by Gregory Stiles

At this site, we discuss ideas relating to synergetic geometry, which will undoubtedly be new to many. While it is true that one can readily learn of synergetics directly from R. Buckminster Fuller either by reading his books or perusing them online (see the Links page), his writing style may not always be the most inviting. As an aid, I thought I would write an introduction to synergetics enriched with thoughts from our research group so that outsiders might have a better appreciation for any work pertaining to this geometry they may encounter here.

R. Buckminster Fuller, more often referred to as Bucky, developed synergetic geometry over the course of a lifetime. Indeed, one could even argue he had found the fateful seeds of his mathematics as early as kindergarten. As Bucky tells the tale, he was given some toothpicks and dried peas to play with architecturally in kindergarten, but while his optically superior peers built the typical cubes reflecting common houses, Bucky, with his poorer vision, felt his way to constructing an array of alternating tetrahedra and octahedra--which are the very essence of the grid used in synergetic geometry as he would present it so many years later.

Bucky grappled with his mathematical ideas for many decades. At one point he took them to a favorite old professor and asked if the professor happened to be aware of any similar work. The professor said that he was not, and encouraged Bucky to proceed, which he then did with his customary enthusiasm.

He was driven largely by a sense of beauty, clarity, and elegance. To those who would later admire him, such as the members of this research group, Bucky was not exactly one who delighted in matching his math to his reality...for the most part, he seemed to think if something was breathtakingly beautiful, it must be true and have a secure place in the actual world. To be sure, this attitude left many challenges for those scientists who consider themselves more driven by data than abstraction when choosing a mathematical foundation upon which to rest a theory. Nevertheless, it is bemusing to consider just how prophetic this visionary has been to those who learn the concepts with which he grappled.

Nearing the last years of his life, Bucky finally felt he could wait no longer to share his ideas. He had felt some resistance to his different ways all his life, and he always seemed to be rather at arm’s length to the official academic community (he joked of having been “fired” from Harvard twice), though one cannot help but ask whether that was more their fault or his. The body of synergetics was, though, finally given to the public in two books: Synergetics in 1975, and a collection of strategic supplementary paragraphs in Synergetics 2, offered in 1979. The timing may have been critical, as Bucky then died in 1983.

Synergetics is, first and foremost, a geometry. Given that many are not even aware that there could exist more than one geometry, it might be helpful to some to back up and consider the history of the concept of geometry. Once we have a working definition, I shall offer examples of the way a geometry such as synergetics has advantages over another geometry, such as Euclidean/Cartesian, and finally I shall conclude by trying to restore some sanity to geometric discussions by getting rid of one the most misused terms in all geometric discussions: dimension.

For centuries, the body of work collected in the book that would be known as the Elements of Euclid defined for literate men the one and only geometry...the geometry, as it were, of God. The book began with simple ideas, or axioms, and advanced from there to more technical propositions. The logic was considered immaculate and inspiring to many, though a tiny few noticed flaws and questioned aspects of the foundations.

One of the main disputes arose over a central idea, which would become popular as the “parallel postulate.” In simple terms, the Elements stated that two parallel lines, if extended to infinity, will never meet. Various mathematical rebels questioned this conclusion, and soon proposed alternatives, as recounted in Morris Kline's history, Mathematics: The Loss of Certainty.

The parallel postulate may have been the focus of much debate, but it is curious to note that several other propositions, which should seem initially far more dubious than the parallel postulate, have received fairly little criticism. The first of these was the definition of a point as that which has no dimension. Though this definition cannot help but be sheer nonsense, it has seemingly been unquestioned for centuries. Even sillier, one is then told that a line, which is something, is somehow supposed to come into existence by arranging several points, which are all nothing, in a given direction. But Bucky wisely asked: how can a sum of nothings be something? Clearly they cannot. It would seem the very foundations of Euclidean geometry were absurd, yet for the most part unquestioned in ancient times, much as they are today. It would seem reasonable to call any geometry that abandons dimensionless points, as synergetics does, non-Euclidean.

In the nineteenth century, three other geometries came to be known. Yet even those johnnies-come-lately would not be able to monopolize the concept of geometry. In modern times, Benoit Mandelbrot became rather famous for offering a set of ideas that would be called fractal geometry, though so far as I know Mandelbrot had not offered us any thought whatsoever about his own stance on the parallel postulate.

If all of this were not enough confusion about the concept of geometry, we had Bucky toss his hat into the ring with Synergetics, also called a geometry. In much the same way that the term “love” has come to mean so many things to so many people, we see that too many people using a term can stretch the meaning such that outsiders, hoping for a simple explanation of complex matters they can digest over morning donuts, become a little peeved and impatient.

Hence we arrive at the question: under what circumstances may one call a set of concepts a geometry? I suppose there could be as many answers to that question as there are people, but for our research group’s purposes we have chosen our own criteria, and we share them in the hope that others might find them as useful as we do. If these concepts are not useful to you, or if you like them and have already heard them elsewhere, we ask your forgiveness--researching in some fifty odd disciplines does not always leave time to read every mathematical treatise ever offered, and we are doing the best we can as far as giving credit where credit is due and suggesting helpful approaches.

The first idea that presents itself to us is that, as geometry comes from roots which mean “earth measure,” to qualify as a geometry the concepts must have a specific way of measuring space. One might think that space itself is such a basic concept it needs no explanation to civilized people. Alas, once intellectuals start tearing at a piece of exposed flesh there are few who will recognize where it came from after the feeding frenzy, so I shall add the next paragraph to further clarify what is meant by space, which the less abstract of you might wish to blissfully skip.

I have seen several definitions of space, such as: 1) Space is nothingness, a void 2) Space is negative space, as described, for example, by artists. If you imagine a framed painting of a man shown standing in the center of a blank white canvas, an artist would say the man represents positive space, and the space between the man and the frame represents negative space. 3) Space is a subdivided grid, like the 3 axis, 90 degree grid offered in Cartesian geometry. The grid has no objective reality, but is merely a handy aid to mapping events. 4) There is an inflexible grid as before in 3), but the grid is real and tangible, with compartments that restrict movements through its corridors. In the same way one might not be able to go from the kitchen to the bathroom, but must pass through room compartments before arriving at the eagerly sought destination, space allows movements in certain ways only. 5) Finally, I have heard of a great many who believe in a real grid that actually moves around all over the place and interacts with both matter and energy. Probably, if one enjoyed torture, one could find further definitions, each with a frothing, fierce advocate. But for our purposes here, I wish only to speak of definition 3--an inflexible, imaginary grid that can come in handy when doing certain practical things such as making measurements of space. To appease the many devotees of Einstein who cherish definition 5, I shall touch upon relativity towards the end of this paper as I sort through the abused concept of dimension. But for now I return to my sheep, and speak of geometry as a way of measuring space as described in definition 3.

So how does one go about measuring space?

We choose to use two criteria here that will satisfy our desire that a geometry measure space in a meaningful way: 1) the geometry must supply a set of coordinate axes at definite degrees from each other and 2) the fundamental volumetric unit of the grid must have some demonstrable relevance to Earth, rather than being a useless fairy tale mathematical construction, or why call it a geometry at all, if the roots for geometry mean “earth measure”?

Undoubtedly some examples of these ideas will help a lot. In Figure 1 we see the Cartesian grid. Criterion 1 is met by stating that there are three straight line axes at 90 degrees to one another, and criterion 2 is met by taking the primary volumetric unit of space--the cube--and showing that cubes do show up in some real life situations, such as in a few crystals at least.

How would one define synergetics in these terms? In Figure 2 we see the synergetic grid, called the isotropic vector matrix (iso = same, tropos = place/shape, vector = line segment noting a length and direction, matrix = grid; or I guess you could call it "the line segments look the same everywhere grid"). For criterion 1, we would say that we have 6 axes at 60 degrees. For criterion 2, we would say we have one fundamental volume, called the quanta module, and that these modules come in two forms, the a and the b, and each has both positive and negative forms. These modules are all tetrahedra. They combine to form the grid network of alternating tetrahedra and octahedra that fill all of space: the synergetic isotropic vector matrix grid. The volumetric quanta modules, called the a and b modules, can be used to construct all known manifestations of matter and energy.

The last sentence may seem rather bold, but I offer an open challenge to anyone who feels they can disprove it with a clear mathematical argument after reviewing the whole of synergetics and papers such as the ones offered here. For those who want to understand why Bucky developed the idea of the quanta modules, I eagerly invite you to read onward and enjoy the powers of this graceful geometry.

As long ago as the ancient Greeks and Egyptians, five fundamental regular solids came to be known--the five platonic solids: the tetrahedron, the hexahedron (cube), the octahedron, the pentagonal dodecahedron, and the icosahedron Figure 3.

To have stumbled upon the shapes of the 5 platonic solids and the geometric reasons that permitted just these 5 to exist is, in my eyes, a great achievement in the course of civilization. (By the way, I am often asked of the connection between synergetic geometry and something called “sacred geometry”. From what I have seen, insofar as sacred geometry shares an interest in the regular and semi-regular polyhedra, the two geometries have some things in common. However, synergetics has far less interest in perfect circles and spheres as actual building blocks than sacred geometry appears to have. In any case, I personally find the idea of calling a geometry “sacred” rather disturbing, as it evokes images in me of madmen slaughtering thousands in the name of creating a perfectly geometric homeland or some other such gobbledygook, and thus I shall merely cautiously note that sacred geometry and synergetics may have a few things in common).

Alas, right at this point the conquerors and butchers were merrily ransacking and pillaging the globe in egotistic campaigns of conquest, and what great steps may have been taken from this critical artistic point without the stupidity of warfare can only be speculated upon while sifting through the ashes of Alexandria and feeling the pangs at each reference to a text in ancient writings that no one has seen.

Nevertheless, had anyone bothered, the key to synergetics has always been just exactly there, namely, with those 5 platonic solids...for all anyone had to do was find a way of relating each to the other, and likely synergetic geometry might have dominated mankind rather than the Cartesian gridding that came much later in the middle of the second millennium.

It would seem all philosophers have searched for something like a “monad”--some fundamental unit of existence that could be used to build all the larger ones. Bucky was no different, and his offering of a fundamental unit was exactly 1/24 the original volume of a tetrahedron: the quanta module.

As shown in Figure 4, the tetrahedron can be subdivided into 24ths, and the irregular tetrahedra thus formed are called the a quanta modules. Bucky then wanted to see if he could use these puzzle pieces to build the other platonic solids. What he found out when he tried to build the octahedron was that a modules alone could not do the job, but if he added another piece that had the identical volume as the a modules (the volume being the same because both modules can be cut from a right angled prism by making cuts along prismatic vertices and midpoints only--in other words, both are exactly half of a same original volume, and thus the same) he could complete his task. Thus he came to define the b modules, as shown in Figure 5. Was Bucky going too far...starting to make his work more complicated than simple? I do not think so, for now that he had both a and b modules, he found that he could combine them in such a way as to form another of the platonic solids, the hexahedron (cube)(see Figure 6).

No further puzzle pieces were necessary. Though undoubtedly disappointed that the a and b modules could not be used to construct the pentagonal dodecahedron and the icosahedron, he found a few sources of consolation. First, though he could not build the pentagonal dodecahedron from the modules, he could construct one of the more popular polyhedra (“poly” means many, “hedra” means sides--polyhedra are many sided figures) found in nature, namely, the rhombic dodecahedron. The rhombic dodecahedron was made of exactly 144 quanta modules, no more, no less, and is shown in Figure 7.

At this website, we have shown the rhombic dodecahedron as a fundamental shape of the atomic nucleus. For those who wonder how Bucky's, or anyone else’s, sense of beauty could allow them to consider the rhombic dodecahedron as being beautiful by gazing at its surface alone, I ask you to consider Figure 8, where we again see the interior as well as the exterior of the shape. Here we see several close packed rhombic dodecahedra constructed of quanta modules. For those of you who may think we deliberately added our own fancy geometric designs to create this beauty, I assure you we did nothing of the kind--that beauty arises quite spontaneously from a rigorous application of the mathematics involved--and frankly, I was as stunned as you might be when I first saw this display on the computer. One might do well to compare the elegance of that figure to imagining what close packed spheres might look like under similar circumstances--which do all of you find more beautiful?

The second form of consolation was that Bucky did indeed find a clever way of relating all the platonic solids--by using what is referred to as angular topology. With angular topology, we may define a polyhedron not by its volumetric interior, but rather by the sum of the angles around each vertex. As shown in Figure 9, all platonic solids are indeed found to be fundamentally connected by one simple equation that considers the tetrahedron (thus defined as 720 degrees) as fundamental:

S + 720 degrees = 360 degrees times X

Here, S is the sum of all angles around the vertices, and X is the number of vertices. So even though quanta modules could not do all that was hoped, angular topology could.

There is a great deal more that could be said on the subject of quanta modules and angular topology, but as this is an introductory essay, I shall pursue such matters later, and return to the main points.

Thus far, then, we have seen the way geometries arrange space. We have seen the Cartesian grid arrange space in a cubic grid, while the synergetic grid arranged space with alternating tetrahedra and octahedra, which can both be subdivided in turn into quanta modules, which are very useful for building the shapes we often see in reality. With synergetics thus defined more clearly, let us explore further the way the synergetic grid, the isotropic vector matrix, can be put to practical use.

I have seen that in mapping systems in physics, the Cartesian grid seems the toughest competitor with the synergetic grid, and in my view the synergetic way always wins. Let us go over a ton of examples of each approach until you get exhausted--just kidding--but let us still take a deep dive into real life for a moment, and contrast the two.

At the paper on this site, entitled "Proof of Synergetics in the Atom" I wrote about the presence of the vector equilibrium (more commonly known as the cuboctahedron) in the atomic nucleus. So let us contrast mapping the vector equilibrium on the Cartesian grid to mapping on the synergetic grid. Consider Figure 10. It is plain the vector equilibrium is a perfect match on the IVM, and looks somewhat silly on the Cartesian, as on that grid one axis hits a midpoint, another a vertex, and the third the center of a face.

Next, let us consider the nucleons of atoms, which are undeniably prevalent in most of the reality we know. In Figure 11 we first see the rhombidodec on the synergetic grid. It is again obvious from this example that the synergetic grid is superior. On the Cartesian grid, six points are hit; on the synergetic all twelve faces are hit straight on.

Next consider electron shells, described by icosahedra. Though Bucky had not been aware of our group's work in this area as it is very recent, he nevertheless described the mathematics of triangle projection well in his textbook on synergetics. Icosahedra factor heavily in synergetics, but are almost ignored in Cartesian thinking--how many icosahedra have you seen in your math courses?

Next, pulling back the camera a little, consider the lattices of crystals. As presented by Professor Arthur Loeb in his supplement to synergetics, pages 860-875, lattices are plainly far better shown on the synergetic grid than the Cartesian. When atoms close pack in crystals, the packing arrays can be modeled by special building blocks designed by Loeb that are made of various tetrahedra and octahedra. To further clarify and extend the important, fascinating and fantastic presentation of Loeb, we intend to be offering an even more detailed presentation of crystal lattices when our work in this area is further along.

All of these examples show an evident superiority of the synergetic grid to the Cartesian, and there are far more examples that could be offered, such as the tetrahedral arrays of diamond, the alternating hexagons and pentagons of buckyballs, the structures of geology--micas, amphiboles, and feldspars; cell vesicles, virus protein coats, and so much more until, in the end, I believe, everything could be included. It is for these reasons that Bucky and researchers like ourselves would offer synergetics and the IVM as an improvement over the Cartesian grid.

So now that we have seen the usefulness of the synergetic grid, I would like to think my introduction is complete. Yet the public has so often asked me about deeper matters I gather this introduction would not be complete if I did not address the implications of synergetic geometry for one primary issue and its relative (pardon the pun): How would synergetics interpret a term that has been mutilated beyond all recognition--dimension? And how would synergetics impact Einstein’s relativity and space-time continuum concepts?

I have been often asked my opinion of a certain piece of mathematical work that suggests that reality exists in 10 or 11 dimensions, such as described by Michio Kaku in Hyperspace. Do I find this plausible or implausible? My answer is that I find the idea implausible, because I feel the term “dimension” has become so abused at this point I cannot see how any mathematics using a variable for some abstract concept of dimension could make sense. I find Kaku's arguments needlessly theatrical, his analogies weak, and his case not viscerally compelling, though I would still encourage him to explore his ideas fully as his intuition guides him--each to follow their own star. As it is, the idea of dimension has become so diluted it seems a wonder to me that any two people could have a meaningful dialogue involving this term.

First, we have a group asserting that the x, y, and z axes (which synergetics wholly abandons) each constitute a dimension. If those are not enough, we have been given hypercubes, which we are told represent a fourth visual dimension, although they are not based on a 4th axis at 90 degrees to the first 3, as this is impossible given the construction method of the first 3. But before anyone could really assert hypercubes as sole representatives of the fourth dimension, they should have to reckon with the idea elucidated by Einstein, namely, that time is a fourth dimension, even though that concept is radically different from a straight line axis. Still not had enough? Enter fractal geometry, where certain mathematical processes are said to have a dimension--again having no connection to straight line axes whatsoever. Thus far, so far as I know, no bakers have proposed the essential significance of custard cream dimensions, and for this we may be grateful.

Really, how necessary are all of these “dimensions”?

I feel the hypercube concept was a quite useless and confusing addition to our thinking insofar as it made claim to showing another “dimension”. What exactly is a hypercube? Though I am sorely tempted to say that a hyper cube is one built by a hyper person, I shall refrain from doing so. The logic behind hypercubes has been described to me as follows:

Start with a point and make a copy of it--extend the new point and connect it to the first with a line. You have just created the first dimension. Now take the line, make a copy, extend the copy by its own length and connect it with lines--you have just built a square defining a plane, and thus the second dimension. Now make a copy of the square, extend it and connect, to make the third dimension, a cube. Now here is where it gets a little loopy in my view. Next you are told to take the cube and extend it (you have 6 options of where to go all of a sudden, too, as the cube has six sides) and now you have two cubes which you can connect with lines: a hypercube. A hypercube is just two cubes connected by lines? Yep, apparently so. (Others say the cube explodes as in the famous Dali painting). Can two such cubes (or six for that matter) be drawn in space by using the first 3 dimensions, the x, y, and z? Of course they can--so therefore a hypercube cannot lay claim to breaking new ground. We cannot draw a plane inside a line, by definition, nor can we draw a cube in a plane, so that at least the first three “dimensions” were distinct from each other. But when we can draw a hypercube within the axes of the first 3 “dimensions”, hypercubes then become impertinent troublemakers for our thinking as I see it. Whatever use they may have, I see no reason to ascribe a special “dimensionality” to them.

But what of the IVM? Since it has 6 axes rather than 3, does this mean that it proposes that there are 6 dimensions to reality? With all the different arguments I have heard, I suppose one could interpret the case that way. To be sure, Bucky boasted that dimensions beyond the third were quite simple matters for synergetics to handle--yet I must respectfully disagree without going into detail on this matter at this time due to space and time limitations. But I will respond here to a popular question: must one now specify 6 points for every coordinate in synergetics rather than 3--one for each axis? To which I answer most emphatically: NO! No, no, no, no, no.

Why not? Because even in synergetics we can still rely on just 3 coordinates to specify a point in space, and to accomplish this task we simply fall back on another form of coordinates which many of you may have seen in passing: polar coordinates. In my high school math education, I was intrigued to learn that there were two popular ways of specifying coordinates: the Cartesian and the polar ways. Naturally, since I felt much more sympathy for the polar approach, this was the one I was told to abandon...though I always wondered about why the Cartesian approach was better. Now I feel a little smugly vindicated in resurrecting polar coordinates for use in synergetics.

To see how polar coordinates work, return to Figure 1. Another way of reaching point P would be first to move over on the xy plane an angular amount, say, theta (theta is a Greek letter, not to be feared--when intellectuals want to scare people they may also try to speak Greek, try not to let them get to you). Next move up on the xz plane another angular amount phi (another Greek letter, you did not get scared that time, did you?). Finally, you need one last piece of info, how far out along the line where these angles meet one needs to go. Another way to think of this is to imagine two perpendicular 360gons (polygons with 360 sides) that share the same center and size, each with a stick running through their diameter, the vertical having the stick through the center z axis so it may rotate through the xy plane, while the other has a stick through the center y axis so it may rotate through the xz plane. The center point where they meet can serve as our origin. To specify the coordinates of any point anywhere in space, we use as a first coordinate an angle in degrees along the horizontal 360-gon, turning the vertical 360-gon until it intersects the point at that angle. We have now identified the plane in which the point can be found. Next, we rotate the horizontal 360-gon until it also intersects the point--and thus we identify the line on which the point may be found. Finally, to precisely pinpoint our point in space, we have only to specify how far out along this line the point is located. Thus we see that any point in space can be specified with 3 polar coordinates. Normally, polar coordinates are used with perfect circles, but as synergetics relies on polygons and abandons pi, polygons are substituted by artistic license of our research group. If 360 degrees are insufficient for a given case, the number of sides (or overall frequency as we might say in synergetics) of the polygon may be increased, just exactly as is done when circles are used. Thus may we use three coordinates without agonizing over possible added dimensions caused by using 6 axes.

But now having seen the way coordinates can work in synergetics, what of Einstein and relativity? Does time not constitute another dimension? When we consider an object, must we not take time into account in synergetics as well?

Einstein mentioned time in several respects, but the two that are of most interest to us here are 1) that objects integrate and disintegrate and 2) that time measurements may be essential to understanding spatial “dimensional” measurements. Let us consider both instances.

As Paul Hewitt writes in Conceptual Physics of a box defined by the first three dimensions:

“The box was not always a box of a given length, width and height. It began as a box only at a certain point in time, on the day it was made. Nor will it always be a box. At any moment it may be crushed, burned or destroyed.”

One certainly have no objection to that logic--yet it still does not compel me to think of time as a fourth dimension. Provided we are intelligent enough to not try building our house out of bricks that do not yet exist, it would not appear that neglecting time as a fourth dimension would be likely to cause us much harm for practical matters.

The second matter, though, concerning the way measurements of time affect measurements of space as described by Einstein in his papers on relativity, is a more serious matter to be addressed. It seems pertinent to review the key concepts here.

As Paul Hewitt has offered the clearest presentation of relativity for the public I have yet seen, I shall again defer to his explanations here (by all means, enjoy Hewitt's full explanations in Conceptual Physics and Conceptual Physical Science). On the subject of the bending of light, which is what got us all into geometric trouble in the first place as it eventually led to the formulation of a nominal “curved space-time continuum,” Hewitt offered the following diagram and explanation, presented in Figure 12.

Hewitt jumps in to stress the official party line here, namely, that light is massless. This is the part of the Einstein theory that makes little or no sense to my colleagues and me. Since light is indeed acting as if it has a mass, why not simply think that it does have mass? Why is it more logical to abruptly leap off the deep end and assume that light is massless and it is instead space that is warped? The idea of warped space itself is one that has been formulated very, very weakly in our view. For if “space” in this context (which has never once been precisely defined by any mathematician or physicist I have questioned) can be influenced by matter and energy, then what is this space made of? Is it not matter or energy, but something else entirely? If so, what is it? Every physicist I have asked about this has given me evasive, cryptic answers, as if they are hiding the fact they really have no idea what they are talking about. Their evasion is enough to convince me their case is groundless, but let us assume for the sake of argument that they are correct, that bending light is best explained by a warped space-time--we must still return to the earlier issue raised by Einstein: that paths of motion are relative to an observer. Thus, even if we want to call “space-time” curved as an observer sees light bending around a large mass, we must still return to the inside observer, who saw the light travel in a straight line. Is the best physics explanation that the inside observer is on drugs? No, we must say, his point of view is equally valid. In this sense, confusion arises from perspective, as in the Doppler effect.

In the Doppler effect, two policemen in a car will hear their siren at the same frequency as it goes down the street. But to an outside observer, the soundwaves bunch up on themselves as the car heads towards them, making the pitch of the sound higher. After the car passes, the waves are stretched out, and a lower pitch is heard. In this case, there is no greater validity in either viewpoint--both interpretations come from different perspectives. Even so, with a geometry like synergetics, one can pull back from either point of view to perceive an absolute truth. We can do this because the IVM grid is defined using definition of space 3, where our grid is abstract, and thus not subject to these local discrepancies.

Was this the only pitfall to be overcome with relativity? No. Einstein also spoke of another case, again described by Hewitt, and considered in Figure 13. The difference between this case and the last was the phrase “even if no relative motion was involved.” The logic here reminds me of something known as a “traveling error” described by one of our associates, which he coined from playing the game “Master Mind”. Every once in a while, in solving a code, one could interpret a steady run of clues saying that one part of a code was right to mean that one part of the code was consistently right in each guess offered to break that code. However, on certain occasions, it was later found that a clue that said one part of the code was right in the first guess, and another clue that also said a part of the code was right in the second guess, and again in the third, was not due to the same code part being correct in the same place each guess, but actually that something had been correct in each part of the code that was changing each guess. In this sense, the error of believing some idea was correct was traveling from one theory to the next, but in the background, it was the changing parts of the theory that were leading to successful outcomes.

In a similar sense, I believe that light bends not because of curved space time, but because it has mass. If the rulers’ measurements also differed as if space time was bent, I would take issue with the phrase “no relative motion... involved.” While it is somewhat correct to say the Earth moves quite a bit as a solitary object in relation to other objects, it hardly seems logical to say that there is no relative motion on the earth when it spins, especially when it is openly admitted that the outside moves faster than the inside. Further, even if it were still, it seems there has been insufficient accounting of all other possible motions that could lead to these results. As long as there is an ounce of heat in the cosmos, there shall always be some form of relative motion as I understand it, such that the phrase “no relative motion involved” becomes fantastic and mythical for practical beings. Even these are not all the faults we find with thinking about light and time and their impact on geometry, but Einstein’s relativity opens up longer roads than I would negotiate in such a short paper, so I shall have to leave this matter for now. Suffice to say that light having mass and only apparently moving, in addition to the idea that there are many forms of relative motion that might also account for distorted space measurements are enough to at least offer the possibility that thinking of space-time as warped may be completely wrong. In any case, we shall have to pursue the matter in greater detail later.

Having thus dealt with Einstein and relativity, what of fractal “geometry” and its use of the term “dimension”? Though I have a great deal of sympathy for fractal geometry, I wish its originators had chosen their terms more carefully. As noted previously, I believe a geometry should only be considered worthy of the name if it specifies coordinate axes at definite degrees and offers a fundamental volumetric unit related to the earth--fractal geometry fails on both counts. Further, its concept of dimension is again contained within the aforementioned coordinate axes systems. Not only that, but the “dimension” described in fractal “geometry” does not apply to single objects, but to a process involving several objects. Let us consider the definition of fractal “dimension” in more detail.

The fractal dimension is defined by a process I shall not delve into deeply here. I am writing for the general public, and many seem to live in mortal terror of mathematics, and more is the pity (pardon my hyperbole). Fractal dimensions mostly range from values of zero to three. A value close to 1 involves a figure that comes close to filling out 1 dimension--a line. A process that comes close to filling up two dimensions (a plane) will have a fractal dimension near 2, while one that comes close to filling up three dimensions will be near 3. Fractal geometry takes a starting object, called an initiator, then applies a rule (called a generator) for, say, dividing it up. For example, an initiator may be a line, and a rule for dividing the line may be to cut the middle third out of the line and keep the outside ends. Then, in the next step (say “iteration” at the next cocktail party to impress people), you apply the same rule to the end fragments, and so on, and so on. In the end you will not have much left in terms of pieces compared to a full line, so the fractal dimension will be less than one. It is clear from the most cursory examination of the concept that this "dimension", again, does not apply to one object, but rather to a process involving several objects. Under such circumstances, the term dimension can only add confusion. I would have preferred a term like “plenitude” (fullness) be used instead. Why plenitude? The higher the fractal dimension, the greater the ratio of positive to negative space (perhaps we could call the plenitude factor the positivity value?).

All of this may seem unduly harsh on fractal geometry, but the truth is I think fractals have much to offer us. I believe the possibilities suggested by the work of those such as Stephen Wolfram are especially tantalizing, but this subject is too vast to explore here. For now, one thought I feel compelled to mention is that I think fractals, more than any other approach, offer what I consider the best potential definition of time: time is that which marks the iterations (excuse my jargon please, I slipped) of a fractal equation. One iteration of a cosmic fractal equation lets all items in a cosmos move one minimum step. In Einstein’s relativity, time is thought of in relation to a measuring tool, like a clock. In that sense time is truly relative--relative to a clock subject to worldly fluctuations. Fractals, however, concern themselves with time in the same way I feel synergetics concerns itself with space, namely, as abstract, precise, and absolute--it seems a rather natural marriage to me.

Finally, what of all the mystics and paranormal individuals who ask me of other dimensions? I cannot say I know for sure all of the possibilities that may exist in this cosmos, but I can say the term “plane” as used to signify another realm sounds silly to me--we do not live on a plane on earth; why would we live on flat planes elsewhere? In addition the idea of “parallel universes” seems equally poorly developed. If the universe is shaped as a sphere, 12 spheres can surround that sphere, so which of the twelve is to be the parallel, and why? In any case, as for other dimensions, I might use the term “scape” for such situations, and define a scape as a frequency range in the electromagnetic spectrum. The bee sees a world in ultraviolet, for example, that is unknown to humans without equipment. In such a case, I suppose one might reasonably speak of, say, an “ultrascape” in a useful way.

In conclusion, I propose that synergetics is the most powerful geometry for the aforementioned reasons, and that the term “dimension” has been drastically abused such that it should perhaps be abandoned altogether, and I offered alternatives for debate in this paper in an attempt to help rid myself at least, of a recurrent pesky confusion flavoring my interaction with the public. Any with helpful comments on specific points are, of course, always welcome to offer even better solutions, which I will be eager to consider.

I prize clarity in thought and writing, and I believe the clearer our concepts, the more productive the works built upon them. For those who have chosen to read this introduction before moving on to other papers, such as "Proof of Synergetics in the Atom," I encourage you to proceed.

A new world lies just beyond the gate.

Fire in!

You may direct e-mail inquiries and comments to Gregory Stiles at

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