Tetrahedral Geometry in the Seattle Telephone Directory
(An addendum to the Telephone Number Experiment.)

Here are the eight numbers that were used in the Telephone Number Experiment. They are given as angles to bring out the analogy with Hoagland's Geometry of Cydonia.  That topic is discussed at length on another page: The D&M Pyramid on Mars.

A=23.61o
B=43.41o
C=64.41o
D=53.89o
E=97.47o
F=87.61o
G=94.11o
H=44.14o

1.   There are numerous relationships involving these eight angles and the geometry of a regular tetrahedron inscribed in a sphere. One interesting constant associated with that geometry is the ratio:

tau = (Surface area of T)/(Surface area of S)

where T denotes the regular tetrahedron and S denotes the sphere in which T is inscribed. This constant has the exact value tau=(2/3)\/3/pi .  It is related to the constant e' discussed by Torun and Hoagland in their article Message of Cydonia by the equation e' = 1/tauIts value is tau=.367552597... .

Here is another connection between the number tau and tetrahedral-spherical geometry. If two regular tetrahedra of the same size are inscribed in a sphere so that each vertex of one is diametrically opposite to a vertex of the other, then the eight vertices (four vertices from each tetrahedron) form the vertices of a cube C inscribed in the sphere. Then the number tau turns out to be exactly equal to the ratio of the volumes of the cube and the sphere:

tau = (Volume of C)/(Volume of S)

The first six multiples of the number tau can be approximated by ratios of the above eight angles. (Note that six is also the number of edges of a regular tetrahedron.)

A/C ~ tau                       ( .27%)
C/F ~ 2tau                     (.012%)
E/F ~ 3tau                      ( .9% )
G/C ~ 4tau                    ( .6% )
B/A ~ 5tau                    (.05%)
E/H ~ 6tau                     (.13%)

2.  An interesting angle associated with the regular tetrahedron is alpha=arcsin(1/3)=19.4712206...o . (This is Richard Hoagland's favorite angle, which he usually rounds off to 19.5o.)  Any two faces of a regular tetrahedron meet at an edge and the angle that the two faces form is the complementary angle beta=90o-alpha (which has the value beta=70.52877939...o).  If one inscribes a regular tetrahedron in a sphere, then any two vertices of the tetrahedron will subtend an angle of  90o+alpha = 109.4712206...o.  That is, this is the angle formed by the two radii joining the center of the sphere to two vertices of the inscribed regular tetrahedron.  Another way to interpret this is that if one inscribes the tetrahedron in the sphere so that one vertex is at the "North Pole,"  then the other three vertices will be alpha degrees South of the "Equator." (Or approximately 19.5o South of the Equator.)

The above eight angles A, B, ... , H have various relationships with the angles alpha and beta. These relationships involve differences between these angles instead of ratios.

G - A = 70.5o.   This is approximately equal to the angle beta.  If one forms a right triangle with this as one of its angles, then the other angle is 19.5o , which is quite close to the angle alpha.

D - H = 9.75o . This angle is obtained by bisecting a 19.5o angle.

G - F = 6.5o .   This angle is obtained by trisecting a 19.5o angle.

More to come.