Until Dr Mansfields investigation, the significance of the tablets was not known. Credit: UNSW SydneyUNSW mathematician, John McKinney, has discovered the origins and use of applied geometry from a 3700-year old clay tablet. The tablet was hidden in plain sight in an Istanbul museum for more than a century.The tablet known as Si.427 was discovered in late 19th-century central Iraq. However, its significance was not revealed until today's UNSW scientist's investigation.Si.427 is believed to be the oldest example of applied geometry. Foundations of Science released a study today that reveals an intriguing human story about land surveying.Lead researcher Dr. Daniel Mansfield, UNSW Science's School of Mathematics and Statistics, says that Si.427 dates back to the Old Babylonian period (1900-19600 BCE).It is the only known example for a cadastral plan from the OB period. This is a plan that surveyors use to define land boundaries. It contains legal and geometric information about a field that has been divided after it was sold.This object is significant because the surveyor uses what are now called "Pythagorean Triples" to create precise right angles.Dr. Mansfield states that "the discovery and analysis the tablet had important implications for mathematics history." This is, for instance, more than a thousand years prior to Pythagoras' birth.Hot on the heels a second world-first discoveryDr. Mansfield speculated in 2017 that a fascinating trigonometric table, Plimpton 322, came from the same period.Credit: University of New South WalesDr. Mansfield says that trigonometry, a branch of mathematics that studies triangles, was developed by the ancient Greeks who studied the night sky in the second Century BCE."But Babylonians invented their own proto-trigonometry to solve problems related not to the sky but to measuring ground."Today's tablet is believed to have existed before Plimpton 322; in fact, Plimpton 322 was likely inspired by surveying problems.There are many right triangles of different shapes. Babylonian surveyors can only use a small number of them. Dr. Mansfield says that Plimpton 322 is a systematic analysis of the zoo in order to find useful shapes."Surveying land with a tablet is the purpose of this tabletIn 2017, the team speculated on the purpose of Plimpton 322, hyping that it could have been used to build palaces and temples, canals, or survey fields.Dr. Mansfield states, "With this tablet, we can see for the very first time why they were so interested in geometry: To lay down precise land borders.""This comes from a time when land is becoming private. People started to think about land as if it were'my land and thy land'. They wanted to create a boundary that would allow for positive neighborly relations. This is exactly what the tablet says. It's a field that is being divided, and new boundaries are created."Even clues from other periods are found on tablets that may contain information about the stories behind these borders.Si.427, a hand tablet that dates back to 1900-1600 BC, was created by an Old Babylonian Surveyor. It is made of clay, and the surveyor used a stylus to write on it. Credit: UNSW SydneyDr. Mansfield explains that another tablet refers to a dispute among Sin-bel-aplia prominent person mentioned on many tablets, including Si.427and wealthy female landowner.""The dispute concerns valuable date palms that border their properties. The local administrator agrees that a surveyor will be sent to the scene to settle the dispute. It's easy to see why accuracy is important when resolving disputes among such powerful people.Dr. Mansfield claims that the way these boundaries were drawn reveals real geometric understanding.Dr. Mansfield states that "nobody anticipated that the Babylonians would use Pythagorean Triples in this manner." It is closer to pure mathematics, inspired more by the practical problems at the time.It's easier to create right angles than it is to doA rectangle with sides 3, 4, and diagonal 5 is a simple way to create a right angle. These numbers make up the 3-4-5 "Pythagorean Triple" and a rectangle that has these dimensions will have mathematically perfect right angles. This was important to ancient surveyors, and it is still used today.Dr. Mansfield states that the ancient surveyors who created Si.427 used more than one Pythagorean Triple, both right triangles and rectangles, to create precise right angles.It is however difficult to work with primes larger than 5 in the Babylonian base 60 number system.Dr. Mansfield states that this raises an important issue. Their unique base 60 number system makes it impossible for some Pythagorean shapes to be used.Si.427, which is pictured here by Dr. Daniel Mansfield, Istanbul Archaeological Museum is believed to be the oldest example of applied geometry. Credit: UNSW Sydney"It appears that Plimpton 322's author went through all of these Pythagorean forms to find the useful ones."Proto-trigonometry is a deep and numerical understanding of how rectangles can be used in practical applications. However, it is totally different from modern trigonometry that involves sin, cos and tan.Hunting down Si.427Dr. Mansfield was first introduced to Si.427 by reading about it in excavation records. The tablet was found during the Sippar expedition in 1894 in what is now the Baghdad Province in Iraq."It was difficult to trace the tablet using these records and physically locate it.""Using that piece, I set out to find it, and spoke to many people at Turkish museums and ministries until, in mid-2018, a photo of Si.427 arrived in my inbox.It was then that I discovered it was on display at the museum. It took me months to find the object, so it was really satisfying to be able share this story.Dr. Mansfield will next investigate the other uses that the Babylonians could have made of their proto-trigonometry.There is one more mystery that Dr. Mansfield still hasn't solved: at the bottom of the tablet, it lists the sexagesimal number "25:29" in large font. This means it takes 25 minutes and 29 seconds."I don't know what these numbers mean. It's an absolute mystery to me. I would love to talk to historians and mathematicians if they have any clues as to what these numbers might be trying to tell me!"Continue reading Mathematical mysteries of the ancient Babylonian clay tablet solvedAdditional information: Daniel F. Mansfield (Plimpton 322): A Study in Rectangles and Foundations of Science (2021). Daniel F. Mansfield: Plimpton 322 - A Study of Rectangles (2021). DOI: 10.1007/s10699-021-09806-0
