Ancient Maya Geometry

2nd Jan 2023

Mexicolore contributor Edwin Barnhart

We are sincerely grateful to Dr. Edwin Barnart, renowned American archaeologist who has appeared on History Channel, Discovery Channel, and in multiple documentary films, and Director of the Maya Exploration Center, for this intriguing introduction to the wisdom of the ancient Maya - in this instance, relating to their advanced knowledge of geometrical measurements...

It’s been known for centuries that ancient civilizations were using the geometric proportions found in nature to create aesthetically pleasing art and architecture. The single most common proportion found by an observation of the natural world is the Golden Mean, also known as the Golden Ratio, Golden Section, or the Divine Proportion. The Greeks called it Phi. It’s a ratio of 1 to 1.168… and it’s found in things as small as snowflakes and as large as the swirl of our galaxy. The “…” part is used because it’s an irrational proportion, meaning its decimal approximation goes on forever, never arriving to a final number. The better-known irrational proportion found within circles named Pi works the same way – 3.1415926….

In the ancient Greek world these natural, irrational proportions were first discovered by Pythagoras. They flew in the face of everything the Greeks thought they knew about mathematics. Pythagoras knew this, so he kept it a secret known only to him and the members of his math cult. Any member caught revealing their secrets was brought out on a boat and choked to death! Eventually the Greeks calmed down and accepted the beauty of the Golden Mean, using it and the mathematically related square root of 5 to build some of their most famous architecture – like the Parthenon. The Greeks were not the only ancients to use natural proportions. The Egyptian pyramids of Giza use the Golden Mean. Babylonian and Chinese examples are also known. The European Renaissance was rife with the Golden Mean – encoded in cathedral architecture and celebrated in mathematical thought. The famous Fibonacci Sequence is a progress of ascending numbers, each pair reaching toward a more precise approximation of the ratio 1:1.618…

Perhaps no civilization investigated the proportions of nature more thoroughly than the ancient Hindus. They saw not only the Golden Mean, but also the square roots of 2 and 3 expressed in the diagonals within perfect squares and triangles. Their natural expressions are best found in things like flowers and shells (pic 2). Hindu mandalas are essentially religious mantras of natural geometry with squares, triangles, circles, and pentagons all being derived from overlaying one upon another. The Hindus call this process and the larger set of natural proportions “Sacred Geometry”.
Did the ancient civilizations of Mesoamerica also employ this “sacred geometry”? Some scholars dabbled in searching for it, but it was Dr. Christopher Powell who finally cracked the code in the 1990’s. Beginning by analyzing the art and architecture of ancient Maya cities like Chichen Itza and Palenque, he encountered repeated uses of not just the Golden Mean, but also the square roots of 2, 3 and 5. They all fit together via the diagonals of a set of rectangles (pic 3). He brought his initial findings to renowned Mayanist Professor Linda Schele, who was at first skeptical. She told him, “If the ancient Maya were doing it, then the modern Maya might still be doing it. If you find evidence among them, you’ll convince me”. So that’s exactly what Powell did.

For the next few years, he spent time in the Maya communities of Yucatan and Highland Guatemala. There, he learned that modern Maya used cord to measure out their houses. They were unconcerned with units of measure. Instead, they focused on proportions. The master architects of those communities had their own, specially created cords. The lengths were derived in various ways, but always beginning with their own bodies as the root measurement. Sometimes it was as simple as multiples of their outstretched arms. In other cases, it was more complex, like the length of a shadow cast from a vertical pole of their height cast on the solstice or equinox. In all cases the length of their cord – made of simple rope – was based upon their individual height, not some standardized unit of measure.
Employing these cords, they would use them to establish the dimensions of their houses. Distances between wooden support posts would be determined with their cords. Heights of walls and roofs in the same manner. Diagonals of the squares and rectangles they created would be used to swing arcs like a mathematical compass, pinning one end of the cord to the ground, stretching it tight, and swinging the arc like Greek Planar Geometry. Using these methods, they consistently created rectangular spaces with natural proportions, especially the square roots of 2 and 3. In an interview with a house builder near Oxkintok, Yucatan, Powell was told, “My grandfather always said that the shapes of the flowers are in our houses”.

He brought his findings back to Schele and convinced her. He became her graduate student, eventually writing a 475-page dissertation about Maya sacred geometry – both modern and ancient. It’s called The Shapes of Sacred Space: A Proposed System of Geometry Used to Lay Out and Design Maya Art and Architecture and Some Implications Concerning Maya Cosmology.
It’s free for download (follow the link below...)
The proportions nature uses to build the world are primarily the square root of two, the square root of three, and Golden Mean, which is geometrical tied to the square root of five. In rectangular form, they can all be created by beginning with the diagonal of a square. Look at the initial square in the top left corner of picture 5. If its sides are all a unit of 1, then its diagonal (the yellow dashed line) is 1.141…., or the square root of two. Imagine that yellow dashed line is your cord and you’ve stretched it from corner to corner diagonally across the square floor of the house you’re building.

If you take that diagonal line and swing it down to make a new rectangle from the square, it becomes a square root of 2 rectangle, with a ratio of 1:1.141… The diagonal of that new rectangle is 1.732…., the ratio of a square root of 3 rectangle. Do it again and you have the square root of 4 (or two squares side by side). Once more and you make a square root of five rectangle. Finally, look at the lower half of picture 5. A square can be used to make a Golden Mean rectangle by making a diagonal from the center point of its bottom line to a top corner. Swinging the line down and you create a new rectangle with a ratio of 1:1.618… (highlighted in grey). If you do it again on the opposite side of the initial square, you complete a square root of five rectangle. That’s how the Golden Mean and the square root of five are related.

Five examples from within Christopher Powell’s dissertation

1. A modern Maya thatch roofed house in Yucatan, Mexico
This first example is the simplest way that a traditional Yucatec thatch and pole house is built. It begins with a square, each side being the full length of the cord. That is the dimension marked as “1” in picture 5. Note the curving sides of the home (apsidal sides is the architectural term). The curve is created by folding the cord in half, using it to find the mid points of the square sides, and swinging an arc with the cord at that half-length to create the curve. The wall and roof heights are also created by the length of the cord and the 45-degree pitch is the diagonal of that same size square. Units don’t matter, it’s about proportions.

2. The floor plan of a modern Maya house in the Highlands of Guatemala
This is the floor plan of a humble Maya house in the town of Chiquimula. Houses there are very different from Yucatan, but they’re still laid out with a cord. In this example, the entire home fits in a square root of 2 rectangle. Within, each room is a square or square root of 2 with the living areas separated from the patio space in the way the diagonal of a square creates a square root of 2 rectangle. Note the bottom right corner is pulled in to let people step under the roofline before entering the home.

3. Plan and profile of the Temple of the Sun at Palenque
Sacred geometry pervades ancient Maya art and architecture, but nowhere was it combined in more complex ways than Palenque. Palenque’s Temple of the Cross, completed in 692 CE, combines the Golden Mean, the square root of 2 and the square root of 3 all in one building. In plan, its wall thickness simultaneously allows its exterior to be a Golden Mean rectangle and the interior to be a square root of 3 rectangle. Then while the front gallery is combinations of squares and square roots of 2, the back gallery is multiple expressions of the Golden Mean again. The temple’s profile, both from the front and the side, exhibits square root of two proportions. Split down the middle laterally, one gets the Golden Mean again.

4. Lintel 24 from Yaxchilan
The limestone block upon which Yaxchilan Lintel 24 is carved was cut from the quarry in the square root of 2 proportion. Subdividing it in the way that a Maya cord would reveals a possible methodology behind the image’s layout. The square section contains all the glyph blocks and its diagonal mirrors the torch held by Shield Jaguar. The bottom sections define the space between the two figures.

5. Pages 41 and 42 from the Madrid Codex
These two pages use clever combinations of squares to also include Golden Mean rectangles and square root of five rectangles in the same way that a cord on the ground would. A frame of 20 day name glyphs creates the central square. Diagonals from the midpoint of each side sets the width of the pages by creating Golden Mean and square root of 5 rectangles. Note how a Golden Mean rectangle within the central square neatly brackets the two central figures and how the outer frame separates elements within the overall design.

Conclusion

Despite those amazing examples, a nettling question must be asked – did the Maya mean to do this, or did their methods of using a cord produce the natural geometry in the same way the universe does? The Hindus and Greeks explicitly called out their intentional emulations of the natural world, but Maya hieroglyphs make no such claims. The challenge becomes proving Maya intentionality.

Toward that end, consider these two pieces of evidence. The first is a panel on the façade of a building in Uxmal’s Nunnery. The Nunnery is a quadrangle and its western building’s upper façade contains a complex limestone mosaic design with the body of a feathered serpent winding along in front of diamond pattern textile designs. The diamond textile designs are separate rectangular frames. Some have squares in their centers, others have four pedaled flowers within. But in the section where the serpent’s head passed through, its body divides the two types. Above it the diamonds have squares. Below it they have flowers. The message appears to be that the magical body of the serpent – a very cord-like body – converts flowers into squares!

The second piece of evidence comes from the Maya creation story, the Popol Vuh. Within its opening lines, it says:-

“The Makers and Modelers made
the four corners and the four sides.
Then they halved the cord, stretching it up
to the heavens and down to the Earth.”

Look at picture 13 – that’s the way a cord can be used to start with a square to create a Golden Mean rectangle.

Step One – create a square, all sides the same length – making the four sides and the four corners
Step Two - find the center point of the bottom line of the square - halving the cord
Step Three – stretch a diagonal line from that bottom line’s center point up to a top corner - stretching it up to the heavens
Step Four – anchor bottom point of the diagonal line and swing an arc downward with the line until its parallel with the square’s bottom line, hence establishing the length a new 1:1.618… rectangle – the Golden Mean – and down to the earth.

The opening passages of the Popol Vuh state the formula to create a Golden Mean rectangle!