The Aztec Calendar Stone

 

When a typical technical and artistic alert person looks at the Aztec Calendar Stone, it appears to be very authentic.  There is a lot of detail and a lot of symmetry but off just enough to appear to be just an artistic carving of volcanic stone, not the softest substance in the world.  And to complicate matters it weighs in at about 25 to 30 tons, not the easiest object to move around, requiring a modern day crane or track hoe.  It was found face down deep in the ground during excavation for some earthquake resistant construction in Mexico City.  Whoever buried it, either wanted to preserve it or denigrate it as a rebuff of any religious significance.  Below is a very nice photo of the stone.

 

Note the 8 pyramid shaped images near the outer area. The pyramids at 12, 3, 6 and 9 o’clock positions have a curl at the base. The other four are straight at the bottom. These are all inviting us to connect with a line.  In order to find a precise, repeatable point on each one, a three point circle was drawn on the image imported into a CAD system. There is no way to measure things with a straight edge or common tape measure.  This image must be viewed in totality to collect any useful information.

 

The rings should contain some type of information describing how the overall layout was designed. They seem to delineate different messages that are categorized.  The above mentioned pyramids that have their base on certain rings.  The 12, 3, 6 and 9 o’clock pyramids have their base on the outer blue ring while the other four are on the cyan ring.  All have their top point on the inner yellow ring.

 

The centers could be thought to be a common point somewhere around the nose which was later erased when the face was carved.  The reality is far from this idea. There is always worry that the heavy stone was not placed properly in the museum or that the camera position was not perpendicular enough.  One thing is certain. Distortion cannot make things more precise and usually the distortion is easily determined by the commonality of the shift in precision.

 

The following image is blown up by a hundred times or more.  The overall height represents just a few inches on the stone. The colored lines are lines drawn from the center of one ring to the center of another.  One can see that the centers vary all over the place, though only a couple inches at full scale.  It is clear that the center of the rings were not important or else purposefully varied to hide that type of analysis. Perhaps it was an effort to make the study a little more complicated.

 

The red lines are lines connecting opposite pyramid tops. Obviously, any two lines should intersect somewhere. But to have all four lines intersect at a common point describes precision to an almost unbelievable level. A different report will explain why the ring centers are not common.  The design wants the researcher to look at perimeters and areas, and not just the radii.

 

Note in the image above that when the software zooms up the screen to see the details of the lines, the background image is at the pixel level and not much idea where the button is located. One simply cannot do detail image analysis along with the detail lines at the same screen.

 

The key to understanding the design of the stone is in the green lines. These are not two intersecting lines but are four lines originating from the intersection center which is not common to any of the ring centers or the pyramid connecting lines.  The small circles in the image below are called buttons in this report.  There are five that form the shape of an upside down open envelope.  They are located on a green circle.  Originally, small circles were drawn on each button, using the most clearly defined circular image at each button because the lighting causes the image to distort the button “roundness”. Sometimes the inner groove is clear and sometimes the outer curvature is clear.

 

The buttons are lettered A,B,C,D,E with F at the center. Even with the initial crude centers on the image, the triangle ABF was drawn and measured with four digits of apparent precision. In trying to find the design intent several angular identities were tried and it was found that the tangent of the angle AFB was near to pi squared divided by 10.

 

This was the first indication that there might be precise design intent. There are five buttons and any three determines a circle with a center somewhere. There are 5 x 4 x 3 / (1 x 2 x3) = 10 combinations and thereby 10 circle centers.  Drawing all these in greatly improved the location of the center of the buttons as some circle centers were obviously eschewed from the initial locations. With the center better located and the center angle set to the arctan(pi^2  /10) = 135.3760013 marked in green at the center top above, the buttons A and B were pretty accurately set. The radius of the circle was determined by the best fit to all five buttons.  Zoomed up, the distance from the button center and the angle lines coming out to the radius could be measured and then the mean radius determined.

 

The dashed green lines were drawn downward from the center to the buttons C and E.  There was a lot of disappointment that this angle did not coincide with any angular identity.  However, the difference between it and the upper green angle was 2.11055 and immediately the hydrogen fine wavelength of 2.1106114 was considered. Knowing this would take the design far away from some crude Mexican Indian mentality, very careful consideration of further development along these lines would be necessary to make it stand up in the eyes of potential critics. At this point, it was just a means to establish some precision when nothing else at the time was obvious.

 

With A, B, C, and E defined with more precision for this model than can be determined from visual aspects, the last button D was connected to buttons C and E and the angle measured at 111.23 degrees.  With due experimental discovery, the relationship to the angle EFC = 137.4866141 was very close to 2/phi. As an experimental trial, the EDC was defined to that value = 111.2290073.

 

In the CAD system, various elements like the above mentioned triangles ABF, EFC and EDC can be grouped and moved slightly and the angles are preserved. This movement is so slight that it does not show up in the visual image. It is becoming a theoretical mathematical model. It remained to determine the balance of the rectangle ABCE knowing that the sum must be 360 degrees and 135.37 and 137.22 were already assumed. The remainder at 87.1373846 degrees had to be split between AFE and BFC which were not equal. Measuring the now more refined button locations, the ratio of AFE to BFC was found to be 11/10 + 135.376 x 2 / 100,000 and so each angle could be found at AFE = 44.120046 and 87.13738 – 44.120046 = 43.0173386 BFC. Without the trial precision, this relationship wouldn’t be found.

 

For this trial to be successful, further relationships would need to be found that confirmed the approach. With now much higher precision, angle AED = 123.5866222 was measured and the difference from 200/phi was .0201755 and the square root was .142040 which is six digits of the hydrogen neutral frequency (reciprocal of the wavelength used initially).  The angle BCD =124.1290684 was determined to four digits to be about 47/90 greater than 200/phi but nothing more precise was found as of this writing. It doesn’t really matter since all the other angles have been locked in place.

 

The inside triangle ADB was measured at 67.67530179 and this was determined to be (135.3760013 – 10/ 2.1106114^(8)) =  67.67530222 with 8 digits of precision which was far beyond anything visual. Numbers raised to the eighth power show up several times in the perimeters and areas of the rings. But there still needs to be some type of overall check point on the theory for the buttons.

 

Earlier, when finding the center of the button green circle, the scale was not known with any great precision.  All the angles do is determine the shape of the five buttons, but do not help with the size of the “envelope shape”. The CAD drawing gave the radius as 1.4536048 and the scale was known to about 4 digits1.725(677376) = 2.508 and one of many trials would be to take the square root several times and shift the decimal one place and since that was .08 above 25 and 5 plays such a role in phi, my great elation when the result of 25.08^(1/8) = 1.4959+. Therefore, the scale was set to the full precision of 1.725677376 using the precise value of 1.495979856 [(1718.75/12)^(1/2) / 8] and also from the product of quantum numbers discussed on several blogs. Sun to earth distance in the 2000 Epoch was 149,598,022.

 

The Cad system has an option to check the perimeter and area of any shape so that was done for the entire envelope of ABCDE and the area was found to be quite precisely the angle BCD / 10 at 12.4128. The perimeter was a simple function of A=1.71828, in this case 1.7183+ but close enough to consider the overall trial to be a success.

 

One might now consider the face with the tongue sticking out.  It is hard for me to imagine that this represents a god since the tongue sticking out would be very disrespectful.  However, while working on this model I kept getting the feeling that the figure was “sticking his tongue out at me personally”. There were many times I wanted to cut the tongue out of the picture. On the other hand, it now seems the tongue was indicating there was a message in the stone. In fact, there is a virtual book of events yet to be discovered in this stone. It is great that the images are all over the internet and the Mexican government, similar to the Egyptians, can’t lock it away. However, it would be good if somebody used a 3-D scanner and controlled the lighting so that not only the image but the depth of the carvings could be accurately measured.

 

The actual carving of these buttons would take no more than a simple piece of plywood bought at the local Home Depot Store…..hmmmm….guess not. But it seems likely that templates of some kind were probably used. In early Sumerian era, hard gemstones were known and used for something and gemstones are widely available in Mexico.  The act of carving would not have been that big of problem. The complex design is something else.

 

In another report the perimeters will be examined in greater detail. For now there is an interesting checksum in the Excel spreadsheet summing up all the perimeters of the rings in the following cell formula:  (LN(SUM(N49:N61))*4)^(1/4) = 2.110768968.  This means the low errors offset the high errors to reach this type of precision. It amounts to 0.2 foot in 143 total perimeter feet. There is a similar good checksum on the area of the rings.

In this rabbit hunt, it is questionable whether the prize is the rabbit or the hunt.  During the labors of analyzing the image so far, many unknown relationships in our measuring systems have been discovered.  Seems like an objective might have been to enhance human development.

 

The intention of this report is to develop interest in other people to start studying the stone with modern computer software and electronic hardware. If this mathematical model is an example, there is a lot to learn from this marvelous stone carving and probably a lot to learn from the pyramids and Teotihuacan.

 

On the blogspot one can go to the upper right corner and click on profile and all the other blogs will be listed.  Happy reading.

 

-bransonjim9 at gmail dot com

Jim Branson

Retired Professional Engineering Manager