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Amy C. Edmondson
A Fuller Explanation
Chapter 6, Angular Topology
pages 70 through 73


(1) A measure of spatial extent. (2) Magnitude, size, scope. (3) The number of factors in a mathematical term. (4) A physical property, often mass, length, time, regarded as a fundamental measure. (5) Any of the least number of independent coordinates required to specify a point in space uniquely.

      The above is a sampling of what you will find in English-language dictionaries under "dimension." As you can see, there are a few distinct meanings—essentially falling into three categories:


Physical extent or measurement, as in "what are the dimensions of this room"?


Orders of complexity, in the most general sense, as in "the many dimensions of an issue or problem. This meaning is as common as it is widely applicable.


The specifically mathematical application: the number of independent terms required to specify a point in space. Our conventional system utilizes three independent, mutually perpendicular axes in space to accomplish this task. This is often the first meaning to occur to people, especially when already thinking about geometry. "Space is three-dimensional." However, this assignment—treating the third category as an exclusive definition-seemed unacceptably limited to Fuller. Exposure to the ordered polyhedra of mathematics and also to organic structures and crystals found in nature makes an orientation toward perpendicularity seem quite arbitrary. Although right angles are sprinkled throughout geometric shapes, they are by no means dominant. And, more often than one might expect, ninety-degree coordinates provide an awkward framework with which to describe both naturally occurring and conceptual formations.

      Fuller viewed the Cartesian coordinate system with its three perpendicular axes, conventionally labeled X, Y, and Z, as a remnant of "flat-earth thinking." Early man, finding himself on a huge fiat expanse, assumed that up-and-down and back-and-forth were the fundamental directions of his universe. Ninety degrees was the obvious natural angle with which to segment and measure space. Humankind has had more and more evidence of nature's radial and spherical bias throughout history—from the discovery of the shape of planets to the behavior of radiation and cellular growth. But neither Copernicus's spherical earth nor the vast array of biological and physical phenomena, all suggesting that angles other than ninety degrees would provide more "natural" or convenient standards, succeeded in reorienting the perpendicular bias of mathematics.

      The "three dimensions" of mathematics—length, width, and height-became part of an unshakable convention. That space cannot accommodate a fourth perpendicular direction is just one of its many intrinsic constraints, and yet this limitation is too often seen as the only characteristic of space. While mathematicians postulate hypothetical "hypercubes" in their attempt to describe a spatial fourth dimension, and physicists refer only to "time" as the fourth dimension, Bucky preferred to call attention to the "four-dimensional" tetrahedron. Time is certainly a dimension, but the physicists' progression "x, y, z, and t" seemed not to emphasize sufficiently that time—permeating all space and all experience—is qualitatively unlike the other "three dimensions."

      As we develop an awareness that space has shape, right angles gradually seem less "natural." The XYZ coordinate system often serves to obscure rather than illuminate spatial characteristics. It is a valuable tool, which we can recognize as one alternative superimposed by human minds, not as a framework organic to the shape of space itself. The word "dimension" is used without contradiction to describe the maximally symmetrical arrangement of three lines in space; likewise it can be applied to time, but it's not the end of the story.

One of the above dictionary definitions refers to the number of coordinates required to specify the location of a point in space. Assuming the existence of a previously specified origin, the number of coordinates happens to be three. Does this result reinforce the exclusive use of the XYZ axes? No, for the three coordinates required do not have to be Cartesian; another option is spherical coordinates, in which the location of any point is fixed by specifying two angles and a radial distance. Cartesian coordinates, on the other hand, describe a location as the intersection of three lines originating at given distances along three perpendicular axes. (See Figure 6-1 for a comparison of the two methods.) The spherical approach is more suited to Fuller's radial ("converging and diverging") Universe; its emphasis on angular coordinates encourages thinking in terms of "angle and frequency modulation."

Cartesian coordinate and spherical coordinate systems
Fig. 6-1. Cartesian versus spherical coordinates.
Click on thumbnail for larger image.

      Dimension is a widely encompassing term, and can legitimately refer to numbers of factors in a variety of geometric phenomena. Considerable time can be devoted to unraveling Fuller's different uses of "dimension" in Synergetics, and we shall continue to cite examples throughout our investigation.


Fuller's book takes a firm stand in the opening sections: "Synergetics originates in the assumption that dimension must be physical" (200.02), meaning size. The declaration is soon reinforced: "There is no dimension without time" (527.01). Firmly imbedded in reality now: it takes time to embody a concept. Everything ties together, so far.

      It would be uncharacteristically clear-cut if that were Fuller's only use of dimension. Synergetics may start with dimension as size, but other applications of the multifaceted term are sprinkled throughout the book. (Identification of space as three-dimensional is not one of them.) Keep in mind that this mathematical convention has a firm hold; it's difficult to think otherwise about space—and consequently not easy to view Fuller's material objectively.

Planes of Symmetry

Both the tetrahedron and the octahedron—two of the simplest structures—incorporate four nonparallel planes. The faces of the tetrahedron present four distinct directions, just as the faces of the cube provide three. The octahedron has four pairs of opposing parallel triangles, and it can be demonstrated that they are parallel to the tetrahedral faces (Fig. 6-2). Fuller refers to this geometric trait as "dimension," and through repeated observation places considerable emphasis on the inherent fourness of the "minimum system in Universe." "The octahedron's planar system is four-dimensionally referenced, being parallel to the four symmetrically interacting planes of the tetrahedron..." (527.31). The icosahedron, on the other hand, exhibiting various fivefold symmetries, embodies "five-dimensionality" (527.50) in Fuller's unorthodox appropriation of terms.

Octahedron's four dimensions inside tetrahedron's four planes

Fig. 6-2
Click on thumbnail for larger image.

Other Applications of Dimension

Another twist: Fuller also refers to the three structural parameters-vertices, edges, and faces-as different dimensions of structure. In a later section covering the concept of dimension, Fuller reintroduces "constant relative abundance" (as explained in Chapter 4) under the heading "527.10 Three Unique Dimensional Abundances"-namely, vertices, edges, and faces. This and other ambiguous-if not contradictory-usages of certain terms can often obscure the mathematical statement being made. In this case, Fuller's point about the consistent arithmetic relationships between vertices, edges, and faces in closed systems is lost amid confusion about the meaning of "dimensional abundances."

      Another unorthodox usage involves pairs of opposites. At one point in Synergetics, for example, a magnet, with its positive and negative poles, is called a two-dimensional system. Along the same lines, "Polar points are two dimensional: plus and minus, opposites" (527.21).

      Finally, "dimensional aggregations" in the opening quotation of this section refers to numbers of layers in certain clusters of closely packed spheres. We shall explore these patterns in Chapter 8. Fuller's different uses of "dimension may be confusing, but they are not, strictly speaking, incorrect-at least not in terms of the dictionary. Mathematical convention is another issue.

      Fuller does mention the historical precedent for conceiving of space as exclusively "three-dimensional," thereby explaining his license to reevaluate our concept of dimension; however, the reference is too late in the book to clear up early confusion:

      ... The Greeks came to employ 90-degreeness and unique perpendicularity to the system as a ....... dimensional requirement for the...unchallenged three-dimensional geometrical data coordination. (825.31)

So, while he does justify his usage with this reference to the word's flexibility, the clarification is obscured by the book's sequence. The reader seeking a quick reason to dismiss Synergetics might focus on Fuller's extravagant citation of other dimensions early in the book. His apparent familiarity with "the fourth dimension" provides just cause for suspicion; however, a simple change of article—from "the" to "a fourth dimension"—gives the term a very different effect.

      Like most subjects in synergetics, "dimension" cannot be neatly presented in one complete package; boundaries are never that clearly defined. In addition to the fact that different subjects overlap, there can always be new twists. The trick is to leave ourselves open to exploration, free to evaluate each new application without bias.

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