E a r t h / m a t r i X
Science in Ancient Artwork and Science Today

The Geometry of Ancient Sites
[Giza, Easter Island, Poverty Point, et. al.]

by Charles William Johnson

Dedicated to Carl P. Munck

Table of Contents
The Geometry of Ancient Sites
The Problematic
The Geography and the Geometry
The Corridors/Pathways on a Timeline
The Great Pyramid Design

The Geometry of Ancient Sites

by Charles William Johnson


Today, we think of art and science as representing two diametrically opposed fields of human endeavor. With counted exceptions, the people who are dedicated to one field or the other are at opposite ends of the personality chain. One either does science or art, but seldom both. The artwork of the ancient past is thought to be wholly on the side of art, with little or nothing to do with science. At best, it may be conceded that some architectural constructions may have required a high level of engineering skill, but even that is thought to have been within the realm of technology and not science as such. Furthermore, there are those who persistently refuse to consider any contribution to science by the ancient cultures.

In our studies of science in ancient artwork, we have come to realize that we know very little about the inner workings of the ancient past. We have no knowledge of the computational math behind much of the ancient artwork and ancient reckoning systems. Such documents as the Rhind Papyrus, deal with common household computations, although it is cited as an example of the computational math involved in mathematics of ancient Egypt. Hardly anything is known about the mathematics of other cultures, such as the maya system.

The absence of such knowledge does not impede the dissemination of wide held views about the worth (or lack of worth) of ancient science. In fact, the very concept of science is all but thought to have been non-existent then by most scholars now. There are many exceptions, whereby scholars and laymen alike are attempting to unveil the way in which ancient cultures functioned, not only at the level of archaeological digs, but at the level of abstracted, theoretical thinking.

Many theories and ideas exist regarding a possible relationship among the different archaeological sites around the world embracing many different cultures. Those societies that constructed pyramidal structures are often thought to have come from the same source. The reasoning questions the coincidence of all those pyramids having been built without there having been some kind of shared source of knowledge. Simply, why did all those ancient cultures think of building pyramids; why didn't some of them build structures into the ground instead of upwards to the sky; or, why didn't some of them build vast flat platforms as far as the eye could see; and so on. The similar characteristics of the pyramidal structures around the world cause one to think that they all came from the same school of thought.

For that reason many of these structures are researched in an isolated fashion, as of the culture in which they exist, without relating a structure of one culture with another from a distinct culture. Comparative analyses are made at times, but usually with the purpose of denying any relationship among the cultures being compared. Exceptions, however, do exist, as we shall review briefly below.

No historical evidence exists that proves these ancient pyramidal structures were related to one another, other than the historical fact that they all existed in the framework of the ancient past. But, the shared trait of having existed in the past is nothing more than circumstantial evidence to many scholars, ergo, meaning no evidence at all.

The first person we ever heard speak about a grid system for the entire planet was Hugh H. Harleston, Jr, back in 1974 in Mexico City. His analysis of the site of Teotihuacan and the subsequent discovery of markers placed throughout the Western Hemisphere, lead him to propose the idea that theTeotihuacan site, in fact, was based on a geodesic grid. In our analyses of the Teotihuacan site some twenty-odd years later, we visualized a grid system based on the geometrical designs within the pyramidal structures of that site. More recently, we have published other considerations regarding basic designs of a grid system within the Teotihuacan site and that of Giza. (Cfr., Earth/matriX, Essay Nos. 83, 85).

Our analyses are based mainly on the numbers of the series of perfect right triangles (Cfr., Earth/matriX, Essay No. 64: "the 3-4-5 Perfect Right Triangle: 5-12-13 and 6-8-10). The perspective that we have followed is that of considering the pyramidal structures within a specific site to other pyramidal structures within that same site. We have compared this internal view of the layout of the structures, without any regard to the placement of the sites on the longitudinal and latitudinal coordinates of the planet.

Carl P. Munck, in his work entitled, The Code, analyzed and compared the coordinate system of the different pyramidal sites around the world. In his analysis, he has assigned the site of Giza, of ancient Egypt (kemi) the starting point for his grid system or, pyramidal matrix as he refers to it. In time, his work is being recognized as having posited the idea of a matrix corresponding to the layout for all or most of the major pyramidal sites around the world, irrespective of the particular culture involved. Personally, I know of no one else who has studied in detail the maps of these sites and related their coordinates on the globe in the manner in which Carl P. Munck has done so.

We shall examine, in a much less detailed manner, possible relationships and characteristics among some of the ancient pyramidal structures and sites around the world. Even if one shows the pyramidal sites to be related as of the geometry of a circle and a sphere (in terms of degrees, radians and the concept of pi), the logic of numbers is insufficient to convince most readers of the conscious design behind the placement of those sites. In our discussion, we shall rely more on the geometry of design than upon the mathematics to discern possible relationships among the different sites. Obviously, we cannot prove that the ancient sites were related, but the discussion may be suggestive of the need for such a proof. Most scholars believe that the different sites are unrelated to begin with, and therefore need no proof.

The Problematic

In order to write this essay, we had to confront a series of problems, which may not have been solved as such. The main problem is to write about relationships in space-time/movement on a single plane while conceptualizing the shape of a sphere and elements upon the surface of that sphere: the Earth. The problem concerns one of perspective. We shall be viewing a series of lines on a flat surface, when the surface is actually spherical in shape. In this sense, any relationship that we may ponder may easily be misunderstood simply by not assuming the suggested perspective. The suggestion to the reader, then, is to look at the illustrations as they appear here on paper, then imagine them as best one can on a sphere. That is no easy task. And, just as we are quite unable at times to do such imagining, we can further imagine just how difficult this must have been for the ancient peoples who did not benefit from the knowledge of knowing that the Earth is not a perfect sphere. For this concept of visualizing the Earth's surface, one is thereby referred to Cahill's Butterfly World Map, in which the world is divided into 4 lunes or, eight curvilinear triangles. [Cfr. Greenwood, p.120.]

Carl P. Munck (The Code), as we mentioned earlier, has discerned a pyramidal matrix which links most of the pyramidal sites around the world through a grid system based on the numerical division of the Earth's spherical-like shape. Mr. Munck's work has proven to be quite extensive and intensive, reflecting nonetheless the difficulties in conceptualizing such a system for the planet Earth. Devising such a grid system based on a number series and constants poses at times insurmountable problems.

1) The measurement of the Earth does not accommodate numbers easily because planet Earth is not a perfect sphere. Therefore, the detailed sequential ordering of coordinate points on the Earth enters into conflict with this numerical straight jacket.

2) The very act of measurement itself defies the characteristics of space-time/movement in terms

of carrying numbers to ten-place decimal endings and beyond. In other words, there comes a physical point where things/events cannot be measured as such.

3) Numbers can be related, as of the logic of numbers themselves, without any relationship actually existing in reality in terms of space-time/movement.

4) The procedure in The Code produces relational numbers as of the concept of p (3.1415926535). The author takes double p (6.283185307) as a given or constant. Then, he further relies upon the cube root of double pi (1.845270149). This choice of numbers stems from the belief that the ratio of the perimeter-to-height of the Great Pyramid of Giza is foretelling of a double pi relationship. However, part of the problem in discerning ancient reckoning is to know the height of the Great Pyramid, about which we can only speculate at this late date given the deteriorated condition of the Great Pyramid itself. Therefore, one of the sought-after results of research in ancient artwork is to discern the height of this particular pyramid; to begin the analysis with a given figure (480.3471728 feet) weakens the analysis.

5) The procedure of The Code employs symbolic numbers selectively discerned from the ancient artwork itself. For example, the number "nine" is chosen (representing "4 sides, 4 lower corners and apex" to the Great Pyramid) in order to propose using the base number 1.845270149 to the 9th power (= 248.0502134). This number then becomes the constant for the entire grid system of The Code. The 248.0502134 number (the cube of double pi), serves then as a way of making all numbers relational within the grid system. Understandably, any number could have been used in this manner with similar results.

6) The Code also begins its analysis with the perimeter measurement of the Great Pyramid (3018.110298 feet), which obviously does not result from measurement, but from computations based on pi and double pi considerations of the presentation. This can be easily understood when we then observe how this number when divided by the given height of 480.3471728 just happens to produce exactly the 6.283185307 figure for double pi.

7) All numbers are filtered through the double pi cube number (248.0502134), and therefore are relational in terms of pi itself. The numbers of longitude and latitude are multiplied times this particular constant (2pi-cube).

8) The grid system is then made to float by changing the coordinate of the Great Pyramid so that it occupies the 00o 00' 00"0 coordinate on the grid itself. All coordinates of the system are then modified accordingly by adding/subtracting to them the original coordinate of the Great Pyramid (31 o 08' 00"8).

9) The Code obtains numbers (measurements) by multiplying unlike elements and dividing degrees and minutes into decimal place numbers in order to obtain supposedly exact representations of seconds then to be expressed in the degree-minute-second system. Consider the procedure for the vector of Teotihuacan:

Example: 55043.7314 divided by 38 degrees divided by 39 minutes equals 37.14151916 minutes, which is expressed as follows in The Code:

55043.7314 : 38o : 39' = 37.14151916"

It is impossible to divide a decimal-like number (55043.7314) by the degree-minute sixty-based number (38 degrees and 39 minutes) in order to obtain once more a decimal-like number expressed now in minutes (37.14151916"). In any case, the latter expression is generally written as: 38o 39' 37"14151916, without the seconds being expressed to such incommensurable decimal places.

10) The Code takes for granted that the ancients utilized only a system based on the 360-degree circle. In our studies, we have seen that a circle may have been divided into any number of reckoned numbers (260c, 364c, 365c, 374.4c, 378c etc.). This possibility also has implications for the concept of pi, which may have been relational and proportional in these different instances, whereby the Nineveh number 2268 may have been a fractal expression for pi on a 260c circle. For, the fractal number 2.268 is relational to pi as of a 260-degree/segment circle.

11) The Code also posits the fixed idea that there is only one answer for each computation. Our studies have shown that the ancient reckoning system may have been relational, with multi-layered meanings to a given element. Depending upon the perspective assumed, a single element may have enjoyed many different meanings. Instead of a fixed grid system, the system may have been flexible, relational and, proportional. Precise meanings are avoided in such a system. In other words, there is no single answer to a particular computation; all meanings become inter-dependent.

In spite of all of these negative-sounding observations, we offer them in a positive tone. We do believe and affirm that Carl P. Munck has discovered the reasoning behind the pyramidal matrix of the ancient archaeological sites around the world. We do consider his work to represent the mathematical basis of computation that may have been involved in situating the different sites around the world.

Professor Carl P. Munck's numbers are certainly correct to a degree as of his detailed computations. And, were he to consider the possibility that the ancients also employed a 260-degree circle/sphere, a 378-degree circle/sphere, or even a 374.4-degree circle/sphere, then all of the numbers would change accordingly. The pyramidal matrix of the ancient past was undoubtedly much more complex than we may imagine. Depending upon whether one is computing sidereal orbital times or, synodic orbital times, the abstracted circle/sphere involved could undergo a theoretically distinct division of segments (i.e., degrees). Contemporary society has grown up with an inherited 360-degree circle/sphere, and it is difficult for us to imagine any other option. In fact, when we imagine such an option in grade school, our teacher would invariably suppress that by telling us "a sphere has three-hundred-and-sixty degrees". That is what we have always been taught. The ancients appear to have been more flexible in their outlook. The historically chosen numbers coming out of their reckoning systems allow for computations that are more flexible and seem to have been based on other day-counts (i.e., 260c, 364c, 378c, 374.4c, etc.).

Many more comments could be in order, but the theme that occupies us at the moment is not how others have attempted to resolve the interrelationships of the ancient sites around the world, but how we might view them from the perspective of geometry. Obviously, geometry and mathematics march hand in hand; one is implied in the other. Nonetheless, given the very monumental nature of the ancient structures and the obvious geometrical designs therein, we shall concentrate on distinguishing relationships based on interconnecting points, lines, circles, triangles, etc., which might exemplify the underlying designs of the ancient artwork itself.

No doubt we are all attempting to lay bare those suspected designs behind the ancient artwork. It is inconceivable for anyone to think that structures such as the pyramids of Giza and Teotihuacan, and those of so many other sites, were constructed without any pre-conceived design in mind. All of us are attempting to comprehend the apparent design elements in order to understand better the very intentions of those who built the pyramidal structures. We suspect that these structures mean something, and we all want to know what those meanings represent. And, as in any search, we must explore each and every pathway until the search is complete. In the following procedure, surely many mistakes are ignored by this author, but we are confident that by reviewing our visualization of the theoretically abstracted lines and inter-relationships of the design elements, we may conclude at least one thing. These monumental structures around the world were constructed with a definite purpose in mind, and possibly a common or shared purpose.

So many scholars have attempted in the past to convince the world of the inter-related nature of past accomplishments in different ancient societies and cultures. But, resistance has been tenacious. Few wish to admit the obvious; many cling to the fixed idea that the different ancient cultures were completely separated from one another by the distances separating them on this immense planet. Most authors are convinced that the ancient societies never came to even know the extent of the planet, its spherical boundaries. The only way that this idea can continue to be fixed in our minds is to continue to believe that all of the ancient populations and their monumental works were each one an isolated example of the human being facing its environment alone.

If it were the case that each ancient structural site were an isolated example of human ingenuity created and re-created as a response to that environment, then, one would expect to find a completely unrelated existence for each site. No two sites should have any kind of visible relationship between themselves. As writers like Harleston and Munck have attempted to show, each in his own reasoning and analysis, the many varied sites are related, but at times in most unsuspecting ways. Aside from the mathematics of it all, the abstracted geometry may offer a distinct insight into the math. Carl P. Munck has made triangulations among the different sites in his extensive work. We shall build upon such triangulations in a similar manner, although without the specifics of the math. Hopefully, that would be an aspect that professor Munck may soon lay bare in his writing.

The Geography and the Geometry

In the Earth/matriX series of essays and extracts we have offered our share of analyzing the numbers of the ancient reckoning systems. In this essay, we wish to concentrate on a visualization of the inter-relationships of the ancient sites identified around the globe. Our analysis, from the start, does not purport to reflect any preciseness developed to the umpteenth scale of points and lines. For all of the preciseness that the construction of the Great Pyramid reflects, the ancient system would appear to be based on approximations of varying degrees and relationships. For example, the concept of a connecting-line, in fact, would appear to be rather the idea of a path or corridor, and not a line limited to the molecularly narrow thin line imagined by infinite numerical preciseness. The idea of a precise point (dot) or line, taken to the contemporary view of the concept of a singularity, leads one to an abstracted point/line that is virtually incommensurable.

One has only to recall that the monuments of ancient sites generally are grouped together within a specific boundary or complex. Even the site of Giza is bordered six miles to the north by the Pyramids of Abu Roash and about eight miles to the south by the Pyramids of Abusir. Although one may measure points and lines to an infinite minute level, it is understood that the space occupied by the site is itself significant. An entire site or group of sites may represent a corridor, a path for the forces thus conceived.

In this sense, there is no single, all-determinant point or line, which may be defined to the minute degree. Rather, each point and line are determined as of the relationships invoked. And, a single point or line may function as one meaning in relationship to a particular design element, while functioning with another meaning in relation to some other design element. The same point may be conceived as the center at one moment and on the perimeter at another moment. "The Center" does not exist as such, but exists in and of each particular event and relationship discerned.

In this way, the ancient artwork appears to reflect all things conceived, which means that any one event-point or event-line may have been conceived in every imaginable way, which means that all other imaginable and unimaginable ways of conceiving that event still remain.

For our analysis, we shall employ an outline drawing of the continents and land masses on a Mercator map of the world. We have shown only some of the ancient sites and a few of the main rivers of the world. We shall speculate with specific relationships by drawing interconnecting lines among the ancient sites, and discuss possible meanings thereof. We shall be discussing some numerical relationships, but only insofar as they are related to the geometry of the abstracted points and lines. The geometrical illustrations are simply approximations and represent an attempt to offer an overall view of the possibility that the different ancient sites were relationally placed as of a grid system or matrix.

Let us begin our presentation with a connecting line between the Poverty Point area within the state of Louisiana, near the Mississippi River basin, and the Giza site of ancient kemi, near the river basin of the Nile.

Poverty Point and Giza (1)


A discerning feature of these sites is that they are both situated near a major river basin. And, if we consider them to be situated along a latitudinal pathway, although not on the exact same coordinate, an apparent straight line may be drawn to connect them, with each of the nearby river basins forming a windmill image. One could even envisage this geometrical pattern to be reflective of the ollin design for movement that is profusely employed in ancient Mesoamerican artwork.

Easter Island (Chile) to Giza (2)


In the previous illustration, we have simply connected the site of Easter Island to the site of Giza, understanding that this may represent another pathway or corridor along the grid. It should be noted that this pathway runs almost parallel to another major river basin, that of the Amazon River.

Now, let us draw a line at a right angle to the site of Poverty Point towards the line between Easter Island and Giza. From the perspective of geometrical concepts, the very idea of a right triangle is commonly found in many of the designs of ancient artwork, and should not surprise us to represent a viable concept on the planetary grid system or matrix. At first, one might think that the drawing of such a line is entirely arbitrary, given the fact that it breaks with the idea of having a relationship to a major river basin area. Yet, as we shall see below, its conceptualization is significant for other dividing points and lines along the relationships of the two continents of the Western Hemisphere.

A Perpendicular Line between Poverty Point and the Easter Island/Giza Corridor (3)


The Counterclockwise Motion of the Flow of the Three Major River Basins (4)


Counterclockwise direction of flow of river currents

From the previous illustration, one may observe the direction of flow of the three major river basins involved in the design. The flow of the currents of these three rivers establishes a counterclockwise motion on the face of the planet along the different lines of the corridors drawn between the sites.

Now, let us draw a line from the area of Poverty Point to the site of Easter Island, along with another perpendicular line from the line between Easter Island and Giza to the area of Poverty Point.

Connecting Lines between Poverty Point Area, Easter Island, and Giza (5)


As we mark off the mid-point of the perpendicular line on the previous illustration, we then observe how another perpendicular line would pass across the tip of the southern continent of the Western Hemisphere as indicated in the following illustration. It seems all too apparent that by joining these three ancient archaeological sites, with interconnecting lines, and marking off the perpendicular lines as shown, one would not expect to see any geometrical relevancy to the natural landscape of the continents. However, as may be observed, the abstracted geometry fits extremely well with natural cut-off points such as the tip of the southern continent. Too many elements seem to coincide in this illustration to call them mere coincidences obtained through happenstance or chance. The placement of the sites in relation to one another, not only reflect a relevancy in numbers as determined by the analyses of Carl P. Munck, but the natural function of geometrical figures is entirely reflected in these same sites and their interconnected lines.

The Mid-Point between Poverty Point Area and the Easter Island/Giza Corridor (6)


Now, let us extend the line of the corridor joining the Poverty Point area and the site of Giza beyond to the East. The extended corridor between Poverty Point and Giza passes along the mouth of the river basin of the Euphrates River, traverses the river basin of the Ganges River, and runs along the basin and into the mouth of the Yangtze River in China. And, furthermore, if we now draw a line between the area of the mouth of the Yangtze River and aster Island, we see that other major river basins lie upon this widen corridor. The mouth of the Ganges River in Bangladesh and the mouth of the Niger River in Nigeria fall almost directly within this particular corridor.

The Extended Corridors of Poverty Point, Easter Island and the Mouth of the Yangtze River (7)


The Four Main Corridors and Their Relation to the Major River Basins of the Area (8)


From the previous illustrations, simply from the perspective of abstracted geometry, one obtains the impression that what writers like Hugh H. Harleston, Jr., and Carl P. Munck have been stating for decades now in mathematical terms is certain. Some kind of geodesic or matrix design exists within the placement of the different archaeological sites around the world. Furthermore, the geometrical designs reveal that the paths or corridors between the different sites appear to have relevancy for the natural make-up of the continents. In other words, the humanly determined sites reflect features and characteristics apparent in nature. In the previous cases we have seen how the geometry appears to reflect a direct relationship with the major river basins of the world.

Now, if we extend some of these geometrically relevant lines (paths or corridors) even further, then we may observe more relationships with other features of the planet Earth. For example, if we extend the mid-point line as drawn below, we may then see a relationship appear with pre-historic sites within Spain and Europe.

The Extension of the Mid-Point Line Toward Pre-Historic Sites in Spain (9)


Besides the relationship, then, with other sites throughout ancient history, we may also observe how these lines (paths or corridors) reflect the tectonic patterns of Earth. In other words, the corridors appear to reflect the inert movement of the tectonic plates of the continents. Consider the following.

Nazca lies between Easter Island and Giza. The base of the triangle, it should be noted, aligns the sites of Easter Island, Nazca and the Giza complex on almost a straight line. The baseline of the triangular alignment appears to fall between the Nazca-Palpa region and Lake Titicaca, in Peru. Further study is required to illustrate the relationship, then, of the Nazca-Palpa region to the other sites around the world and hydrography.

(Next: part two, The Geometry of Ancient Sites)

Copyrighted © 1995- 2012 by Charles William Johnson. All rights reserved. Reproduction prohibited.

E a r t h / m a t r i X
The Geometry of Ancient Sites
By Charles William Johnson

Published by: Earth/matriX P.O. Box 231126 New Orleans, Louisiana 70183-1126 USA
Branch: Earth/matriX-México Jorge Luna /Director - Mexico, Apartado Postal 70-257, Ciudad Universitaria, México, D.F., 04510, México
August, 1999.
ISBN 1-58616-186-5

Copyrighted © 1995-2012 by Charles William Johnson. All rights reserved. Reproduction prohibited. Printed in the United States of America. Published simultaneously in Mexico. This publication, or parts thereof, may not be reproduced in any form of photographic, electrostatic, mechanical, or any other method, for any use or purpose, including information storage or retrieval, without written permission from the author, except for the inclusion of brief quotations in a review.

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