224.00 Principle of Angular Topology

224.01 Definition: When expressed in terms of cyclic unity the sum of the angles around all the vertexes of a structural system, plus 720 degrees, equals the number of vertexes of the system multiplied by 360 degrees.

224.02 All local structural systems in Universe are always accomplished by nature through the elimination of 720 degrees of angle. This is the way in which nature takes two complete 360-degree angular tucks in the illusory infinity of a plane to render systems locally and visibly finite. The difference between visually finite systems and illusory infinity is two cyclic unities.

224.03 Structural systems are local, closed, and finite. They include all geometric forms, symmetric or asymmetric, simple or complex. Structural systems can have only one inside and only one outside. Two or more structures may be concentric and triangularly interconnected to operate as one structure.

224.04 The difference between the sum of all the angles around all the vertexes of any system and the total number of vertexes times 360 (as angular unity) is 720, which equals two unities. The sum of the angles of a tetrahedron always equals 720. The tetrahedron may be identified as the 720differential between any definite local geometrical system (such as Greek “solid” geometry; and finite universe.

224.05 Line: A line has two vertexes with angles around each of its vertexial ends equal to 0. The sum of these angles is 0. The sum of the vertexes (two) times angular unity (360) is 720. The remainder of 0 from 720is 720, or two unities, or one tetrahedron. Q.E.D.

224.06 Triangle: The three angles of one “face” of a planar triangle always add up to 180as a phenomenon independent of the relative dimensional size of the triangles. One-half of definitive cyclic unity is 180. Every triangle has two facesits obverse and reverse. Unity is two. So we note that the angles of both faces of a triangle add up to 360. Externally, the sum of the angles around each of the triangle’s three vertexes is 120, of which 60is on the obverse side of each vertex; for a triangle, like a line, if it exists, is an isolatable system always having positive and negative aspects. So the sum of the vertexes around a triangle (three) times 360 equals 1080. The remainder of 360from 1080leaves 720, or one tetrahedron. Q.E.D.

224.07 Sphere: The Greeks defined the sphere as a surface outwardly equidistant in all directions from a point. As defined, the Greeks’ sphere’s surface was an absolute continuum, subdividing all the Universe outside it from all the Universe inside it; wherefore, the Universe outside could be dispensed with and the interior eternally conserved. We find local spherical systems of Universe are definite rather than infinite as presupposed by the calculus’s erroneous assumption of 360-degreeness of surface plane azimuth around every point on a sphere. All spheres consist of a high-frequency constellation of event points, all of which are approximately equidistant from one central event point. All the points in the surface of a sphere may be interconnected. Most economically interconnected, they will subdivide the surface of the sphere into an omnitriangulated spherical web matrix. As the frequency of triangular subdivisions of a spherical constellation of omnitriangulated points approaches subvisibility, the difference between the sums of the angles around all the vertex points and the numbers of vertexes, multiplied by 360 degrees, remains constantly 720 degrees, which is the sum of the angles of two times unity (of 360 degrees), which equals one tetrahedron. Q.E.D.

224.08 Tetrahedron: The sum of the angles of a tetrahedron, regular or irregular, is always 720, just as the sum of the angles of a planar triangle is always 180. Thus, we may state two propositions as follows:

224.081 The sum of the surface angles of any polyhedron equals the number of vertexes multiplied by 360minus one tetrahedron; and

224.082 The sum of the angles of any polyhedron (including a sphere) is always evenly divisible by one tetrahedron.

224.10 Descartes: Descartes is the first of record to have discovered that the sum of the angles of a polyhedron is always 720less than the number of vertexes times 360. Descartes did not equate the 720with the tetrahedron or with the one unit of energy quantum that it vectorially constitutes. He did not recognize the constant, whole difference between the visibly definite system and the invisibly finite Universe, which is always exactly one finite invisible tetrahedron outwardly and one finite invisible tetrahedron inwardly.

224.11 The Calculus: The calculus assumes that a sphere is infinitesimally congruent with a sphere to which it is tangent. The calculus and spherical trigonometry alike assume that the sum of the angles around any point on any sphere’s surface is always 360 degrees. Because spheres are not continuous surfaces but are polyhedra defined by the vectorially interconnecting chords of an astronomical number of event foci (points) approximately equidistant from one approximate point, these spherically appearing polyhedrawhose chords emerge from lesser radius midpoints to maximum radius convergences at each of the spherically appearing polyhedra’s vertexes, ergo, to convex external joiningmust follow the law of polyhedra by which the sum of all the angles around the vertexes of the polyhedra is always 720 degrees less than 360 degrees times the number of vertexes. The demonstration thus far made discloses that the sum of the angles around all the vertexes of a sphere will always be 720 degreesor one tetrahedronless than the sum of the vertexes times 360 degrees ergo, one basic assumption of the calculus and spherical trigonometry is invalid.

224.12 Cyclic Unity: We may also say that: where unity (1) equals 360, 180equals one-half unity (\sfrac{1}{2}), and that 720equals two times unity (2); therefore, we may identify a triangle as one-half unity and a tetrahedron as cyclic unity of two. As the sum of a polyhedron’s angles, 720is unique to the tetrahedron; 720is the angular name of the tetrahedron. 720 is two cyclic unities. The tetrahedron is the geometrical manifest of “unity is plural and, at minimum, is two.” The tetrahedron is twonessbecause it is congruently both a concave tetrahedron and a convex tetrahedron.

224.13 Where cyclic unity is taken as 360 degrees of central angle, the difference between infinity and finity is always exactly two, or 720 degrees, or two times 360 degrees, or two times unity. Cyclic unity embraces both wave and frequency since it represents angles as well as cycles. This is topologically manifest in that the number of vertexes in any structural system multiplied by 360 degrees, minus two times 360 degrees, equals the sum of the angles around all the vertexes of the system.

224.20 Equation of Angular Topology: S + 720= 360*Xⁿ*Where: S = the sum of all the angles around all the vertexes (crossings) X ⁿ= the total number of vertexes (crossings) No. of VertexesSum of
Angles around
each VertexSum of
angles
multiplied byNo. of
Vertexes.
De-FiniteNo. of Vertexes multiplied kr360 . Finite minus Finite De-FiniteLine

image

224.30 Polarity: Absolutely straight lines or an absolutely flat plane would theoretically continue outward to infinity. The difference between infinity and finity is governed by the taking out of angular sinuses, like pieces of pie, out of surface areas around a point in an absolute plane. This is the way lampshades and skirts are made. Joining the sinused fan-ends together makes a cone; if two cones are made and their open (ergo, infinitely trending) edges are brought together, a finite system results. It has two polarpoints and an equator. These are inherent and primary characteristics of all systems.

224.40 Multivalent Applications: Multiple-bonded bivalent and trivalent tetrahedral and octahedral systems follow the law of angular topology. Single-bonded monovalent tetrahedral and octahedral arrangements do not constitute a system; they are half systems, and in their case the equation would be: $S + 360\degree = 360\degree
X^$

224.50 Corollary: Principle of Finite Universe Conservation: By our systematic accounting of angularly definable convex-concave local systems, we discover that the sum of the angles around each of every local system’s interrelated vertexes is always two cyclic unities less than universal nondefined finite totality. We call this discovery the principle of finite Universe conservation. Therefore, mathematically speaking, all defined conceptioning always equals finite Universe minus two. The indefinable quality of finite Universe inscrutability is exactly accountable as two.

224.60 Tetrahedral Mensuration: The sum of the angles around all vertexes of any polyhedral system is evenly divisible by the sum of the angles of a tetrahedron. The volumes of all systems may be expressed in tetrahedra.

224.70 Equation of Tetrahedral Mensuration:Sum of face angles = n Tetrahedra

720

Where:

720= one tetrahedron