902.00 Properties of Basic Triangle

902.01 Subdivision of Equilateral Triangle: Both the spherical and planar equilateral triangles may be subdivided into six equal and congruent parts by describing perpendiculars from each vertex of the opposite face. This is demonstrated in Fig. 902.01, where one of the six equal triangles is labeled to correspond with the Basic Triangle in the planar condition.

902.10 Positive and Negative Alternation: The six equal subdivision triangles of the planar equilateral triangle are hingeable on all of their adjacent lines and foldable into congruent overlays. Although they are all the same, their dispositions alternate in a positive and negative manner, either clockwise or counterclockwise.

902.20 Spherical Right Triangles: The edges of all spherical triangles are arcs of great circles of a sphere, and those arc edges are measured in terms of their central angles (i.e., from the center of the sphere). But plane surface triangles have no inherent central angles, and their edges are measured in relative lengths of one of themselves or in special-case linear increments. Spherical triangles have three surface (corner) angles and three central (edge) angles. The basic data for the central angles provided below are accurate to \sfrac{1}{1,000} of a second of arc. On Earth

1 nautical mile = 1 minute of arc
1 nautical mile = approximately 6,000 feet
1 second of arc = approximately 100 feet
\sfrac{1}{1,000}second of arc = approximately \sfrac{1}{10} foot
\sfrac{1}{1,000} second of arc = approximately 1 inch

These calculations are therefore accurate to one inch of Earth’s arc.

902.21 The arc edges of the Basic Disequilibrium 120 LCD Triangle as measured by their central angles add up to 90as do also three internal surface angles of the triangle’s ACB corner:

$$\begin
BCE = 20\degree 54\prime18.57\prime\prime = ECF\ ECD = 31\degree 2 2\prime 38.
53\prime\prime = DCE\ DCA = 31\degree 43\prime02.9\prime\prime = ACD\ \hline
\ 90\degree 00\prime 00\prime
\end$$

902.22 The spherical surface angle BCE is exactly equal to two of the arc edges of the Basic Disequilibrium 120 LCD Triangle measured by their central angle. BCE = arc AC = arc CF = 20\degree 54\prime 18.57\prime\prime.

902.30 Surface Angles and Central Angles: The Basic Triangle XCB can be folded on the lines CD and CE and EF. We may then bring AC to coincide with CF and fold BEF down to close the tetrahedron, with B congruent with D because the arc DE = arc EB and arc BF = arc AD. Then the tetrahedron’s corner C will fit exactly down into the central angles AOC, COB, and AOB. (See Illus. 901.03 and 902.30.)

image

902.31 As you go from one sphere-foldable great-circle set to another in the hierarchy of spinnable symmetries (the 3-, 4-, 6-, 12-sets of the vector equilibrium’s 25-great-circle group and the 6-, 10-, 15-sets of the icosahedron’s 31-great-circle group), the central angles of one often become the surface angles of the next-higher-numbered, more complex, great-circle set while simultaneously some (but not all) of the surface angles become the respective next sphere’s central angles. A triangle on the surface of the icosahedron folds itself up, becomes a tetrahedron, and plunges deeply down into the congruent central angles’ void of the icosahedron (see Sec. 905.47).

902.32 There is only one noncongruence: the last would-be hinge, EF is an external arc and cannot fold as a straight line; and the spherical surface angle EBF is 36 degrees whereas a planar 30 degrees is called for if the surface is cast off or the arc subsides chordally to fit the 90-60-30 right plane triangle.

902.33 The 6 degrees of spherical excess is a beautiful whole, rational number excess. The 90-degree and 60-degree corners seem to force all the excess into one corner, which is not the way spherical triangles subside. All the angles lose excess in proportion to their interfunctional values. This particular condition means that the 90 degrees would shrink and the 60 degrees would shrink. I converted all the three corners into seconds and began a proportional decrease study, and it was there that I began to encounter a ratio that seemed rational and had the number 31 in one corner. This seemed valid as all the conditions were adding up to 180 degrees or 90 degrees as rational wholes even in both spherical and planar conditions despite certain complementary transformations. This led to the intuitive identification of the Basic Disequilibrium 120 LCD Triangle’s foldability (and its fall-in-ability into its own tetra-void) with the A Quanta Module, as discussed in Sec. 910.00, which follows.