Grid of the Gods

by Farrell, Joseph P.

  The basic intuitive or experiential technique behind all the more formal principles and techniques of higher dimensional geometry may be glimpsed by the sentence that begins the seventh chapter of Regular Polytopes, “Ordinary Polytopes in Higher Space.” One may, at this point, be asking, “What is a ‘polytope’ anyway?” Coxeter answers the question, and in doing so, reveals the intuitive or imaginative technique in play: “POLYTOPE is the general term of the sequence point, segment, polygon, polyhedron…”15Remember the “dimensional analysis” of the Pythagorean Tectratys of chapter 12? Coxeter is here saying the same thing: as one adds dimensions, one is able to describe more and more things, so that in zero dimensions, one can only describe a point, in one dimension, only a segment of a line, in two dimensions, polygons such as triangles, squares, octagons, and so on, and in three dimensions, polyhedra, such as tetrahedra, cubes, octrahedra, and so on. Coxeter puts this imaginative point more precisely as follows:

  In space of no dimensions the only figure is a point, Πo. In space of one dimension we can have any number of point; two points bound a line-segment, Π1, which is the one-dimensional analogue of the polygon Π2 and polyhedron Π3. By joining Πo to another point, we construct Π1. By joining Π1 to a third point (outside its line) we construct a triangle, the simplest kind of Π2. By joining the triangle to a fourth point (outside its plane) we construct a tetrahedron, the simplest Π3. By joining the tetrahedron to a fifth point (outside its 3-space!) we contruct a pentatope, the simplest Π4… The general case is now evident: any n +1 points which do not lie in an (n-1)- space are the vertices of an n-dimensional simplex…16

  The term “polytopes” (Π) thus denotes what happens to polyhedra when they are rotated into more than three dimensions. Or to put it differently, the term “polytope” denotes the common mathematical elements of the same figure in various dimensions, such as the progression in the above quotation from Coxeter, of “triangle, teatrahedron, pentatope” and so on. Notice also, that in each of the two previous examples — polygons in two dimensions and polyhedra in three — that the shapes of the polyhedra result from performing what geometers call an “orthorotation,” or a rotation from two dimensions into a third dimension that is perpendicular or at 90 degree angles from the other two. The same thing happens in the rotation from three to four, and from four to five dimensions: each dimensional orthorotation simply adds another spatial dimension to the previous ones, in a dimension at right angles to all the previous dimensions.

  Obviously, this cannot exist in a three dimensional space any more than a three dimensional polyhedron can exist in a two dimensional space, but they can be pictured or represented in a two dimensional space, and more importantly, they can be exactly mathematically described by certain specific techniques that we will get to in a moment. To put Coexter’s point about polytopes of certain types being analogues — there’s that crucial word again! — somewhat differently, all polytopes of a specific type — say, triangles, tetrahedra, and pentatopes — share certain properties in common regardless of how many dimensions in which they occur; they are, in “physics terms,” all coupled harmonic oscillators to one another across dimensions, and the “coupling” occurs in the numbers themselves, numbers that in turn result from the regular geometry of their shape. To boil it all down, all polytopes of a particular species have a “shape” in n-dimensions, and that shape isprojected into the dimensional space in which it exists — say, two, three, or four dimensions — by certain regular mathematical laws.

  Now let us observe yet another implication of Coxeter’s always deceptively simple remarks, namely, that the simplest way to describe the orthorotation of any polytope or “shape” from one type of dimensional space to another is tetrahed rally. Coxeter puts this point with his customarily deceptively simple statements: “Two-dimensional polytopes are merely polygons… Three-dimensional polytopes are polyhedra…”17

  A “triangular” shape in two dimensions is a natural oscillator of the same shape in three, four, and more dimensions, and this is the real encoded secret in the Sacred Tectratys and Pentactys of the Pythagoreans, for the simple symbol is also a symbol of this mathematical technique, yet another legacy from High Antiquity, for it is essential to recall that the Pyragoreans regarded it as a key cosmological secret:

  The Pythagoreans, in other words, were preserving a secret of hyper-dimensional geometry, whether they knew it or not, and given all the indications that they knew the image concealed a great a various multi-leveled meaning, they may, indeed, have had an inkling of it.

  c. The Essential Mathematical Techniques

  With these things in mind, we are ready to address the next question: what are the exact mathematical techniques used to describepolytopes in more than three, or >3, dimensions? Coxeter beginsRegular Polytopes by describing one such technique in detail:

  To be precise, we define a p-gon as a circuit of p line-segments A1A2, A2A3,…, ApA1, joining consecutive pairs of p points A1, A2, …, Ap. The segments and points are called sides and vertices. Until we come to Chapter VI we shall insist that the sides do not cross one another. If the vertices are all coplanar we speak of a plane polygon, otherwise a skew polygon.

  A plane polygon decomposes its plane into two regions…

  (Remember our topological metaphor, and that initial differentiation of the dimensionless Nothing into two regions, joined upon a common surface? Coexter is now describing the same process in two dimensions, via a process of differentiation of a two dimensional Nothing, a “plane”, by means of a regular polygon whose surface joins its finite interior with its infinite exterior! In other words, you have been doing higher-dimensional geometry all along, in the topological metaphor, the only difference between topology and geometry being, that topology is not dealing with geometrical objects, but the spaces themselves! To return to Coxeter):

  …

  A plane polygon decomposes its plane into two regions, one of which, called the interior, is finite. We shall often find it convenient to regard the p-gon as consisting of its interior as well as its sides and vertices. We can then re-define it as a simply-connected region bounded by p distinct segments.18

  Before we distill all this lingua mathematica arcana, there is one more statement, again from the beginning of the book, worth citing:

  A polyhedron may be defined as a finite, connected set of plane polygons, such that every side of each polygon belongs also to just one other polygon, with the proviso that the polygons surrounding each vertex form a single circuit(to exclude anomalies such as two pyramids with a common apex). The polygons are called faces, and their edges sides. Until Chapter VI we insist that the faces do not cross one another. Thusthe polyhedron forms a single closed surface, and decomposes (three dimensional)19 space into two regions, one of which, called the interior, is finite.20

  Again, Coxeter has described a process of three-dimensional differentiation of a three-dimensional space by means of a polyhedron.

  So what is the first basic mathematical principle in evidence in the transition of any polytope from one system of dimensional spaces to another?

  It’s so simple that you, the reader, know it already, and have known it since elementary school: one counts the

  1) points, or vertices of an object;

  2) the edges, or lines of an object; and

  3) the faces of an object.

  Thus, a triangle has three vertices, three edges, and one “face”; a teatrahedron four vertices, six edges, and four faces, and so on. A square has four vertices, four edges, and one “face”; a cube has eight vertices, twelve edges, and six faces, and so on.

  To this technique there is added yet another, and this one is a bit more complicated, but it is also the crucial technique. We begin, once again, with Coxeter’s own summation of this technique:

  A regular polygon is easily seen to have a centre, from which all the vertices are at the same distance oR, while all the sides are at the same distance 1R. This means that there a
re two concentric circles, the circum-circle and in-circle, which pass through the vertices and touch the sides, respectively.

  And, notes Coxeter in the very next sentence, anticipating a physics application:

  It is sometimes helpful to think of the side of a p-gon as representing p vectors whose sum is zero.21

  In other words, for any regular polygon, such as a square, it is possible to draw a circle whose center shares the center of the square, and whose circumference touches upon, or is tangent to, the four vertices of the square, which is the circumscribing circle, and it is also possible to draw a circle whose circumference touches upon, or is tangent to, the edges of the square, which is the circuminscribed circle.

  But note that Coxeter is describing a process that, like the polygon itself, can be orthorotated into three or more dimensions, in which case, the circuminscribing and circumscribed circles, become cicuminscribing and circuminscribed spheres, and hyper-spheres. But the numbers will be preserved in all dimensional spaces.

  Now we consider the next most difficult component of these circumscribing and circumscribed n-circles.22 If we imagine a circle in two dimensions, and a square within it, obviously the square will touch on four points of a circumscribing circle. If we now instead circuminscribe an octagon, there will be eight points on touching on the circle. Dividing the octagon again will produce sixteen touching points, and with each such division, the regular polygon assumes a shape closer and closer to the circuminscribing circle. Similarly a process in three dimensions with regular polyhedra will more and more approximate the shape of the circuminscribing sphere. Notably, the circumference of the circle is, as everyone knows, C=2πr, where C is the circumference and r is the radius of any given circle. Thus, as the regular polygons circumscribed in a circle more closely approximate the circle itself, the closer they get to that crucial relationship of 2πr. In other words, the relationship of 2π and its multiples becomes a crucial component of rotations into more than three dimensions.23

  This point about “squaring the circle” is also quite an important technique for this type of higher dimensional mathematical technique, for it allows geometers to determine the numerical relationships of objects in more than three dimensions to their circuminscribing and circuminscribed hyper-spheres. It is, along with the counting of vertices, edges, and faces (in order to determine what type of object one is dealing with), the essential technique.

  It is this fact that, at least with respect to the Great Pyramid, also means that the structure was deliberately conceived as a higher- dimensional analogue, for as most investigators are aware, the Great Pyramid is built as an example of “squaring the circle” and “cubing the sphere,”24 it is built, in other words, according to the very technique of higher-dimensional geometry.25

  d. Tetrahedral and Octahedral Groups

  We noted, previously, that there was a peculiar connection between Sumerian notation and the type of notation used by geometers in higher-dimensional mathematics. Before exploring that connection, it is worth mentioning that there is another deep connection between the Sumerian sexagesimal numerical system, based upon multiples of the number 6, and that of higher dimensional geometry. Coxeter notes that in the rotation groups of polyhedra, that there are three groups in particular:

  …(we) have the tetrahedral group of order 12, the octahedral group of order 24 (which is also the rotation group of the cube) and the icosahedral group group of order 60 (which is also the rotation group of the dodecahedron).26

  While consideration of rotation groups would far exceed the technical limitations of presenting higher-dimensional geometries to a general audience, it is worth noting that all these numbers are “Sumerian” in that they are all multiples of 6! And this brings us at last to the other peculiar Sumerian connection.

  e. Schl㥬i Numbers, the Platonic Solids, and Their Extensions

  Ludwig Schl㥬i (1814-1895) was the Swiss mathematician who first investigated regular polytopes in more than three dimensions, deriving a simple method of representing the counting of the numbers of vertices and faces.27 Schl㥬i’s notation convention, for regular polyhedra in three dimensions, is a symbol comprising two numbers, p and q, which looks like this:

  {p,q},

  where p is the number of sides of a face of a regular polygon and q is the number of faces around each vertex. A cube would thus look like this:

  {4,3}

  with the four denoting the equal sides of a square’s face, with three such faces around each vertex.28 The notation convention is strongly reminiscent of Schwaller de Lubicz’s understanding that numbers represent geometrical functions, and of the Sumerian notation convention where, similarly, the numbers denote functions of cubing or squaring, or of some other function of multiplication.29 As more dimensions are added, the number expands: {p, q, r…}.

  C. Counting Faces and Vertices In Mexico and Meso-America

  We have already observed that the Great Pyramid in particular is an actual analogue of the technique of squaring the circle, a crucial step in the kind of analysis and technique geometers utilize to describe higher-dimensional objects. But expressed in terms of a Schl㥬i number, there is nothing so unique about them; they have five vertices, four triangular faces, and one square face. One would have to “adjust” the notation to reflect this fact, but it could be easily done.

  It is when one turns to the pyramidal structures in Mexico and Meso-America that one is confronted with something very interesting, as the following charts and diagrams of the various pyramids of Teotihuacan and Tikal demonstrate.

  Harrelson’s Overview of the Pyramid of the Sun at Teotihuacan30

  Overview of Small Pyramid At Tikal31

  Small Pyramid At Tikal: Front View

  Perspective View of Small Tikal Pyramid

  Tikal: Temples I & II: Front, Side, and Overhead Views

  Tikal Temple I: Overhead

  Tikal Temple II: Overhead

  Tikal Temple III: Overhead

  Tikal Temple IV Overhead

  Tikal Temple V Overhead

  What is immediately apparent in all these examples is that these are not true regular pyramids, they are elongated, in many cases their vertical orientation is not symmetrical, being skewed off center, and most importantly, as Carl Munck observed, they also have numerous corners, edges, and faces.

  Why is this so important?

  For one thing, just as the two large pyramids at Giza, the skewed off-center vertical alignment suggests once again that they were deliberately conceived as structures with a twist or rotation, in short, as structures analogous to torsion.

  And it is important for another reason, because, as Coxeter pointed out, higher dimensional polytopes — and we realize a pyramidal structure is already a departure from a regular polytope — have numerous vertices, faces, and edges.32 And atop each of these structures is a “temple” that, if one looks at them closely, appear to be designed as some sort of resonant cavity.33 What this strongly suggests or implies is that, just as a tetrahedron can be represented or “squished” into a two-dimensional representation or analogue, so too can higher-dimensional pyramidal objects or constructs be “squished” into a three-dimensional structure that is an analogue of them. This fact, coupled with Mr. Richard Hoagland’s simple torsion experiments conducted at Tikal near these very pyramids, strongly suggests these structures were deliberately designed as hyper- dimensional analogical, alchemical structures designed to manipulate the physical medium, and to respond to it.

  With this in mind, let us return to Sir William Flinders Petrie’s observations concerning the Second Pyramid at Giza, remembering the context that the plan of the compound is to rotate, producing all those tetrahedral structures seen in the previous section of this book. Petrie notes that “The lower two courses of the casing” of the Second Pyramid “are of granite, very well preserved where it is not altogether removed.”34 In addition to this, “the builders made the face(of the granite casing stones)35 drop down fo
r some depth vertically from the edge of the slope, building the pavement against the vertical face.”36 Consider what this means.

  It is known that granite possesses very active piezoelectric properties, given all the small quartz crystals embedded in it. Thus, in a certain sense, one has the piezoelectric analogue of the primary to a Tesla magnifying impulse transmitter, for the enormous weight pressing down on these casing stones places them under constant stress. Furthermore, just as Tesla stated of his own technology in the trial transcript cited as an epigraph at the beginning of this chapter, it was necessary for his Wardenclyffe Tower to “grip the earth” in order to make it quiver. It is equally the case that it is necessary to grip it in order to respond to its natural vibrations.

  So what do we have, when we combine all these observations about Giza, Mexico, and the wider cosmological myths we have examined in this book?

  1) Present in the Mexican and Giza Pyramids are structural analogues of rotations and torsion;

  2) The Mexican pyramids also appear, with their multi-cornered edges and faces, to be some sort of analogues of irregular higher-dimensional objects, contained within an overall normal pyramidal structure;