1007.20 Invalidity of Plane Geometry

1007.21 We are dealing with the Universe and the difference between conceptual thought (see Sec. 501.101) and nonunitarily conceptual Universe (see Scenario Universe, Sec. 320). We cannot make a model of the latter, but we can show it as a scenario of meaningfully overlapping conceptual frames.

1007.21 About 150 years ago Leonhard Euler opened up the great new field of mathematics that is topology. He discovered that all visual experiences could be treated as conceptual. (But he did not explain it in these words.) In topology, Euler says in effect, all visual experiences can be resolved into three unique and irreducible aspects:

vertexes, faces, and edges (Secs. 223.04 and 1006.20) or, as unique dimensional abundances:

points, areas, and lines (Sec. 527.11) or, as structural identifications:

joints, windows, and struts (Sec. 986.053) or, as we say in synergetics topology:

crossings, openings, and trajectories (Sec. 524.30) or the more generalized:

events, nonevents, and traceries or more refined as:

fixes, discontinuities, and continuities or in most refined synergetics:

events, novents, and even interrelatabilities (Sec. 269.05).

1007.23 In topology, then, we have a unique aspect that we call a line, not a straight line but an event tracery. When two traceries cross one another, we get a fix, which is not to be confused in any way with a noncrossing. Fixes give geographical locations in respect to the system upon which the topological aspects appear. When we have a tracery or a plurality of traceries crossing back upon one another to close a circuit, we surroundingly frame a limited view of the omnidirectional novents. Traceries coming back upon themselves produce windowed views or areas of novents. The areas, the traces, and the fixes of crossings are never to be confused with one another: all visual experiences are resolved into these three conceptual aspects.

1007.24 Look at any picture, point your finger at any part of the picture, and ask yourself: Which aspect is that, and that, and that? That’s an area; or it’s a line; or it’s a crossing (a fix, a point). Crossings are loci. You may say, “That is too big to be a point”; if so, you make it into an area by truncating the comer that the point had represented. You will now have two more vertexes but one more area and three more lines than before. Euler’s equation will remain unviolated.

1007.25 A circle is a loop in the same line with no crossing and no additional vertexes, areas, or lines.

1007.26 Operationally speaking, a plane exists only as a facet of a polyhedral system. Because I am experiential I must say that a line is a consequence of energy: an event, a tracery upon what system? A polyhedron is an event system separated out of Universe. Systems have an inside and an outside. A picture in a frame has also the sides and the back of the frame, which is in the form of an asymmetrical polyhedron.

1007.27 In polyhedra the number of Vs (crossings) plus the number of F’s, areas (novents-faces) is always equal to the number of L’s lines (continuities) plus the number 2. If you put a hole through the system as one cores an apple making a doughnut-shaped polyhedronyou find that V +F =L. Euler apparently did not realize that in putting the hole through it, he had removed the axis and its two poles. Having removed two axial terminal (or polar) points from the inventory of “fixes” (loci-vertexes) of the system, the V +F L + 2 equation now reads V +F =L, because two Vs have been deducted from the inventory on the left side of the equation.

1007.28 Another very powerful mathematician was Brouwer. His theorem demonstrates that if a number of points on a plane are stirred around, it will be found after all the stirring that one of the points did not move relative to all the others. One point is always the center of the total movement of all the points. But the mathematicians oversimplified the planar concept. In synergetics the plane has to be the surface of a system that not only has insideness and outsideness but also has an obverse and re-exterior. Therefore, in view of Brouwer, there must also always be another point on the opposite side of the system stirring that also does not move. Every fluidly bestirred system has two opposed polar points that do not move. These two polar points identify the system’s neutral axis. (See Sec. 703.12.)

1007.29 Every system has a neutral axis with two polar points (vertexes- fixes). In synergetics topology these two polar points of every system become constants of topological inventorying. Every system has two polar vertexes that function as the spin axis of the system. In synergetics the two polar vertexes terminating the axis identify conceptually the abstractsupposedly nonconceptual function of nuclear physics’ “spin” in quantum theory. The neutral axis of the equatorially rotating jitterbug VE proves Brouwer’s theorem polyhedral ly.

1007.30 When you look at a tetrahedron from above, one of its vertexes looks like this:

image

Fig. 1007.30 View of Tetrahedron from Above: There are four triangles: three surround the top vertex; the fourth is implicit in the base.

You see only three triangles, but there is a fourth underneath that is implicit as the base of the tetrahedron, with the Central vertex D being the apex of the tetrahedron. The crossing point (vertex-fix) in the middle only superficially appears to be in the same plane as ABC. The outer edges of the three triangles you see,XCZ), CDB, ADB, are congruent with the hidden base triangle, ABC. Euler assumed the three triangles ACD, CDB, ADB to be absolutely congruent with triangle ABC. Looking at it from the bird’s-eye view, unoperationally, Euler misassumed that there could be a nonexperienceable, no-thickness plane, though no such phenomenon can be experientially demonstrated. Putting three points on a piece of paper, interconnecting them, and saying that this “proves” that a no-thickness, nonexperiential planar triangle exists is operationally false. The paper has thickness; the points have thickness; the lines are atoms of lead strewn in linear piles upon the paper.

1007.31 You cannot have a something-nothingness, or a plane with no thickness. Any experimental event must have an insideness and an outsideness. Euler did not count on the fourth triangle: he thought he was dealing with a plane, and this is why he said that on a plane we have V +F =L +1. When Euler deals with polyhedra, he says “plus 2.” In dealing with the false plane he says “plus 1.” He left out “1” from the right-hand side of the polyhedral equation because fie could only see three faces. Three points define a minimum polyhedral facet. The point where the triangles meet in the center is a polyhedral vertex; no matter how minimal the altitude of its apex may be, it can never be in the base plane. Planes as nondemonstrably defined by academic mathematicians have no insideness in which to get: ABCD is inherently a tetrahedron. Operationally the fourth point, D, is identified or fixed subsequent to the fixing of/4, B, and C. The “laterness” of D involves a time lag within which the constant motion of all Universe will have so disturbed the atoms of paper on which A, B, and C had been fixed that no exquisite degree of measuring technique could demonstrate that A B, C, and D are all in an exact, so-called flat-plane alignment demonstrating ABCD to be a zero-altitude, no-thickness-edged tetrahedron.