Back to . . .  Curve Bank Home Calculus Index Curve Bank Index Sums of Infinite Series ~ Baravelle Spirals ~ Deposit # 69 Robert LaiJonathan Sahagun

Legend for the Figures
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t0 = 1
t1 =  number of inscribed
triangles after the first iteration
Series

4 triangles

8 triangles

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 6 1/24  =  0.042 3/4 = 0.75 General Formulas

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.

 JAVA  applet contributed by Robert Lai oakeymini@gmail.com 2006. Javascript update contributed by Jonathan Sahagun jonathansahagun93@gmail.com 2018.