Abstract Series Num. 1 - 9

The maya 360c calendarical system may be achieved through addition and the method of duplatio, without multiplication.

Tables of numbers do exist within the maya culture in ancient Mexico. One has only to examine the Dresden Codex which has been interpreted as a registration of the movement of Venus and corresponding eclipses. Although we may never find exact example of the Tables of Numbers cited herein, the tables reflect the procedure implied in the 18 x 20 and 13 x 20 calendar reckoning. In fact, the 260c and the 360c calendar reckonings are themselves already the product of related calculations.

Multiplication is ultimately a form of addition. The Tables of Numbers, as illustrated in this essay, through their method of calculation, do not require a multiplication procedure; simple addition is sufficient to effect the calculations. The counting of time-cycles seems to take for granted a system of counting; something that is often overlooked. Furthermore, the ancient concept of time-cycles and the comparison of whole cycles, becomes more profounds we contemplate the relationship of one year to that of millions of days. Such a comparison, in itself, reflects a profound degree of human consciousness, that we still have difficulty in comprehending ourselves. In other words, if the ancient peoples represented numerically millions of days, then they consciously thought in terms of thousands of years.

The pointer of the Aztec Calendar functions as an indicator of the first and last days of the 260c and 365c day-count calendars.

The pointer of the Aztec Calendar and the day-glyph concentric ring therein are intimately related through design and allow for rapid, symbolic calculations. Again, the calculations may be made as of the relationships of the rings, or mentally executed without any reference to the rings themselves. Significantly, there is an apparent design and pattern behind the arrangement of space within the Aztec Calendar based on a counting system for the day-glyphs yet to be comprehended fully.

The day-glyph ring of the Aztec Calendar reveals a design that reflects the quincunx sign; a symbol for the planet Venus.

In this essay we shall examine the patterns within the arrangement of the day-glyphs around the pointer of the Aztec Calendar. The patterns obtain by connecting lines among specific day-glyphs on the concentric ring, and fall within the realm of the pointer.

The design of the Aztec Calendar appears to obey a scientific set of rules that fall within its counting system that is interrelated to artistic expression. The placement of space does not appear to be random, but reflects deep thought and consideration where possibly nothing was left to chance. The measurements, the lines, the angles, their significance, follow rules that are constantly conveying new information about the time-cycle and their relationships to one another.

The Aztec Calendar may be divided according to five and seven segments.

In this essay we shall examine the perceived divisions of space created by the concentric circles (or rings) of the Aztec Calendar. The complexity of the Aztec Calendar's design stands out immediately.

One may view the calendar's component elements from different perspective. The rings may be, and have been, grouped together in different ways as the following drawing shows. The rings of the Aztec Calendar have been ordered in many other ways. In a later essay we shall analyze some of those distinct possibilities in an attempt to comprehend their meaning. However, in this essay, we are concerned with the way in which space is divided in the overall design of the Aztec Calendar. The precise meaning of the rings and their different elements is unknown. In order to be able to discern their meaning, we must first examine the logic of the general design of the Aztec Calendar, and for that reason we have chosen the theme of spatial division.

The Aztec Calendar's spatial divisions reflect the day-counts.

In an earlier essay we have shown the examples of the 260-day count (260c) and the 360-day count (360c) on the Aztec Calendar. Now, let us look briefly at other possibilities regarding the 365c, the 584c, and the 780c.

Different methods for calculating the "squaring of the circle" are here in illustrated in relation to the pyramids of Giza and Teotihuacan.

The author attempts to illustrate distinct possibilities for squaring the circle with straightedge and compass in terms of the equivalency of the area and the perimeter/circumference of a square and circle respectively. The method and procedure employed may have served as a similar way in which computations of proportion could have been achieved in the ancient artwork without employing mathematics, while using a floating system of measurements. Angles and numbers that are relevant to measurements of different ancient sites are employed in order to illustrate specific relationships and solutions of computation. The relation of the 4 x 5 rectangle is also explored in terms of the squaring of the circle.

A comparative analysis of the Aztec Calendar and the Floorplan of the pyramids of Teotihuacan, which reveal a common design and distinct patterns based on astronomical knowledge.

Since I was very young, it has always intrigued me as to why the ancient peoples of Mexico employed numbers such as 52, 104, 520, 1040, etc., to identify time-cycles. A possible answer may lie in the mathematical reasoning that I have attempted to explain in this essay.

It would appear that those numbers reflect significant percentages of the Great Sun Cycle of 26,000 years, or the Precession of the Equinoxes. A 52-year cycle, for example, represents 1/500th of a Great Sun Cycle. However, the reasoning behind this is far more elaborate in nature.

The ancient peoples discerned such a relationship through the orbital day-counts of Venus and Earth. They appear to have related earth's 365 day-count to a 52-year cycle, and Venus' 584 day-count to a 65-year cycle. The complete relationship is: 52/365 : 65/584

This relationship produces the 260-count intervals, as may be observed in the charts presented in this essay. Ultimately, the, the 260-count acts as a translation and calculation device between the 365-count and the 584-count. Whole cycles may thereby be obtained between the day-counts of Venus and Earth.

If this is in fact the mathematical reasoning behind the origin of the 260 day-count, then it would appear that the ancient peoples of Mesoamerica:

1) Knew the Precession of the Equinoxes and the Great Sun Cycle of 26,000 years in detail;
2) developed the 260-count calendar as a product of their already knowing and relating Venus' 584-count and Earth's 365-count orbits; and,
3) knew the exact orbital relationship and configuration of the solar system in this respect.

Such findings would maintain that the order and time of the discovery of their calendars may be reversed. It was not a case of them having used the 260-count calendar, and then having "corrected" it to the 365-count of Earth, but actually knew already the 365-count from which they generated the 260c calendar.

Although there are many scientists who do suspect that the ancient astronomers actually knew more than what has been generally attributed to them, the mathematical reasoning explained in this essay may demonstrate that the ancient peoples knew far more, far earlier than what has been suspected until now.

The Temple of Quetzalcoatl (Teotihuacan) is shown possibly to represent numerical sequences that correspond to the 584 day-count of Venus.

This essay presents a theoretical reconstruction of the method of calculation that may possibly have been employed in the relationship between Venus cycles (65c) and the full lunar cycles (64c). For this exercise, we have chosen the quantifiable elements of serpent head figures (Quetzalcoatl), and those which have been identified as the rain god (Tlaloc), on the Temple of Quetzalcoatl at the pyramid site of Teotihuacan, Mexico.

The exact number of figures that originally adorned the temple remains questionable, due to the fact that only a part of the temple was found relatively intact on the wall facing west.

For the purposes of this analysis, we shall work with the seven-level reconstruction.

The exact number of elements on the temple may be incorrectly reconstructed. Nevertheless, the numerical exercise involves a logic of its own. Although a specific monument has been chosen for this analysis, one could actually carry on this discussion around the numerical quantities themselves, without reference to a particular monument. For the sake of a visualization of the numbers, we have selected the Temple of Quetzalcoatl, and distinct day/year/cycle counts. We have selected the 65c of Venus, alongwith the 64/66c, which appears to pertain to the Earth's moon. We shall also be making reference at times to the 52c cycle relating to Earth's 365 day-count. We expect to show that the 64/66c may serve as a repeat pattern of numbers (or cycles) that relates Venus' cycles to the full lunar cycles.