Deriv; A brickwork lion on the ancient Babylonian Ishtar’s Gate and Pythagorean Proof

Ancient Babylonian use of the Pythagorean Theorem and its Three Dimensions


Very much like today, the Old Babylonians—20th to 16th centuries BC—had the need to understand and use what is now called the Pythagoras' (or Pythagorean) theorem. They applied it in very practical problems such as to determine how the height of a cane leaning against a wall changes with its inclination. This sounds trivial, but it was one of the most important problems studied at the time.

A remarkable Old Babylonian clay tablet, commonly referred to as Plimpton 322, was found to store combinations of three positive integers that satisfy Pythagoras' theorem. Today we call them primitive Pythagorean triples where the term primitive implies that the side lengths share no common divisor.

Old Babylonian clay tablet (known as Plimpton 322) stores combination of primitive Pythagorean triples

Old Babylonian clay tablet (known as Plimpton 322) stores combination of primitive Pythagorean triples: (above) photo of original and (below) translated.

Babylonian clay tablet (known as Plimpton 322) stores combination of primitive Pythagorean triples: Translated

Why was the tablet built?

Unlike what one may imagine, the reason behind the tablet was not an interest in the number-theoretical question, but rather the need to find data for a ‘solvable’ mathematical problem. It is even believed that this tablet was a ‘teacher's aide’ for setting up and solving problems involving right triangles. This sounds like an environment not so different from our classrooms today.

How is this related to us?

As human beings we share the same nature as the Old Babylonians - in solving problems to live and evolve. The problems nowadays are normally more exotic and elaborate than a cane against a wall, but they share the same legacy. Right-angles are everywhere, whether it is a building, a table, a graph with axes, or the atomic structure of a crystal. While these are our contemporary challenges, we, like the Babylonians, strive to deepen our understanding of the Pythagoras' theorem, and on the various triples that generate these useful right-angles for our everyday practical applications.

Spherical trigonometry: Three right angles inside a triangle on a sphere

Spherical trigonometry: Three right angles inside a triangle on a sphere (Public Domain)

Were the Babylonians so different from us?

Babylonians may have used algorithms to compute side lengths of right-angled triangles into areas, and vice versa, similar to our contemporary numerical methods of analysis. These areas were farming fields, while the side lengths were canals for irrigation. Maybe the canals were structured to distribute a certain amount of water per canal.

Nowadays, in a very similar manner, computers are used to find the distribution of properties (e.g., stress, deflection, etc.) along a material (e.g., a metal beam), or even the displacement of fluids through volumes (i.e., computational fluid dynamics). 

To find a solution to a problem, analytical solutions are often not available. Hence, numerical methods are employed. These consist of splitting the volume that is being analyzed (say the material of a beam, or the air in a room) into small elements (typically Platonic solids like prisms or tetrahedral). It is interesting to think that this so-called “meshing” in the engineering world, or splitting a calculation into small portions, was already applied by the old Babylonians.

Overall, one could say that the tendency to split a problem into pieces, and solve them individually to find the answer, is a human characteristic that we share with the people of Babylon. From this perspective, the Babylonians were not so different from us.

Babylonian problems employed calculations similar to our contemporary numerical challenges

Babylonian problems employed calculations similar to our contemporary numerical challenges

Did the Babylonians know the Pythagoras’ Theorem?

Some argue that scribes in Old Babylonian period knew Pythagoras’ theorem 1,000 years before he did [2]. The most famous tablets here – one showing a square with two diagonals, and Plimpton 322 containing a table of numerical symbols – suggests that the Babylonians knew at least some of the consequences of the theorem [3]. Whether they derived the proof as Pythagoras did, it is unknown.

abylonian tablet bearing a rough sketch of a square and its diagonals, and Pythagoras and his theorem

A Babylonian tablet bearing a rough sketch of a square and its diagonals, and Pythagoras and his theorem

It is clear that the Pythagoras' theorem has greatly helped humankind to evolve. It is universally applicable, but still it is exclusively binding to two dimensions. Since we live in a three dimensional world, the awareness of this gap in knowledge poses the question…

What does the Pythagoras' Theorem look like in three dimensions?

Until recently, this was not known. For the past five years, Luis Teia has conducted a one-man quest to shed light on this mystery. Based on his recent paper, Pythagoras triples explained via central squares published in the 2015 edition of the Australian Mathematics Senior Journal, he derived the proof. It is not philosophically farfetched to say that even the simplest things escape great minds. For example, Teia shows how to hand draw all the Pythagoras’ triples starting from the first (3,4,5) together in one outward evolving infinite spiral. This is something Pythagoras himself could have drawn some 2500 years ago, but the fact is, it eluded him and his followers, until now. Similarly, the three dimensional version of his theorem also eluded him, and is now presented.

Progressive construction of the Pythagoras family of triple as an outward evolving infinite spiral

Progressive construction of the Pythagoras family of triple as an outward evolving infinite spiral

Why is the 3D Pythagorean Theorem important?

Considering the radical technological leap that Pythagoras imparted on mankind with his theorem some 2.5 millennia ago, the implications of this new 3D version can be equally significant today. Perhaps, it leads to an “upgraded” three-dimensional version of our current trigonometry. Who knows?

As for Teia’s next dream, he will be exploring the physical meaning of the three dimensional theorem (ergo the 3D “cane against the wall” problem), and the practical impact it has on contemporary science.

A Pythagorean fractal tree

A Pythagorean fractal tree

Featured image: Deriv; A brickwork lion on the ancient Babylonian Ishtar’s Gate. (CC BY-SA 4.0) and Pythagorean Proof (Public Domain)

Unless otherwise noted, Images courtesy author, [Luis Teia]

By: Luis Teia

References

Friberg, J. (1981). “Methods and traditions of Babylonian mathematics.” Historia Mathematica. Vol. 8, pp. 227-318

CNN online article. “Pythagoras, a math genius? Not by Babylonian standards” by Laura Allsop http://edition.cnn.com/2010/WORLD/meast/12/17/old.babylonian.math/ [cited 05.01.2016]

New York Times online article. “Masters of Math, From Old Babylon” by Edward Rothstein

http://www.nytimes.com/2010/11/27/arts/design/27tablets.html?_r=2 [cited 05.01.2016]

Teia, L. (2015). X3+Y3=Z3: The Proof. Book available at www.amazon.com › X3-Y3-Z3-The-Proof

Teia, L. (2015). “Pythagoras triples explained via central squares.” Australian Senior Mathematical Journal, Vol. 29, No. 1, pp. 7-15

Register to become part of our active community, get updates, receive a monthly newsletter, and enjoy the benefits and rewards of our member point system OR just post your comment below as a Guest.

Top New Stories

(1) Knotted tanned hide bundle before extraction of contents; (2) & (4) gold dinars; (3) signet ring with intaglio; (5) contents of knotted tanned hide bundle.
In mid-September, a large treasure was unearthed during a dig at the Abbey of Cluny, in the French department of Saône-et-Loire: 2,200 silver deniers and oboles, 21 Islamic gold dinars, a signet ring, and other objects made of gold. Never before has such a large cache of silver deniers been discovered. Nor have gold coins from Arab lands, silver deniers, and a signet ring ever been found hoarded together within a single, enclosed complex.

Human Origins

Deriv; Ancient Celtic dolmen from Poulnabrone, Ireland and carved Egyptian deity Thoth
When ancient Egypt and Ireland are spoken about in the same breath it usually results in the rolling of eyes, polite exits and the sound of murmurs citing pseudo-history and new age babble. At least...

Ancient Technology

Grinding stone, Dendera Temple, Egypt.
Most people know of the great construction achievements of the dynastic Egyptians such as the pyramids and temples of the Giza Plateau area as well as the Sphinx. Many books and videos show depictions of vast work forces hewing blocks of stone in the hot desert sun and carefully setting them into place.

Ancient Places

El Caracol Observatory at Chichen Itza (Wright Reading/CC BY-NC 2.0) and Composite 3D laser scan image of El Caracol from above
In 1526, the Spanish conquistador Francisco de Montejo arrived on the Yucatan Peninsula of Mexico and found most of the great Maya cities deeply eroded and unoccupied. Many generations removed from the master builders, engineers, and scientists who conceived and built the cities, the remaining Maya they encountered had degenerated into waring groups who practiced blood rituals and human sacrifice.

Our Mission

At Ancient Origins, we believe that one of the most important fields of knowledge we can pursue as human beings is our beginnings. And while some people may seem content with the story as it stands, our view is that there exists countless mysteries, scientific anomalies and surprising artifacts that have yet to be discovered and explained.

The goal of Ancient Origins is to highlight recent archaeological discoveries, peer-reviewed academic research and evidence, as well as offering alternative viewpoints and explanations of science, archaeology, mythology, religion and history around the globe.

We’re the only Pop Archaeology site combining scientific research with out-of-the-box perspectives.

By bringing together top experts and authors, this archaeology website explores lost civilizations, examines sacred writings, tours ancient places, investigates ancient discoveries and questions mysterious happenings. Our open community is dedicated to digging into the origins of our species on planet earth, and question wherever the discoveries might take us. We seek to retell the story of our beginnings. 

Ancient Image Galleries

View from the Castle Gate (Burgtor). (Public Domain)
Door surrounded by roots of Tetrameles nudiflora in the Khmer temple of Ta Phrom, Angkor temple complex, located today in Cambodia. (CC BY-SA 3.0)
Cable car in the Xihai (West Sea) Grand Canyon (CC BY-SA 4.0)
Next article