The Elamite Formula for The Area of a Regular Heptagon

TL;DR

This paper analyzes an ancient Babylonian formula for the area of a regular heptagon, providing geometric explanations and demonstrating its superior accuracy compared to other historical formulas.

Contribution

It offers a geometric interpretation of the Susa mathematical text's formula and compares its accuracy with other ancient and Greek methods.

Findings

The formula is more accurate than contemporaneous Babylonian formulas.

The approximation surpasses Heron's formula from Greek mathematics.

Possible methods of applying the formula in ancient construction are discussed.

Abstract

In this article, we study the inscription on the reverse of Susa Mathematical Text No.\,2, a clay tablet held in the collection of the Louvre Museum and thought to date from between 1894--1595 BC. We focus on the formula given in this text for the approximate area of a regular heptagon. We give a geometric explanation for the formula and show that this approximation is more accurate than other contemporaneous formulas in Babylonian mathematics and even that of Greek mathematician Heron who proved it almost 1800 years later. We also consider the possible ways the Susa scribes might have applied this formula to construct the regular heptagon for inscription on a clay tablet.