Legend for the Figures
All viewers of this material will
join the National Curve Bank  A
MATH Archive in thanking Robert Lai of CS 491 for developing
this project.

For the
Novice . . . .
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 7^{1}+7^{2} +7^{3} + 7^{4}
+ 7^{5}
= 19,607. While not a converging series, as in the case of
Baravelle
Spirals, we appreciate the early Egyptian fascination with sums of
series.
Rhind
Papyrus Problem # 79 


Houses

7



Cats

49

1
3801


Mice

343

2
5602


Sheaves
(of wheat ?)

2401

4
11,204


Hekats
(measurers of grain)

16,807

Total
19,607 


Total
19,607

Note: 1 + 2 + 4 = 7




Much fame has
been awarded mathematicians, e.g., Euler, Leibniz,
Taylor, Maclaurin, etc., for investigating infinite series. Please
see a streaming video and derivation of the formula for the sum of a
geometric series (NCB # 44) for other illustrations of convergent
series.

