Appendix B

for

The Geomorphology and Geometry of the D&M Pyramid

Erol O. Torun

added: August, 1989

Tetrahedral Geometry

The possibility that the geometry of the D&M Pyramid might serve as a secondary representation of other geometric forms was considered but not yet discussed. For example, the bilaterally symmetrical D&M Pyramid appears, at first glance, to have five-fold symmetry. More specific mathematics which are inherent in five-fold symmetry are also suggested by the prevalence of proportions containing the square root of five, a factor in the Golden Mean proportion and the Fibonacci series.

A more recent observation along these lines is that the internal geometry, orientation, and siting latitude of the D&M Pyramid all combine

to represent the geometry of a circumscribed tetrahedron. A tetrahedron is the simplest of the "Platonic solids", a polyhedron having six edges, four vertices, and four sides where each side is an equilateral triangle.

A circumscribed tetrahedron is a tetrahedron that has been enclosed in a sphere that just touches each of the four vertices.

The first observation that led to this discovery is that the "front" of the D&M Pyramid (the part towards the Face) forms two equilateral triangles. In Synergetics, R. Buckminster Fuller demonstrates how two equilateral triangles may each be opened at one corner and combined to form a tetrahedron. Fuller defends the opening of the triangles by stating that two points (or lines) cannot occupy the same place. Thus one corner of a triangle must be the overlapping edges of a line, making the triangle topologically a very flat spiral. Two of these flat spirals (triangles) can be stretched out and assembled into a tetrahedron.

Another observation involves the apparent representation of the number e (the base of the natural logarithm) in the geometry of the D&M Pyramid, and the occurrence of this number in ratios with pi and the square root of five. These two ratios involving e are produced by all three methods used to analyze the D&M Pyramid's geometry: angle ratios, trig functions of individual angles, and radian measure of individual angles. The occurrence of e in ratio with two other mathematically significant numbers increases the likelihood that e is one of the numbers intended for representation.

The presence of 60 deg angles produces an ambiguity: the Sine of 60 deg is defined as (sqrt 3)/2, and this is very close, but not equal to, the ratio of e/pi:

(sqrt 3)/2 = 0.866025

e/pi = 0.865256

It is this ambiguity that is resolved by the geometry of a circumscribed tetrahedron.

The surface area of a sphere, divided by the surface area of the tetrahedron circumscribed by it, yields a very close approximation of e which shall be termed e':

e = 2.718282

e' = 2.720699

When the ratio e/pi is evaluated using this "tetrahedral approximation of e", the result is precisely equal to (sqrt 3)/2:

e/pi = 0.865256

e'/pi = 0.866025 = (sqrt 3)/2

The two values listed above also appear to be generated by the orientation and siting latitude of the D&M Pyramid. Recent geodetic information from Dr. Mert Davies of RAND Corp. indicates that 40.868 N latitude (Arc Tan e/pi) passes through the D&M Pyramid, with the estimated error range staying within the base. If the 40.868 deg N latitude (Arc Tan e/pi) passes precisely through the reconstructed apex, then the 40.893 deg N latitude (Arc Tan (sqrt 3)/2) cuts through the only two points on the D&M Pyramid that, when connected, form a line of latitude. These are the left front corner (towards the City Square) and the upslope terminus of the flat protuberance on the front edge (towards the Face).

This flat protuberance, when measured from either front corner, subtends 19.5 deg. This angle is the latitude angle generated by a circumscribed tetrahedron; when one vertex of the tetrahedron is coincident with the sphere's axis of rotation, the other three vertices will touch the sphere at 19.5 deg latitude.

Another potential, but less certain, expression of tetrahedral geometry involves the lines of latitude for the entire Cydonia complex. There are a few potential ways to plot these lines of latitude while staying within the error figures provided by Davies. One approach would place a line of latitude through the center of the City Square and tangential to the periphery of the southern side of the Tholus. If this line of latitude is used to build a grid, the western ground level edge of the D&M Pyramid is oriented exactly north-south, and the eastern edge (facing the Tholus) is offset from the east-west direction by 19.5 deg.

Reference:

Fuller, R. Buckminster; 'Synergetics', pp.4-6, Macmillan,

New York and London (1975).

END