(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=23.61^{o}

B=43.41^{o}

C=64.41^{o}

D=53.89^{o}

E=97.47^{o}

F=87.61^{o}

G=94.11^{o}

H=44.14^{o}

**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/*tau** . *Its 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
~
2*tau ** *
(.012%)

E/F
~
3*tau *
( .9% )

G/C
~
4*tau *
( .6% )

B/A
~
5*tau *
(.05%)

E/H
~
6*tau ** *
(.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 19.5^{o}.) Any two faces of a regular tetrahedron meet
at an edge and the angle that the two faces form is the complementary angle
*beta*=90^{o}-*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 90^{o}+*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 19.5^{o} 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 = 70.5^{o}.
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
19.5^{o} , which is quite close to the angle *alpha.*

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

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

*More to come.*