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.
4
triangles
8
triangles
