comparative analytical drawings of the pyramids of...

Comparative AnalyticalComparative AnalyticalDrawings ofDrawings of

the Pyramids ofthe Pyramids of Teotihuacan Teotihuacan,,Mexico and the Pyramids of theMexico and the Pyramids of the

GizaGiza Complex, Egypt Complex, Egypt

Charles William Johnson

Earth/Earth/matriXmatriX Editions EditionsSCIENCE IN ANCIENT ARTWORK

©2011 Copyrighted by Charles William Johnson

Earth/matriX: SCIENCE IN ANCIENT ARTWORK

Charles William Johnson

Earth/matriX Editions, P.O. Box 231126, New Orleans, Louisiana 70183-1126, USA

Earth/matriX Mexico, Apartado Postal 70-257, Ciudad Universitaria,México, D.F., México 04510

Founder: Charles William JohnsonDirector/Mexico: Jorge Luna MartínezInternational Associate: Marco IcazaISBN 1-58616-471-6

Comparative Analytical Drawings of the Pyramids of the Comparative Analytical Drawings of the Pyramids of the Giza Giza Complex Complex andandthe Pyramids ofthe Pyramids of Teotihuacan Teotihuacan, Mexico, Mexico

In this brief study, I have offered only a few select analytical comparative drawingsand overlays between the Giza Complex of ancient Egypt and the pyramidal site ofTeotihuacan, Mexico. In the Earth/matriX collection, there are thousands of similaranalytical drawings that shall be posted.

Many of these analytical drawings constitute the theoretical approach that I havefollowed over the past nineteen years in studying the math and geometry behindancient artwork from different cultures around the world.

From my studies, it would appear that there exists a sacred proportion employedin the ancient design of the pyramidal sites around the world. In this case, the sitesof the Giza Complex and that of the pyramidal site at Teotihuacan, Mexico, bothappear to share a common, basic geometric design. The basic design appears tobe based on fractal proportions utilizing the side measurements of perfectright triangles.

For questions and comments, please, contact me at the following address:[email protected]

January 16, 2011

The Teotihuacan pyramidal site is one of the most organized archaeologicalsites in my view, based on an apparent pattern of squares and rectangles.From the old steoreoview presented earlier of the Sun Pyramid, it is obviousthat the exactness of the measurements of the Teotihuacan site has been lostlong ago. Yet, the reconstruction of the pyramids and the secondary pyramidalstructures establishes a reflection of the site’s original layout. Authors, such asHugh Harleston have studied the site in detail and projected it precisemeasurements through a standard Teotihuacan unit. This study simpleexamines a geometric pattern that appears to reflect the design layout of thevarious primary and secondary pyramids on the site.

There are numerous ways to conceptualize the geometry of the Teotihuacansite, even though it may be apparently based on squares and rectangles.

The Giza complex shares similarly suggested layout based upon squares andrectangles. For this reason, it would seem appropriate to consider both of thesepyramidal sites as basic concepts of geometry. And such has been the case overmany decades of scholars studying, and even triangulating the layout of thepyramidal complexes at Teotihuacan, Mexico and Giza, Egypt.

A few slides are presented that show imaginary geometric concepts for theTeotihuacan site and how one may derive certain patterns based upon basicgeometric considerations. Then a distinctive pattern based on the sidemeasurements of perfecto right triangles is offered in the form of a shape thatsimulates ancient Egyptian hieroglyphs, and even Maya glyphs.

Throughout my studies, it has become clear that the ancient people around theglobe, and especially those of ancient Egypt and Mexico, share similar aspectssuch as calendar reckoning and language systems. The fact that both of thesedistant cultures share possibly the same or similar design behind theirpyramidal structures should not be surprising in my mind.

A centerpoint in front of the Pyramid of the Sun at Teotihuacan, Mexico

Sun Pyramid

Moon Pyramid

La Ciudadela

A Centerpoint

It is possible to distinguish countless center points on the floor plan ofthe Teotihuacan pyramidal site.

©2005-2011 Copyrighted by Charles William Johnson

Teotihuacan, Mexico

A centerpoint in front of the Pyramid of the Sun at Teotihuacan, Mexico


Conceptual center points of the architectural structures at Teotihuacan, Mexico

Sun Pyramid

Moon Pyramid

La Ciudadela

A Center pointOne can imagine all of the primary and secondary architectural constructions,pyramids and mounts, as each one having its own center point. There simplyexist too many center points to list, as one may consider the actual center of asquare design or the apparent point to which each set of square/rectangular

shapes seem to be pointing.©2005-2011 Copyrighted by Charles William Johnson

Teotihuacan, Mexico

A radial baseline between two significant center points of the Sun Pyramid.

Imaginary Circles

One may imagine connecting two center points.For example, two center points may be connected by a radial line of a circle.

In this case, the imaginary center point on the axis in frontof the Sun Pyramid may be connected to the center point

derived from a small mount located at the rear of that samepyramid.


Teotihuacan, Mexico

Note the relationships of a radial baseline of the Sun Pyramid.

The Five WorldsImaginary CirclesImaginary Lines

©2005Copyrighted by Charles William Johnson

By connecting different center points along the axis of the Teotihuacan site, manydifferent radial lines may be drawn. It would appear that it is by design that the

radial distance between the two significant points map out the entire length of theTeotihuacan complex and determine the distance that separates certain structures.


Teotihuacan, Mexico

The radial baseline of the Sun Pyramid may be employed fractally.

The Five WorldsImaginary Circles

Note the coincidence of the radial lines and those between each one with regard tothe surrounding pyramids and mounds along the entire length of the axis of the

Teotihuacan complex.©2005-2011 Copyrighted by Charles William Johnson

Teotihuacan, Mexico

The diametian baseline of the Sun Pyramid may be employed fractally.

Imaginary Circlesand

Baseline Proportion

Note the coincidence of only a few of the diametian lines to other structureswithin the Teotihuacan complex. There are too many to list.


Teotihuacan, Mexico

Other fractal design measurements may be identified and employed to mark offdistances between and among various architectural structures a the Teotihuacansite. For example, consider the linear distance between two connecting circles,

and how that same distance relates to the same distance between variousstructures.

Imaginary Triangles ©2005-2011 Copyrighted by Charles William Johnson

Teotihuacan, Mexico

Multiples of the baseline of 51.5 degrees as of the 60:60:60 trianglesreveal this particular pattern. Note how the extended lines intersect

with the secondary monumental structures of Teotihuacan.

51.5 degrees51.5 degrees

Triangles


Teotihuacan, Mexico

Here the design relationship of the angle of inclination of the Great Pyramid of Giza in Egypt is used to draw lines at an approximate 51.5-degree

angle as of the center point on the axis in front of the Pyramid of the Sun at Teotihucan, Mexico. The 51.5·51.5· 77 degree triangle is drawn within

the 60·60·60 degree triangle as shown.

Triangles


Teotihuacan, Mexico

Other geometric forms may be usedto analyze these ancient works, such

as circles, triangles and hexagonalfigures.

Note the coincidence of the outlineof the concentric circles of the

Aztec Calendar with the differentoutlines of the various monumental

structures at the site of Teotihuacan,Mexico.

The center point in thisdrawing falls right infront of the Pyramid ofthe Sun at Teotihuacan,on the runway, Lacalzada de los muertos.


Teotihuacan, Mexico

The design elements of theAztec Calendar, also reflectthe layout of the design ofthe site at Teotihuacan, Mexico.

By superimposing the AztecCalendar’s design over thesite of Teotihuacan, a directrelationship among the elementsof the calendar and the pyramidal site becomes visible.

Note the fractal hexagonal design in the Aztec Calendar from the boundaries of the Teotihuacan site [green lines].

Hexagonals51.5 51.5

degreesdegrees ©2005-2011 Copyrighted by Charles William Johnson

Teotihuacan, Mexico

Hexagonals

Cutaway of the hexagonal designin the Aztec Calendar, based on the

design boundaries of the site ofTeotihuacan.


Teotihuacan, Mexico

The Ring of Flames detailed in themultiple views of the Aztec Calendar.

The different geometric forms serve toportray movement in the ancientartwork, as one contemplates thewave-like path of the ring of flames.

There are many different ways toanalyze fractal proportion in theancient artwork. A few have beenpresented as illustration of the wealthof fractals in geometry.

Now, let us assume a slightly differentgeometric approach, one that is notgenerally found in the literature.

Hexagonals

OllinOllin (Movement) (Movement)


The 1 • 6 • 8 • 9 right triangle1 • 6 • 8 • 9 right triangle establishes a manner in which the ancientreckoning counts may be computed as of the side measurements of a 3 • 4 • 53 • 4 • 5perfect right triangle. Further, it allows for computations to be based upon thecube, as in the extension of the Pythagorean Theorem, instead of the squares ofthe terms.

The reasoning behind the use of the 1•6•8•9 triangle, and variations, is illustratedin relation to the design of the Great Pyramid.

The 1 • 6 • 8 • 9 Triangle and Ancient Reckoning

Profile ofthe Great PyramidGiza, Egypt

54

3

©2002 Copyrighted by Charles William Johnson. All rights reserved.

Begin with a perfect 3•4•5 right triangle.

108

6

Double its sidemeasurements:

3 · 4 · 56 · 8 · 10


Divide the hypotenuse in the following manner.

98

61

1³ + 6³ + 8³ = 9³

1 + 216 + 512 = 729

729 = 729

A relation of equivalencysimilar to the Pythagorean

theorem for a triangle,based on

four side measurementsinstead of three.

The 1 · 6 · 8 · 9 Triangle


In this analysis, the side measurements of perfect right triangles are employed,but in a manner that is not observed as far as I know in the literature analyzing

ancient artwork. Now, look at the right triangle as of the figure of a square.

98

61

Profile ofthe Great PyramidGiza, Egypt

1 · 6 · 8 · 9Swing out the two sectional

lines of the hypotenuseat 90-degree angles

to their adjacent linesas shown.


The 1 · 6 · 8 · 9 Triangle

98

611

99


The 1 · 6 · 8 · 9 Triangle With the resulting1 · 6 · 8 · 9

geometric figure, a square-like

design appearsthat resembles an ancient

Egyptian hieroglyph for “house”.

See next slide.8

611

99

The The KawilKawil

In the ancient Aztec and Maya cultures one findsa similar geometric design referred to as the Kawil.

1· 3 · 6 · 8 · 9

Consider the Maya Companionnumber, 1366560 in

relation to the 13689 fractal

1368900 minus 1366560 = 2340doubles to...

46809360

18720

The Maya Long Count fractalas difference...1872000

11·3·6·8·9·3·6·8·91· 3 · 6 · 8 · 9

Consider the reciprocal of 13689 fractal

1 / 1.3689 = 0.730513551halves to...

0.365256775

The 365.256 fractal countsuggests the sidereal

orbital time ofthe Earth

1

3

6

8

9

1·3·6·8·9

The design may be modifiedbased upon the ancient Mayareckoning number [1368900] in relation to a Maya companionnumber [1366560].

1³ + 3³ + 6³ + 8³ = 9³

1 + 27 + 216 + 512 = 756

756 = 756756 = 756

A relation of equivalencysimilar to the Pythagorean

theorem for a triangle,based on

four side measurementsinstead of three

for the 756c baseline756c baselineof the Great Pyramid.of the Great Pyramid.

11·3·6·8·9·3·6·8·988

99

33

66

11


From the previous commentaries, we may understand that the 1•6•8•9 trianglerelates directly to the construction of the design of the Great Pyramid. Themanner in which this may occur concerns simple geometry and mathematics. Thedifferent measurements of the Great Pyramid may bededuced from elementary math and geometry. For example, the sidemeasurements (755.7909764 feet), as we have shown in earlier essays, deriveseasily from essential geometry of the division of a rectangle by diagonal lines.Something similar appears to obtain from the use of the extension of the 3•4•5pyramid to its divisions into a 1•6•8•9 triangle. The use of four terms wouldappear to be contradictory for the reference of a triangle. However, as we haveshown above, the extensionextension of the Pythagorean Theorem to the cubeof the Pythagorean Theorem to the cube involvesequations based upon four terms.

In the following slides, points and lines of coincidence between the lineardesigns and the placement of the mounds and pyramids shall be illustrated.From the following analysis, it will become evident that apparently allelements, such as primary and secondary mounds and pyramids arerelational to one another through the geometry of the design s based onperfect right triangles.

33·4·5·6·4·5·61 1 · 6 · 8 · 9· 6 · 8 · 91 1 · 6 · 8 · 36· 6 · 8 · 36

Variations onthe perfect righttriangles reflect

the design of theTeotihuacan site.


1-6-8-101-6-8-101-6-8-101-6-8-10

1-6-8-361-6-8-36

1-6-8-361-6-8-36

Fractal Proportion

A perfect right triangle with side measurements of1·6·8·10 as fractal triangle on the progression.

TeotihuacanTeotihuacan, Mexico, Mexico

1-6-8-101-6-8-10

1-6-8-101-6-8-10

Fractal Proportion

1

6

8

9

1 · 6 · 8 · 936 216 288 32472 432 576 648144 864 1152 1296288 1728 2304 2592576 3456 4608 51841152 6912 9216 130682304 13824 18432 207364608 27648 36864 414729216 55296 73728 8294418432 110592 147456 16588836864 221184 294912 33177673728 442368 589824 663552147456 884736 1179648 1327104294912 1769472 2359296 2654208589824 3538944 4718592 53084161179648 7077888 9437184 106168322359296 14155776 18874368 21233664... ... ... ...infinitely so...

Fractal Proportion in Number Series

1

69


1-6-8-10-36 1-6-8-10-36

A perfect right triangle with side measurements of1·6·8·10 as fractal triangle on the progression.

Fractal Proportion


1-6-8-10-361-6-8-10-36

Fractal Proportion

1-6-8-10 1-6-8-10 Teotihuacan Teotihuacan

1-6-8-10 1-6-8-10 1-6-8-10 1-6-8-10 1-6-8-10 1-6-8-10

Fractal Proportion

1-6-8-10-36 1-6-8-10-36 Teotihuacan Teotihuacan

Fractal ProportionFractal Proportion

13

6

8

9

36

1-3-6-8-9-361-3-6-8-9-36Teotihuacan Teotihuacan

13

6

8

10

36

1-3-6-8-10-36 1-3-6-8-10-36 Teotihuacan Teotihuacan

Two BasicTwo BasicGeometric Geometric DesignsDesigns

La La ciudadelaciudadelaTeotihuacanTeotihuacan, Mexico, Mexico

A perfect right triangle with side measurements of 3·4·5 plus 6 as the commencement side of a fractal triangle on the progression.

3·4·5·66·8·10·12

3 3·4·5·6 ·4·5·6

LaLa ciudadela ciudadela at atTeotihuacanTeotihuacan, Mexico, Mexico

A perfect right triangle with side measurements of 3·4·5 plus 6 as the commencement side of a fractal triangle on the progression.

3 3·4·5·6 ·4·5·6

In order to demonstratethe fractal nature ofthe numerical valuesinvolved in the designfigure, the 1·3·6·8·9design is placed overthe Ciudadela in Teotihuacan. Noticehow each segment ofthe design touches upona significant architecturalelement.

LaLa ciudadela ciudadela at atTeotihuacanTeotihuacan, Mexico, Mexico

The Ciudadela of the Teotihuacan pyramidal site in Mexico iscompared to a fractal expression of the pyramids of the Giza

Complex

©2006 C.W. Johnson

Note how the boundarylines of specific

mounds|pyramids liejuxtaposed to one

another. The complexityof smaller mounds in theTeotihuacan site share a

fractal coincidence inthe design with the

design of the pyramidGiza Complex.

The Ciudadela of the Teotihuacanpyramidal site in Mexico is comparedto a fractal expression of the 1·3·6·8·9 Design

©2006 C.W. Johnson

Note how the boundarylines of specific

mounds|pyramids liejuxtaposed to one

another. The complexityof smaller mounds in theTeotihuacan site share a

fractal coincidence inthe

1·3·6·8·9 layout.

1 3

6

8

9

The square root of The square root of

the reciprocal of seven the reciprocal of seven

142857.142857142857…142857.142857142857…

isis

377.964473009377.964473009


377.964473009doubles to...755.928946018755.9289460181511.857892033023.71578406

Side measurement of the baseline of the Great Pyramidis officially recognized to be 755.79 feet.


5

12

13

10

A perfect right triangle 5A perfect right triangle 5·12·13 with its sides at right angles,·12·13 with its sides at right angles,plus the first side measurement in the following triangle onplus the first side measurement in the following triangle onthe progression of perfect right triangles: 5the progression of perfect right triangles: 5·12·13·10.·12·13·10.

55·12·13·12·131010·24·26·24·262020·48·52·48·524040·96·104·96·104fractal multiplesfractal multiplesetc.etc.

5

12

13

10


A perfect right triangle 5A perfect right triangle 5·12·13 with its sides at right angles,·12·13 with its sides at right angles,plus the first side measurement in the following triangle onplus the first side measurement in the following triangle onthe progression of perfect right triangles: 5the progression of perfect right triangles: 5·12·13·10.·12·13·10.

5

12

13

10

The site of Teotihuacan, Mexico reflects a conscious design in itslayout, as I have shown in previous Earth/matriX essays. Here, wesee how the measuements of a perfect right triangle, 5:12:13 plusits double of five (10) on the next triangle in the progression revealsthe design placement of the monumental structures at Teotihuacan.

512

13

10

A perfect righttriangle 5:12:13with its sides atright angles tothemselves:


A geometric design based on a perfect right triangle

with side measurements of5 ·12·13·(10).


Placement of the 5·12·13·10 geometric design coincides with everyaspect of the geometric placement of the Teotihuacan mounds and pyramids.

The Giza Complex

The same geometric design of a perfect right triangle

with side measurements of5 ·12·13·(10).

The line of 10 unitsis the number of the sidemeasurement of the next

perfect right triangle doubledon the fractal progression of

triangles.5 ·12·13·(10)10·24·26·(20)

TheThe Giza Giza Complex Complex

TheThe Giza Giza Complex Complex

1-6-8-10 1-6-8-10

1-6-8-10 1-6-8-10

Fractal Proportion Giza Complex

The Sphinx

The Giza Complex

The Giza Complex

Placement of the Sun Pyramidover the Sphinx at GizaAlignment of small juttingbar on back wall of the SunPyramid to the small pyramidat Giza.

Placement of Giza XXXX Pyramid over the Sun Pyramid in Mexico

Khufu

Moon

CiudadelaSun

Mid-point

Khufu

Fractal Proportion

The Sphinx

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