Geometry of
The Tablet of Shamash



Michael S. Schneider





This Babylonian Tablet of Shamash was created nearly three millennia ago in the early in the 9th century BC in Sippar, now southern Iraq. Today it resides in the British Museum on display in beautiful condition.

From the British Museum's description: "This stone tablet shows Shamash, the sun-god, seated under an awning and holding the rod and ring, symbols of divine authority. The symbols of the Sun, Moon and Venus are above him with another large sun symbol supported by two divine attendants. On the left is the Babylonian king Nabu-apla-iddina between two interceding deities.

"The cuneiform text describes how the Temple of Shamash at Sippar had fallen into decay and the image of the god had been destroyed. During the reign of Nabu-apla-iddina, however, a terracotta model of the statue was found on the far side of the Euphrates and the king ordered a new image be constructed of gold and lapis lazuli. The text then confirms and extends the privileges of the temple."

Shamash, the sun, along with Sin, the moon, and Ishtar, the life-giving force of the earth, symbolized the three great forces of nature. In particular, the role of Shamash was to oversee justice everywhere the sun's light shone. This is reminiscent of the role of the Egyptian Ma'at, daughter of the solar deity Re, and goddess personifying justice, truth, righteousness and mathematical harmony. To ancient understanding, justice was related to mathematics, weighed out by the different mathematical means (geometric, arithmetic, harmonic, Golden mean) as well as musical intervals and other ratios which demonstrate harmonious relationships to serve as models of balance and justice.

In my 1994 book A Beginner's Guide To Constructing The Universe, in Chapter 5 in a section about the Golden Mean in art, I showed this illustration (below) of the Tablet of Shamash with Golden Ratio calipers dividing the lengths of the side and top at the Golden Mean. (It was actually in 1969 that I first realized this.) Indeed, if we look at the accurate, officially published dimensions of the whole tablet (length 29.210 cm, width 17.780 cm (although I wish they were measured in ancient Babylonian units)), their ratio is 1.642.... This is only a few mm from the ideal Golden Ratio of 1.618... and, depending on where it was measured across its uneven rectangle (and the crack must be considered), this is within the range of tolerance to consider this ratio. But confirmation must come from its consistent appearances in the details.



In his 2003 book "The Golden Ratio: The Story of PHI, the World's Most Astonishing Number", Mario Livio debunked much art which had been claimed to be based on the Golden Ratio. But he graciously called attention to this simple illustration from my book as an ancient example. He mentioned that I wrote that there were many more Golden Ratio relationships in it. So, I'll try to expand on that statement here. First, the points of the calipers indicate the points found in the traditional construciton of the Golden Rectangle (below): a large square at the bottom, with a Golden Rectangle above it which itself has been subdivided into a small square and a smaller Golden Rectangle. We can see more clearly how some elements of the tablet were guided by these lines.



The steps of the geometric construction which produce Golden Rectangles (below) require swinging arcs up from the midpoints of the sides of a square. The whole, as well as the upper parts are Golden Rectangles (yellow). Then, squares are marked out leaving a trail of smaller and smaller Golden Rectangles.



Below (left), the small Golden Rectangle has been subdivided into even smaller squares and Golden Rectangles. Now we see the edge and top of the god's seat. This subdivision is reminiscent of the traditional construction of "whirling squares" and its Golden Ratio Spiral (right).



Being a sacred object, it was designed with great aesthetic and symbolic care. The justice of Shamash was found in the eternal truths of mathematics, which would be found in the tablet's design. The placement of every element of this scene was determined by a symbolic and "magic" formula involving mathematics and mythology. Perhaps the Golden Ratio was chosen for it's self-referential harmony, or its ubiquitous appearance in nature. Below, a yellow square was drawn on the inside right of the upper Golden Rectangle. It runs up one edge of the offering table and tangent to the solar disc. A white arc, equal to the side of the square, maintains this length across it's diagonal corners. Notice how it crosses the red diagonal of the rectangle at the center of the solar disc. Notice also how the blue diagonal of the square skims Shamash's headdress, passes through the sacred circle, and crosses the white arc at the corner of the table. What other diagonals can you imagine? There is more to discover in this scene if you subdivide it into squares, Golden Rectangles and diagonals, which I'll leave for you to explore.



One other aspect of the tablet should be pointed out: the division of the bottom text into three equal columns. This is accomplished by a very easy construction (below) also known around the world (it works for any rectangle, not just the square). It simply involves drawing the square's two diagonals (red), and then "half-diagonal" (black) lines to the midpoints of the top and bottom. Where the red and black lines cross is the natural division into thirds, and vertical white lines have been drawn between them. Upon close inspection, the text is actually separated by a space between two parallel lines. The constructed white lines run up the right line of each pair of lines. These were probably drawn first, and their corresponding left lines were probably drawn a unit distance apart. I've extended the white lines up beyond the text to see if this sacred triple division reveals anything there, and indeed we do see a separation into the group of three visitors, central solar disc plus column (like the god's sacred ring and rod), and the division of the god's garment and ring (except for his toes which dip into the middle realm, as do the hands of the front attendant). Mathematical design was vital in the ancient world, especially in sacred works, and perhaps more than is realized.



Below is an animated version of some aspects of geometric construction.



This geometric exploration was adapted from an activity found in Constructing The Universe Activity Book Volume 3.


If you're interested in purchasing a Shamash Tablet and Golden Spiral puzzle, click on the picture below


If you're interested in purchasing a high quality Shamash Tablet Wall Relief, click on the picture below


Images and text (c) 6/2007 Michael S. Schneider





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