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Re: Ariadne's thread
Posted:
Nov 24, 2011 7:21 AM
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Hypothetical Maya measures of length:
1.69375 centimeters -- 1 finger(breadth)
16 fingers or 27.1 cm -- 1 foot
49 fingers or 83 cm -- 1 rod
The Aztec mapilli 'finger of the hand' measured
1.74 cm, the xocpilli 'foot print' 27.86 cm, and
the yollotli 'heart' 83.59 cm, from the center of
the chest to an outstretched hand, called vara
'rod' by the Spanish (John E. Clark, Aztec
dimensions of holiness, also in The Archaeology
of Measurement, op.cit.).
If the diameter of a circle measures 39 feet or
624 fingers, the circumference measures 40 rods
or 1960 fingers, implicit pi value 245/78 from
the sequence
3/1 (plus 22/7) 25/8 ... 245/78
Draw a cross, and a circle of radius 13 around
its center. The four arms mark 4 points on the
circumference. The triple 5-12-13 marks 8 more
points on the circumference. Divide the radius
into 13 x 13 finer parts and a new triple emerges,
119-120-169 that marks again 8 more points on
the circumference. Divide the radius into 13 x 13
x 13 = 2197 still finer parts and a new triple
appears, 845-2028-2197 that marks again 8
further points on the circumference. And so on.
The four arms of the cross and the triples of
the ever finer radius 13 x 13 x 13 x 13 x 13 ...
mark 4 plus 8 plus 8 and 8 and 8 and 8 ...
rational points on the circumference. If you
combine them with straight lines you get an
ever rounder polygon of 4 12 20 28 36 44 ..
sides. There are always two or three different
lengths of the sides (never more), and they
can be given as whole number multiples of
the square root of 2 and/or 13 and/or 2 x 13.
Let us calculate the second polygon inscribed
in the circle of radius 169. The triples are
65-156-169 (5-12-13) and 119-120-169.
The polygon has 20 sides: 8 sides of 13 times
the square root of 26; 8 sides of 18 times the
square root of 13; and 4 sides of the square
root of 2. Considering that the circumference
of the circle is a little longer than the periphery
of the inscribed polygon, we may use values
for the roots that are a little above the actual
values: 3/2 for the square root of 2 and 29/8
for the square root of 13 and 41/8 for the
square root of 26
8x13x41/8 plus 8x18x29/8 plus 4x3/2
= 533 plus 522 plus 6 = 1061
The rounded periphery divided by the diameter
338 yields 1061/338 = 3.139053... for pi, from
the sequence
41/13 (plus 204/65) 245/78 449/143
653/208 857/273 1061/338
The value 245/78 can be turned into the formula
diameter 39 feet circumference 40 rods.
Another formula involving the foot and the rod
concerns the square: side 3 rods diagonal 13
feet, or side 13 feet diagonal 6 rods. The values
for the square root of 2 are 208/147 and 147/104
respectively, from the number column whose
first lines are (4 3 8) / 7 11 14 / 18 25 36 /
43 61 86 / 104 147 208 / ...
The second triple of the above sequence,
119-120-169, occurs in the Venus Pyramid
whose ideal base measures 238-240-338 feet
(119-120-169 doubled). The circumference
of an imaginary circle around the base would
measure 1061 feet, according to the above
pi value 1601/338.
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