Australian mathematician discovers applied geometry engraved on 3,700-year-old tablet

7th August, 2021.      //   General Interest  // 

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On a 3,700-year-old Babylonian clay tablet, an Australian mathematician found what may be the oldest known example of applied geometry.

Si.427 is a tablet with a field plan that measures the borders of certain land.

The tablet was unearthed in what is now Iraq in the late 19th century and dates from the Old Babylonian era, between 1900 and 1600 BCE. Before Dr. Daniel Mansfield of the University of New South Wales found it down, it was kept at the Istanbul Archaeological Museum.

Another Babylonian tablet had previously been identified by Mansfield and Norman Wildberger, an associate professor at UNSW, as holding the world’s oldest and most precise trigonometric table. They hypothesized at the time that the tablet had a practical application, such as surveying or building.

Plimpton 322, for example, used Pythagorean triples to represent right-angle triangles: three whole integers where the sum of the squares of the first two equals the square of the third – for example, 32 + 42 = 52.

“You don’t just accidentally come up with trigonometry, you’re usually doing something practical,” Mansfield explained. Plimpton 322 inspired him to look for more Pythagorean triples-containing tablets from the same time period, which led him to Si.427.

“Si.427 is about a piece of land that’s being sold,” Mansfield explained. The tablet portrays a field with marshy regions, as well as a threshing floor and adjacent tower, in cuneiform script with its characteristic wedge-shaped indentations.

According to Mansfield, the rectangles showing the field had opposing sides of identical length, implying that surveyors at the time found a technique to construct perpendicular lines more accurately than before.

“Much like we would today, you’ve got private individuals trying to figure out where their land boundaries are, and the surveyor comes out but instead of using a piece of GPS equipment, they use Pythagorean triples.”

Despite the fact that Plimpton 322 and Si.427 both employ Pythagorean triples, they are almost 1,000 years older than the Greek mathematician Pythagoras.

“Once you understand what Pythagorean triples are, your society has reached a particular level of mathematical sophistication,” Mansfield said.

Three Pythagorean triples are found in Si.427: 3, 4, 5; 8, 15, 17; and 5, 12, 13.

Working with prime numbers bigger than five was challenging for the Babylonians because they employed a base 60 number system, which is comparable to how we record time today.

Si.427 was discovered at an era of growing private property ownership, according to a research published in the journal Foundations of Science. “Now that we know what problem the Babylonians were solving, that recolours all the mathematical tablets from this period,” Mansfield said. “You see mathematics being developed to address the needs of the time.”

One aspect of Si.427 that perplexes Mansfield is the sexagesimal figure “25:29” engraved in big letters on the back of the tablet, which is comparable to 25 minutes and 29 seconds.

“Is it part of a calculation that they performed? Is it an area that I haven’t come across yet? Is it a measurement of something?” he stated. “It’s really annoying to me because there’s so much about the tablet that I understand. I’ve given up trying to figure out what that one is.”