Mayan Geometry and 5,12,13 Triangle
By Stan Cizek, August 15, 2022
Abstract
Once you commit to express your ideas and knowledge in terms of architecture and geometric shapes,
you are also committed to follow a basic set of rules governing the understanding of geometry and
interpretation of geometric shapes regardless of whether you are an ancient or present-day observer.
I have collected a number of explicit examples reflecting the application of the 5,12,13 triangle as the
key element of Mayan structural design and present them in this paper for consideration.
Introduction
The 5,12, 13 triangle definition is based on shape partitioning and counting method, and does not
depend on any known unit of measure system. However, some of the examples refer to actual structural
dimensions in metric meters.
The emphasis of this paper is using the 5 x 12 rectangle or two 5 x 12 x 13 triangles as a tool to match to
a specific set of artifacts. The associated mathematical implications are noted but not addressed here.
The basic set of this tool is described in the following example:
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The first example represents the triangle/rectangle set with dimensions 5,12,13 where the unit
of measure is equal to non-dimensional one.
The second example is based on an analysis of the Chichen Itza Great Ball Court footprint with
dimensions 70,168, 182 in meters.
The third example is subset of the Chichen Itza Great Ball Court footprint defining the 4
quadrants as 35, 84, 91 each in meters.
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Architectural structures examples
I base my work only on documents available on the Internet on-line format. The Chichen Itza Great Ball
Court structure was analyzed first because of the availability of the Carnegie map [1].
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Chichen Itza Great Ball Court
The rectangle tool 5 x 12 will correctly overlay the structure on the map. In addition, the El Castillo
Pyramid shape can be overlayed by two of these rectangles as indicated in the picture. It also illustrates
that the rectangle can be scaled to accommodate patterns of different dimensions. The measurements
can be compared to the map’s external source:
“Archeologists have identified thirteen ballcourts for playing the Mesoamerican ballgame in Chichen
Itza,[53] but the Great Ball Court about 150 meters (490 ft) to the north-west of the Castillo is by far the
most impressive. It is the largest and best preserved ball court in ancient Mesoamerica.[44] It
measures 168 by 70 meters (551 by 230 ft).[54 “[2].
In summary, there are 49 5 x 12 rectangles (98 triangles 5x12x13) in a quadrant 84 x 35. The 5x12x13 is
also a unique triangle where perimeter = area, in this case perimeter = 30 = area.
Is this a coincidence that this unique form of triangle has been selected to be the Great Ball Court
footprint or does it represent a standard method applied in general?
At this time, I don’t have an access to any maps that would allow a second match to be demonstrated.
However, there is a drawing of an E-Group in Uaxactun that has been considered to be an astronomical
observatory since it was published.
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Uxcaltun E-Group Observatory
Uaxactun E Group (Ricketson & Ricketson 1937:figure 68) [3]
The 5 x 12 rectangle closely matches the structural layout on the platform. It also explicitly defines the
observation point (intersection of the two purple lines) which is stela 19. Considering Occam’s razor,
that is a superior solution to the currently prevailing definition of the observation point associated with
structure VII. It also provides a reasonable explanation for the jagged lines where the position of the
Sun should be observed. In principle this defines the minimum and maximum position where the Sun
would supposedly be observed on the given dates. This is something that should be considered by
today’s astronomers.
The next group artefacts to be compared to this triangle/rectangle geometrical pattern are examples of
well-known stone panels. The first one is the Pacal’s sarcophagus lid in Palenque; both the drawing and
the actual stone artefact. The second one is a stone panel featuring ruler K’an Joy Chitam in Palenque.
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Panels Examples
The first key observation is the stacking/folding format of the 5 x 12 scaled rectangle within the frame.
The second is the concept of counting the half panels in addition to the full pattern count. It should be
noted that the outer dimension is not following the same pattern. However, the double lines in the
frame are generally aligned with each 5 x 12 rectangle.
On the Internet I also found additional stone artifact images that the rectangle method could be applied
to. The matching accuracy is fascinating. In the two examples on below, four 5 x12 rectangles are
stacked up along the long edge of the artifact and the long edge of the rectangle is aligned with the
bottom of the artifact. The bottom left example is matching one 5 x12 rectangle standing on the short
edge.
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Stelae Examples
Examples of Stelae on this page demonstrate two panels matching two 5 x 12 rectangles, one on top of
the other, with the short edge at the bottom.
The following two examples demonstrate the combination of five 5 x 12 rectangles with the long edge of
rectangle aligned with the bottom.
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Examples on this page are showing the following stelae and triangle/rectangle match:
Copan Stela B
Tikal Stela 31
Quirigua Stela K.2
Quirigua Stela D
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Dresden Codex Pages Examples
The original copy of the Dresden Codex sustained heavy water damage during WW2. To obtain correct
measurements based only on this copy is rather difficult. Thus, I also used facsimiles of this document
that were made prior to the water damage. I used the Gates facsimile version as the primary document
for this analysis.
The method of collecting measurements was based on scaling and matching the geometric rectangles to
selected pages of the Dresden Codex. Microsoft Visio software [6] was utilized to execute these
operations of scaling and matching in a controlled manner. Outside of these two processes, there was
no individual side stretching or shrinking or otherwise distorting the measurements.
Using the predetermined rectangle geometric shape also allowed for automatic error detection /
correction where there was substantial damage to the upper portion of the original facsimile pages.
Dresden Codex Analysis
The Kingsborough version pages were used first to establish an approximate pattern as shown
in Picture 1.
Picture 1: Kingsborough version of the Dresden Codex
The pattern in these three pages was scaled and matched against the grid of graph paper which
indicates that visually the format is a 5 x 12 rectangle or two 5,12,13 triangles. The rectangle’s perimeter
is in blue and the area is highlighted in red. I also checked arithmetically that the long edge to the short
edge ratio is always 2.4. The additional subdivision within the rectangle to 2,3,4 parts also provided a
very close match.
The next step involved applying the same process to the Gates facsimile version and the damaged
Forstemann facsimile version for comparison, as shown in Picture 2. The Gates version borders were
clearly defined in red and the matching process with the geometric rectangle pattern in blue was a
simple task confirming that it is indeed a 5x12 rectangle as defined by red boarders.
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The complicated part was to match the Forstermann version damaged pages. I had to select pages
where at least two bottom corners of the border in red were identifiable in addition to some remnants
of red border along the long edge. Once I aligned the inner corners of the bottom edge, the geometric
rectangle would match the page format with a reasonable degree of accuracy. As a note; the red border
area measured here is not identical to the size of the page from one folding line to the next.
Picture 2: Forstemann version and Gates version comparison.
Conclusion
Are all these examples of simple pattern observations the result of a coincidence or is this intentional
repeating pattern?
This method matching the known geometrical pattern to actual artefacts should be one way of
determining the unit of measure and its definition used in Mesoamerica. The folding/stocking pattern of
the core triangle/rectangle is a strong indication of a well-defined broader metrology system.
Developing a simple classification system using present-day technology could help to advance our
knowledge of ancient Mesoamerica architectural concepts.
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Reference Index
[1] Chichen Itza Resources: Carnegie Map (mesoweb.com)
[2] Chichen Itza - Wikipedia
[3] 35 (famsi.org)
[3] AChase-et-al.-2017-intro.pdf (caracol.org)
[4] Dresden Codex Format and Metrology | Stan Cizek - Academia.edu
[5] FAMSI - Maya Codices - The Dresden Codex
[6] Microsoft Visio 2021 Standard | MyChoiceSoftware.com
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