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
