November 11, 2016

The spiral is a curve traced by moving either outward or inward about a fixed point called the pole. A Baravelle Spiral is generated by connecting the midpoints of the successive sides of a regular polygon. Triangles will be formed. The process of identifying and repeatedly connecting the midpoints is called iteration. Mathematically, the Baravelle Spiral is a geometric illustration of a concept basic to the Calculus: The sum of an infinite geometric series - an unbounded set of numbers where each term is related by a common ratio, or multiplier, of "r" - converges to a finite number called a limit when 0 < r < 1. Much time in the Calculus curriculum, and its applications in the sciences, focuses on whether a particular mathematical expression has a limit and thus be highly useful. Historically, one of our oldest mathematical documents, the Rhind Papyrus (ca. 1650 BC), offers a set of data thought to represent a geometric series and possibly an understanding of the formula for finding its sum. In this case, the common ratio of r = 7 is obviously NOT less than 1 and leads to 71+72 +73 + 74 + 75 = 19,607. While not a converging series, as in the case of Baravelle Spirals, we appreciate the early Egyptian fascination with sums of series.