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Sums of Infinite Series
~ Baravelle Spirals ~


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Deposit # 69


Robert Lai




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.



t0 = 1
t1 =  number of inscribed
triangles after the first iteration
Series
Triangle spiral
Four triangles4 triangles
Equation for triangles
Square spiral
Eight triangles8 triangles
Equation for triangles
Pentagon spiral
Pentagon spiral 
Equation for pentagon
Hexagon spiral
24 triangles24 triangles
Equation for 24 triangles



a  and  r  values for  n-gon  Baravelle Spirals
n a r
3
1/4  =  0.25
1/4  =  0.25
4
1/8  =  0.125
1/2  =  0.5
5
Equation pentagon
Equation for r
6
1/24  =  0.042
3/4 = 0.75
General
Formulas
General formula
General formula for r

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  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. 


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.



References
Choppin, Jeffrey M.  "Spiral through Recursion."  Mathematics Teacher  87 (October, 1994), pp. 504-8.
Stewart, James.  Calculus, 5th ed, THOMSON Brooks/Cole,  2003, p. 751.
Venters, Diana and Elaine Krajenke Ellison.  Mathematical Quilts:  No Sewing Required.  Key Curriculum Press, 1999.
This link is to NCB Deposit #51 and has other illustrations from their wonderful book.
Wanko, Jeffrey J.  "Discovering Relationships Involving Baravelle Spirals."  99 (February, 2006), pp. 394-400.

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JAVA  applet
contributed by


Robert Lai
oakeymini@gmail.com

2006.