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## 25Aug17

## Interview: Aussie mathematician cracks code of ancient "Indiana Jones" tablet

The mystery of a famous 3,700-year-old Babylonian clay tablet, once owned by the real "Indiana Jones," has been unlocked by an Australian mathematician, it was revealed Friday.

Dr. Daniel Mansfield's findings from the University of New South Wales in Sydney, is now the oldest example of an accurate trigonometric tablet, beating out Pythagoras and the better known Greek mathematicians by more than 1,000 years.

The tablet was originally discovered in the Ottoman Empire (now modern day Iraq) between 1900-1910 by Edgar Banks and has baffled the world's leading historians and researchers for over a century.

"Edgar Banks is a very famous character in history, he was an American academic, an obtainer of rare antiquities and an adventurer," Mansfield explained to Xinhua.

"He is really the person on which the fictional character Indiana Jones was based, and he even looks like him, he has the same hat."

Shortly after acquiring the tablet, Banks sold it to a well-known collector called George Arthur Plimpton.

Over time the piece became known as Plimpton 322, and is now displayed at the Rare Book and Manuscript Library at Columbia University in New York.

Although many experts from an array of different fields have speculated on what the numbers on the artifact might be, the answers remained a mystery.

"There has been lots of theories as to what they thought it was for," Mansfield said.

"People have suggested it's just a piece of pure mathematics, a list of triangles, a teacher's aid for solving quadratic problems, there are loads of theories about this."

But it was not until 1945, when researchers uncovered the tablet contained "Pythagorean triples," which pointed to a strong possibility the artifact might have be an instrument for trigonometry.

"A Pythagorean triple is a set of three numbers which describe the sides of a right angle triangle, but the suggestions are always in terms of angles," Mansfield said.

"But no one, until I came along, had considered that it could be trigonometry in terms of ratios and that is really the fundamental difference."

Mansfield's journey began while he was searching for materials to use in a class he was teaching.

"I came across this picture of Plimpton 322 and I thought, that's interesting, I wonder what that is maybe I can use it?" he recalled.

The two-year process to unlock the mystery began where it had done for most mathematicians who looked into it, Mansfield also believed the artifact was indeed an instrument of trigonometry, but not in its modern understanding.

"I have a colleague called Norman Wildberger who wrote a book on how to do trigonometry without angles."

"I came to Norman and said check out this tablet, I think it's trigonometry without angles and what do you think of it?

"He said it's definitely trigonometry!"

The Sumerian cuneiform script written on the tablet holds four columns and 15 rows that describe a sequence of 15 right-angle triangles, which decrease in inclination.

But unlike modern base 10 systems that are the standardly used in mathematics today, the Babylonians worked with a sexagesimal, or base 60 system.

This lead Mansfield to an idea that had been explored in previous studies, which suggested the left-hand edge of the tablet was broken off and originally had six columns completed with 38 rows.

The Babylonians had developed a measurement for steepness, that modern mathematicians would refer to as a gradient or an angle.

But the Babylonians did not think in the same way, the ancient civilisation measured steepness in terms of how much is was in length, not angle.

For example, a shallow ramp would need to be long in length and a steep ramp would be short in length.

In modern translation, the Babylonians understood, "the longer the length of a ramp, the more the side of triangle would need to eat," Mansfield said.

"How much it eats per unit depth is literally there measurement of steepness."

Although it is not understood what the tablet was actually used for, Mansfield suspects it was likely an implement for surveying, but admits the window into this period of history is so small that it is very difficult to know.

"To me the significance is really about realizing that mathematics isn't just a pure science, it also has a large cultural component to it, that's really just invisible to us," Mansfield said.

"The whole world learns trigonometry in this one single minded way, trigonometry is about angles and to me the contribution here is to show us that, that is a culturally-determined position."

"You don't need to study trigonometry exclusively from angles and in fact, there is a pretty good alternative to it based only on ratios."

## [Source: By Levi J Parsons, Xinhua, Sydney, 25Aug17]

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