition, so that the word all in the definition of a circle is not necessary. However, it is the very word which distinguishes the mathematicians’ definition of the circle from Fuller’s! Indeed, all vertices of Fuller’s polygonal circle are equidistant from a given point, its center, but there are many other points equidistant from that center. Hence Fuller’s circle is not the mathematicians’ circle. Neither is Fuller’s straight line that of the mathematician, namely the shortest distance between two points, because for some points on a ‘‘line’’ having finite width the sum of the distances from the two endpoints is greater than for others, so that the points constituting Fuller’s line cannot correspond to the shortest distance.

16
17 All this may be considered abstract philosophizing without substance, just like the question of how many angels can occupy the tip of a pin. However, the problem of optimizing distances is a very real one when one considers Fuller’s great circles of a sphere or the shortest distance between two points on an irregular surface. Here we need the mathematicians’ concept of limits, hence the concept of infinity, so hateful to those who believe in a discrete universe. Indeed, we tread on dangerous ground when we allow a variable to ‘‘go to infinity,’’ for there is no number infinity: it is just larger than any preassigned value. The fact is that we need to bring Fuller’s chalk circle and line, as we can visualize and draw them, into consistency with the mathematicians’ idealized concept.

18 We recognize a drawn chalk circle as a circle precisely because we do not have X-ray vision: we cannot see the location of the ions in the chalk because our eyes can only respond to a limited range of the electromagnetic spectrum. We can see those objects whose scale conforms to the wavelength of visible light. Our eyes do not respond to radio waves or infrared radiation, whose wavelength is larger than that of visible light, nor to ultraviolet or X-rays, whose wavelength is shorter. As a matter of fact, X-rays used to examine teeth or bones have a much larger wavelength than those used to locate the positions of ions, whose scale is so much smaller. Physical structures are therefore hierarchical: the scale at which we observe them depends on the power of resolution of the tools with which we examine them. Fuller’s chalk circle is identified as a circle precisely because with visible light we cannot distinguish its atomic components. Conversely, X-ray diffraction would not enable us to identify the object as a circle, but would tell us a great deal about the ionic structure of the chalk itself.

19 Fuller considers a circle as a polygon having lots of sides. We can easily distinguish it from a triangle, square, pentagon, hexagon, octagon, and even a triacontagon, which has thirty sides. However, sooner or later a polygon will turn up that has so many sides that we can no longer visually distinguish it from a circle. We can then say that within our power of resolution this polygon is indistinguishable from a circle. No mathematical infinities necessary here, only the finite level of resolution appropriate for the hierarchical level at which we are examining the structure. We can then refine the mathematics as long as we remain within the level of resolution appropriate for the measurements. This concept of resolution is very real in the use of computers, where the density of pixels determines whether or not a curve will be perceived on the screen as smooth. It also plays a role in wave (quantum) mechanics, where the uncertainty principle tells us that energy levels, while discrete, cannot be determined exactly over a finite time interval.

20 Let us examine on this basis Fuller’s rounding off of the dihedral angle of the regular tetrahedron to 72°. When we bring five regular tetrahedra together, each sharing an edge with one of each of the other four tetrahedra, we will find that they do not fill the space around the common edges: a gap of over 5° can readily be observed. The roundoff is therefore not permissible within our power of resolution.

21 These concepts may be applied to the following well-known problem.¹ Four bugs find themselves at the vertices of a square whose sides run from east to west and from north to south. The bug at the northwest vertex faces east, the one at the northeast vertex faces south, the one at the southeast corner faces west, and the one in the southwest faces north. Each bug thus faces the bug clockwise from it along an edge of the square. The bugs have been conditioned to travel at the same speed at any moment, although they may speed up or slow down at any time, as long as each remains traveling at the same speed as every other one. Each bug will now travel to the one it faces, and the problem is whether or not they will ever meet. As each bug travels, the bug it faces will also travel. Therefore each bug will need to adjust its direction of travel in order to keep tracking its target; as a result, the bugs will spiral into the center of the original square. But Buckminster would remonstrate that the bugs would have a finite reaction time, so that their paths would not be smooth spirals, but a series of straight line segments. Whether or not we perceive these paths as smooth spirals depends on the bugs’ reaction time as well as our power of resolution.

23 The first question to be answered is whether the path traveled into the center has finite length. It turns out (cf. ref. 4) that as long as the path is smooth within our power of resolution, its length just equals the initial distance between each bug and its target. This would mean that each bug would easily be able to meet its target at the center of the original square. However, other factors play a role. If the bugs travel at constant speed, then as they approach the center of the square, their angular velocity will increase alarmingly. Now the size of the bugs will begin to play a role: if they are relatively large, the centrifugal forces on their inner and outer shoulders will differ sufficiently so that they will fly apart, although they may touch shoulders before this catastrophe would occur. The smaller they are, the more their angular velocity will increase, but the less the centrifugal forces will differ, so that they well might touch shoulders before critical angular velocity is reached. Buckminster Fuller will not permit us to shrink them to a point, so the outcome is still uncertain.

24 Suppose, however, that we would avoid catastrophe by using the fact that the bugs may accelerate or decelerate together. Let them travel at constant angular velocity so that they will not fly apart. The problem now will be that as they approach the center of the square, their speed will decrease, with the result that they will never reach the center. And consider this embarrassing question: if they do reach the center, from which direction will they do so? A tangent to the spiral path at all times makes an angle of 45° with the radial vector from the bug to the center of the square, so that the bug will never actually be directed into that center. So can the bugs ever meet in the center? Within our power of resolution the answer could be affirmative, but woe to anyone who would let the bugs shrink to a point!

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26 An analogous problem is the following one. An airplane leaves Boston with its automatic pilot set so that it will fly and continue to fly in a northwest direction. It will spiral toward the north pole, and there a landing strip will be readied for the plane. The question from the north pole is how the landing strip should be oriented. The answer is that any orientation is all right, because the plane will not be able to land as long as its automatic pilot remains fixed on a northwest course: the north pole will always remain to the right of the programmed trajectory, but since the plane is not a point, it will, within its power of resolution, find itself direcdy over the pole, and can then land as it pleases.

27 These examples illustrate the indeterminacies inherent in a model for continuous structure, and the pitfalls avoided by Buckminster Fuller by postulating a discrete structure. Nevertheless, mathematical abstractions such as calculus provide idealized constructions whose behavior simulates that of real structures within the limits of their resolution. It would appear that the concept of limited resolution would reconcile Buckminster Fuller’s hierarchical model of real, discrete structures with the idealized ones of the mathematician.

28 Discoveries of Synergetics from Synergetics:

29 Explorations in the Geometry of Thinking

30 250.01 Discovery

31
32 250.02 Discoveries are uniquely regenerative to the explorer and are most powerful on those rare occasions when a generalized principle is discovered. When mind discovers a generalized principle permeating whole fields of special-case experiences, the discovered relationship is awesomely and elatingly beautiful to the discoverer personally, not only because to the best of his knowledge it has been heretofore unknown, but also because of the intuitively sensed potential of its effect upon knowledge and the consequently improved advantages accruing to humanity’s survival and growth struggle in Universe. The stimulation is not that of the discoverer of a diamond, which is a physical entity that may be monopolized or exploited only to the owner’s advantage. It is the realization that the newly discovered principle will provide spontaneous, commonsense logic engendering universal cooperation where, in many areas, only confusion and controversy had hitherto prevailed.

10.

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34 Academic Grading Variables in Respect to Science Versus Humanities250.101 Whether it was my thick eyeglasses and lack of other personable favors, or some other psychological factors, I often found myself to be the number-one antifavorite amongst my schoolteachers and pupils. When there were disturbances in the classroom, without looking up from his or her desk, the teacher would say, ‘‘One mark,’’ or ‘‘Two marks,’’ or ‘‘Three marks for Fuller.’’ Each mark was a fifteen-minute penalty period to be served after the school had been let out for the others. It was a sport amongst some of my classmates to arrange, through projectiles or other inventions, to have noises occur in my vicinity.

11.
12.

20.

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36 Where the teacher’s opinion of me was unfavorable, and that, in the humanities, was—in the end—all that governed the marking of papers, I often found myself receiving lower grades for reasons irrelevant to the knowledge content of my work—such as my handwriting. In science, and particularly in my mathematics, the answers were either right or wrong. Probably to prove to myself that I might not be as low-average as was indicated by the gradings I got in the humanities, I excelled in my scientific classes and consistently attained the top grades because all my answers were correct. Maybe this made me like mathematics. But my mathematics teachers in various years would say, ‘‘You seem to understand math so well, I’ll show you some more if you stay in later in the afternoon.’’ I entered Harvard with all As in mathematics, biology, and the sciences, having learned in school advanced mathematics, which at that time was usually taught only at the college level. Since math was so easy, and finding it optional rather than compulsory at Harvard, I took no more of its courses. I was not interested in getting grades but in learning in areas that I didn’t know anything about. For instance, in my freshman year, I took not only the compulsory English A, but Government, Musical Composition, Art Appreciation, German Literature, and Chemistry. However, I kept thinking all the time in mathematics and made progressive discoveries, ever enlarging my mathematical vistas. My elementary schoolwork in advanced mathematics as well as in physics and biology, along with my sense of security in relating those fields, gave me great confidence that I was penetrating the unfamiliar while always employing the full gamut of rigorous formulation and treatment appropriate to testing the validity of intuitively glimpsed and tentatively assumed enlargement of the horizon of experientially demonstrable knowledge.My spontaneous exploration of mathematics continued after I left Harvard. From 1915 to 1938—that is, for more than twenty years after my days in college—I assumed that what I had been discovering through the post-college years, and was continuing to discover by myself, was well known to mathematicians and other scientists, and was only the well-known advanced knowledge to which I would have been exposed had I stayed at Harvard and majored in those subjects. Why I did not continue at Harvard is irrelevant to academics. A subsequent special course at the U. S. Naval Academy, Annapolis, and two years of private tutelage by some of America’s leading engineers of half a century ago completed my formally acknowledged ‘‘education.’’My Independent Mathematical Explorations230.21 In the twentieth year after college, I met Homer Lesourd, my old physics teacher, who most greatly inspired his students at my school, Milton

37 Academy, and who for half a century taught mathematics at Harvard. We discovered to our mutual surprise that I had apparently progressed far afield from any of the known physio-mathematical concepts with which he was familiar or of which he had any knowledge. Further inquiry by both of us found no contradiction of our first conclusion. That was a third of a century ago. Thereafter, from time to time but with increasing frequency, I found myself able to elucidate my continuing explorations and discoveries to other scientists, some of whom were of great distinction. I would always ask them if they were familiar with any mathematical phenomena akin to the kind of disclosures I was making, or if work was being done by others that might lead to similar disclosures. None of them was aware of any other such disclosures or exploratory work. I always asked them whether they thought my disclosures warranted my further pursuit of what was becoming an ever-increasingly larger body of elegantly integrated and coordinate field of omnirationally quantified vectorial geometry and topology. While they could not identify my discoveries with any of the scientific fields with which they were familiar, they found no error in my disclosures and thought that the overall rational quan- tation and their logical order of unfoldment warranted my further pursuing the search.

38 250.30 Remoteness of Synergetics Vocabulary

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40 250.301 When one makes discoveries that, to the best of one’s knowledge and wide inquiry, seem to be utterly new, problems arise regarding the appropriate nomenclature and description of what is being discovered as well as problems of invention relating to symbolic economy and lucidity. As a consequence, I found myself inventing an increasingly larger descriptive vocabulary, which evolved as the simplest, least ambiguous method of recounting the paraphernalia and strategies of the live scenario of all my relevant experiences.

41 250.31 For many years, my vocabulary was utterly foreign to the semantics of all the other sciences. I drew heavily on the dictionary for good and unambiguous terms to identify the multiplying nuances of my discoveries. In the meanwhile, the whole field of science was evolving rapidly in the new fields of quantum mechanics, electronics, and nuclear exploration, inducing a gradual evolution in scientific language. In recent years, I find my experiential mathematics vocabulary in a merging traffic pattern with the language trends of the other sciences, particularly physics. Often, however, the particular new words chosen by others would identify phenomena other than that which I identify with the same words. As the others were unaware of my offbeat work, I had to determine for myself which of the phenomena involved had most logical claim to the names involved. I always conceded to the other scientists, of course (unbeknownst to them); when they seemed to have prior or more valid claims, I would then invent or select appropriate but unused names for the phenomena I had discovered. But I held to my own claim when I found it to be eminently warranted or when the phenomena of other claimants were ill described by that term. For example, quantum mechanics came many years after I did to employ the term spin. The physicists assured me that their use of the word did not involve any phenomena that truly spun. Spin was only a convenient word for accounting certain unique energy behaviors and investments. My use of the term was to describe a direct observation of an experimentally demonstrable, inherent spinnability and unique magnitudes of rotation of an actually spinning phenomenon whose next fractional rotations were induced by the always co-occurring, generalized, a priori, environmental conditions within which the spinnable phenomenon occurred. This was a case in which I assumed that I held a better claim to the scientific term spin. In recent years, spin is beginning to be recognized by the physicists themselves as also inadvertently identifying a conceptually spinnable phenomenon—in fact, the same fundamental phenomenon I had identified much earlier when I first chose to use the word spin to describe that which was experimentally disclosed as being inherently spinnable. There appears to be an increasing convergence of scientific explorations in general, and of epistemology and semantics in particular, with my own evolutionary development.