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Euler's law states that the number of vertices plus the number of faces in every system (remember Fuller's definition) will always equal the number of edges plus two. It may not sound like much at first, until you reflect on the variety of structuresfrom the tetrahedron to the aforementioned crocodilethat all obey this simple statement. Every system shares this fundamental relationship. The number of vertices can be precisely determined by knowing the number of faces and edges, and so on.
Denote the numbers of vertices, faces, and edges by V, F, and E respectively. Then we have V + F = E + 2. What about that constant 2? The other numbers might be extremely large, or as small as four, yet by Euler's equation the difference between the number of edges and the sum of the number of vertices and faces will always be exactly two. It seems unlikely at first.
To gain confidence in this principle, let's try it out on the regular polyhedra. Remember the four vertices and four faces of the tetrahedron; four plus four is eight, exactly two more than its six edges. Not bad so far. Similarly, we can count to check the other four regular polyhedra. The results are displayed in Table I.
The persistent 2 has led to some controversy. Fuller long ago assigned his own meaning to the recurring number: 2 occurs in the equation to represent the "poles of spinnability." That requires some clarification.
All systems, Fuller explains, can be spun about a central axis. An axis has two poles (e.g., north and south), and thus at any given time, two vertices must be poles. Subtract the two poles from the total number of vertices to get a number exactly equal to the combined number of edges and faces. Simply stated, in Fuller's view, the permanent 2 represents two vertices which, as "poles of spinnability," should be subtracted from the total number of vertices to equalize the equation. It's a puzzling explanation.
There is another way to view that constant element, as you may have guessed. What are the characteristics of the variables included in Euler's Law? Zero-dimensional points, one-dimensional lines, and two-dimensional areaseach a level higher than the last. The law compares all aspects of structurealmost. Something's missing. Three-dimensional space is the next and only absent parameter; its geometrical units (corresponding to vertices, edges, and faces) are cells. Why are cells left out of this fundamental relationship? The other view, as expounded by Loeb, says they're not. (3)
Recall Bucky's definition of a system: a subdivision of space creating an inside and an outside, both equally important. Two cells! Could the constant 2 in the equation be incorporating the otherwise only missing dimension? Indeed it is. Euler's law is actually a special case of Schlaefli's formula for any number of cells. In other words, if the number of cells, C, is substituted for 2, the equation holds true for multicellular structures, that is, arrangements with more than two cells: V + F = E + C, even when C is greater than 2. (4)
Evaluation of significance is a tricky business, but we cannot avoid indulging in it altogether, as Fuller's Synergetics overflows with such speculation. It is ultimately puzzling that Fuller, with his emphatic observation that every polyhedron is a system dividing Universe into two parts (inside and outside), would not connect Euler's constant 2 with the implied two cells. His insistence that both parts of a system must be considered equally important provides a truly new orientation in geometry.
At some point in any discussion about Euler and the "polar two," Fuller would speak of a structural system's inherent "constant relative abundance." The meaning of the term eluded many. Fuller observes in Synergetics that the number of faces in triangulated systems is always two times the number of vertices minus two (the subtraction, he says, again taking account of the two poles). (5) He further states that the number of edges is three times the number of vertices less two.
Turning these two statements into simple equations, we have
F + 4 = 2V, E + 6 = 3V.
- (F + 4 = 2V)
E - F + 2 = V,
So his observations directly substantiate Euler's law. Constant relative abundance refers to the everpresent two faces and three edges for each vertex in triangulated polyhedra, (excepting of course the two "poles," which are not included, according to Fuller's rationale).
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