**Maya Calendrical Geometry. (04/05/2001)**

**Part I: To construct an equilateral triangle
on any given finite straight line cut in extreme/ mean ratio.**

**A first problem that must be solved in the creation of a Euclidean
geometry is the construction of an equilateral triangle. Once
that task is successfully accomplished, every other figure and
proof in The Elements can be achieved. That the Maya were
probably able to conceptualize an equilateral triangular space
during the Classic period using only a compass and a straight-edge,
and do so in a fashion that was uniquely their own without recourse
to Euclid's actual methodology, can be inferred from the fact
that such a construction and proof is preeminently possible, if
not simplistically attainable, using the precise geometrical limitations
and paradigms already in place in the figure identified elsewhere
(Maya Calendrical Harmonics)
as a geometrical expression of the 260-day Tzolkin. In
fact, to accomplish the goal of constructing the equilateral triangle
on that figure, it is necessary to add but a single straight line
to the elements already in place. This is true, and demonstrable,
because any straight line which connects two 18 degree angle intersections
with the outer circle (whose radius is equal to the extreme segment
of the extreme/ mean ratio) separated by 8 x 18 degrees of circular
arc (144 degrees) always passes through the circumference of the
inner circle (whose radius is equal to the lesser segment of the
extreme/ mean ratio) at two points which always subtend exactly
60 degrees of circular arc on that circle's circumference. The
chord of that arc, then, by virtue of the fact that things
equal to the same thing are equal to each other, is always
equal to the length of the radius of the inner circle and to the
length of the lesser segment of the extreme/ mean ratio. To repeat
the important fact here: this is always true when inner
and outer circle have radii equal to lesser and extreme segment,
when the circles are concentric, and when half the mean segment
is used to measure the 18 degree intervals of their respective
divisions into 20 equal parts. This is always true as well
without regard to the length of the initial given finite straight
line that is cut in extreme/ mean ratio. The following figure
illustrates this Maya "proof" of the equilateral triangle,
where the area shaded in red is that triangle:**

**It is also true that the exposure of the equilateral triangle
in this figure is a naturally occurring feature of its calendrical
expression because there are only, and exactly, 13 numerical coefficients
in the structure of the tzolkin. What this means is that there
are always seven "empty" places on the intersection
points between the 18 degree angle lines and the circumference
of the inner circle when all thirteen numbers are deployed on
consecutive spaces for the currently articulated day-names. That
configuration in turn leaves 8 x 18 degree angles between the
first and last numbers deployed in every sequence of thirteen
consecutive days. Hence, every straight line connecting the first
and last 18-degree angle mark which holds the first and last number
always subtends a 60 degree arc on the inner circle. In other
words, the equilateral triangle is always exposed by inference,
at least, as each successive day of the tzolkin is counted.
In the figure above, the number 6 (at 6 Cimi) and the number 5
(at 5 Etz'nab) are the first and last numbers and days in this
sequence of 13 days. When the 6 is moved to its next position
adjacent to Cauac, marking the day-name 6 Cauac, the line subtending
the base of the equilateral triangle shifts to its next location
between 6 Cauac and 7 Manik. This revolving structure of equilateral
triangles delineated by the revolving straight line between first
and last marked days as the count moves forward in the tzolkin
is determined by the necessary conditions of the figure itself
and cannot be termed coincidental to it. To suppose the Maya
were unaware of this feature of their 260-day calendar seems disingenuous.
To prove conclusively that they did know, however, will probably
remain elusive. **

**As noted in the previous discussion of the figure, it was not
drawn with a compass and a straight-edge for the purpose of this
illustration and for that reason the configuration here is only
meant to be approximate. Hence, and in order to verify the validity
of the claim being put forward here, it is necessary to construct
this figure using a compass and a straight-edge.**

**A final thought: since the line which cuts the 60 degree arc
on the inner circle always subtends 144 degrees of arc on the
outer circle, where 8 and 12 divisions of 18 degrees each falls
just short of the exact value of the extreme/ mean ratio (with
8 and 13 being the proper division at .6153846), it might be worthwhile
to contemplate the adjustment Maya astronomers made to accommodate
this shortfall when they fixed the length of their calendrical
Baktun interval at 144,000 days. Counting 13 Baktuns
in each Long Count notational system brings everything back to
its proper point of departure in comparing, as it were, degrees
in a circle (360) to days in a Tun (360).**

**Part II: The absence of binary opposition
in calendrical harmonics.**

**An obvious fact based on simple definition is that opposition
cannot be said to exist in any actual articulation of harmonic
constructs. Harmony is the absence of opposition. In Classic period
Maya conceptualizations of space and time, which are always integrated
by virtue of the geometrical figure under scrutiny here, the idea
that overrides the concept of binary opposition is the insertion
of a mirror between one side of the figure and the other. In the
example given above, a second straight line, which connects the
points on the outer circle identified by 3 Cib and 8 Lamat, can
be added to the figure to create the base-line of a second equilateral
triangle whose apex is also located at the center of the two concentric
circles. Since the second line (3 Cib to 8 Lamat) is parallel
to the first line (between 5 Etz'nab and 6 Cimi), where both lines
subtend opposite and equal values of 144 degrees on the same circumference
of the outer circle, and where the line marked by 4 Caban and
7 Manik exactly bisects the opposite and equal 36 degree angles
between 3 Cib and 5 Etz'nab, on one side of the figure, and between
6 Cimi and 8 Lamat, on the other, that line is also parallel to
the two lines that subtend the equal and opposite base-lines of
the two diametrically opposed equilateral triangles. Saying that
the line between 4 Caban and 7 Manik represents a mirror in Classic
period Maya conceptualizations of space and time is doing nothing
more nor less than acknowledging what is only obvious from the
figure itself. This is true because the two equilateral triangles
occupy exactly the same space in the context of the two equal
but opposite 72 degree angles marked between 7 Ahau and 4 Kan,
on one side of the figure, and by 10 Oc and 1 Ix, on the other.
What is equally apparent here is that the lesser angles, shaded
in green, on one side of the equilateral triangles, and the greater
angles, also shaded in green, on the other side, are exactly reversed
in their locations relative the figure as a whole on opposite
sides of the mirror represented by the 4 Caban and 7 Manik line.
That reversal of location demonstrates what is meant by calling
these two equilateral triangles mirror-images of each other. The
following figure demonstrates these facts:**

**Also true here, of course, is that the lines drawn between
7 Ahau and 10 Oc and 5 Chicchan and 2 Men, the ones which quarter
the two circles along their north-south and east-west axis lines,
perform the same function as mirrors in the figure. The areas
shaded in yellow are meant to illustrate this feature of the configuration.
The point to be taken here is that every 18 degree angle line
in the figure, from one diametrically opposed side of the circles
to the other, always functions as a mirror reflecting opposite
halves of the structure. Individual figures that appear in the
structure by virtue of drawing straight lines through consistently
opposed points always produce equal and opposite figures on the
other side of the configuration. Since there is never any variation
in terms of where diametrically opposed points are located relative
to each other from one side of the mirror to the other, except
to note that they are always mirror-images of each other relative
to the whole, it becomes difficult, if not impossible, to quantify
and qualify any significant differentiation between one diametrically
opposed point or figure from its identical twin on the other side
of the mirror. To say, then, that two points or figures in this
structure are binary opposites (good/ evil, same/ other, here/
there, now/ then) ceases to express any significant distinguishing
characteristics in either quantity or quality as those terms are
usually understood in Western religious and philosophical ideology.
This is true in harmonic figures because Good possesses precisely
the same quantification and qualification possessed by its diametrically
opposed, mirror-imaged twin, Evil. This same observation can be
made for every other binary opposition conceived in an harmonic
structure. Since there is no meaningful way to differentiate one
thing from the other, across the replicating line of the mirror,
there is no ground for assigning anything that exists to one classification
or the other. Any such assignment becomes an arbitrary distinction
based on prejudice, as opposed to fact, in the absence of any
meaningful differentiation.**