505.80 Background Nothingness

505.81 When we draw a spherical triangle on the Earth’s surface (Fig. 812.03), it demonstrates an aspect of geometry apparently not recognized by Euler. The spherical triangle also demonstrates self and otherness. The spherical triangle is the first awareness: there is an inherent twoness in the triangle’s insideness and outsideness; and the axis of the two poles constitutes the two points of self and otherness. The background nothingness of these two points represents an area not contained by a line. Euler apparently did not realize that there could be an area not contained by a line.

505.82 From Euler we know that the number of locally identified minimum entities called points plus the number of separate areas equals the number of lines plus the number two. In minimum awareness we have two identified entities which, being local points, must have directional fixedness against a background of nothingness. Ergo, in minimum awareness two points plus one area of nothingness have one inherent line of most economical interrelationship between the two points. These two points plus one area equal the number of linesin this case “one” plus Euler’s abstractly accommodative two. (The line of interrelationship is another aspect of the prime vector. See Sec. 540.10.)

505.83 The Euler “plus one” abstractly accommodates two in the minimum awareness model:

points + area = lines + 2

2 + 1 = 1 + 2

3 = 3

Three of two kinds = three of two other kinds = six of four kinds = the six vector-edge relationships existing between the four different event-point fixes. Points are subdifferentiable systemsi.e., microsystems of events too close to be resolved from one another. Areas are supradifferentiable systemsi.e., macrosystems of event points too far apart to resolve. The nothingness area is unbounded by any visible closed line. Nothingness is the part of the system unencompassed by the observer. (See Sec. 1052.350.)