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.
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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 |
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Houses
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7
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Cats
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49
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1
3801
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Mice
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343
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2
5602
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Sheaves
(of wheat ?)
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2401
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4
11,204
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Hekats
(measurers of grain)
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16,807
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| Total
19,607 |
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Total
19,607
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Note: 1 + 2 + 4 = 7
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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.
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