The Earth
The geometrical model of the Earth described here is generally a grid of great circles around a cube. These great circles are symmetrically arranged and defined by various axis of rotation about this regular polyhedron.
Rotation of The Cube
Most of our interpretation of ancient architecture is cuboidal. We call the cube a fundamental building block. The cube can also be referred to as a hexahedron, a regular polyhedron with six sides. How does the cube reflect the a geocentric feeling for the Earth?
Let’s learn how to make a spherical model of the hexahedron. All the faces of a hexahedron are squares. Great circle planes are generated by rotating the hexahedron along three different kinds of axes. This is how a spherical model of the hexahedron is made.
A) The rotation of the hexahedron on axes through midpoints of opposite edges define six great circles, each of which passes thru four vertices. The underlying planar chord structure of each of these great circles forms a rectangle with two edges equal to an edge of the square face of the hexahedron and two edges equal to the diagonal of a square face of the hexahedron.
B) The rotation of the hexahedron on axes through the centers of opposite faces define three equatorial great circles. Each of these circles does not pass through any of the vertices. The underlying planar chord structure forms a square. The edges of this square are larger than the edges of the hexahedron.
C) The rotation of the hexahedron on axes thru opposite vertices define four great circles. The underlying chord structure forms a hexagon.
The Characteristic Triangle
These great circles formed by rotation of the hexahedron upon axes through midpoints of opposite edges and centers of opposite faces as described in A and B above lead to a repeating pattern of one triangle on the surface of the sphere. This triangle is called the characteristic triangle and has a right handed and left handed form. The repetition of this triangle fills the entire surface of the sphere with identical pairs of mirrored triangles. In its planar form this right triangle divides the square face of the hexahedron into eight triangles.
Construction of the Characteristic Triangle
If we look closely at the construction of one of these triangles on the hexahedron itself we can see that it is composed of three linear measures. If the hexahedron is assigned an edge length e=2 then half of this is designated as one unit of length. A second edge of the triangle is defined by the radius of the small circle circumscribing the face of the hexahedron and is named r. The apothem, a line drawn from the center of a face to a midpoint of an edge, describes the third side and is named a. The apothem of the hexahedron is always a perpendicular bisector of an edge.
Let’s look at the construction of the geometrical model. A circular arc is drawn as the radius of the circumscribing sphere. A right angle is inscribed on either side of a vertex of the cube cubecorresponding to chord lengths of ½ e = 1 and r = square root of two in the case of a hexahedron. Both right triangles are inscribed in a semicircle of radius (square root of 3) / 2. The projections are drawn on the arc of radius = square root of 3.
The apothem is drawn as part of a right angle inscribed in a semicircle of radius square root of two with the length of the side originating at the endpoint of the side that corresponded to ½ e. In the case of the poygonal hexahedron a = ½ e = 1 but when it is projected onto the circumscribing sphere it is no longer equal to ½e. This is because neither endpoint of the planar apothem, a, of the hexahedron lies on the circumscribing sphere. Each edge of the characteristic spherical triangle generated by projecting the edges of the triangulated hexahedron onto a circumscribing sphere is a different angular measure; a = 45° , ½ e = 35.264° and r = 54.736° .