
Re: Ariadne's thread
Posted:
Nov 24, 2011 7:21 AM


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 51213 marks 8 more points on the circumference. Divide the radius into 13 x 13 finer parts and a new triple emerges, 119120169 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, 84520282197 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 65156169 (51213) and 119120169. 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, 119120169, occurs in the Venus Pyramid whose ideal base measures 238240338 feet (119120169 doubled). The circumference of an imaginary circle around the base would measure 1061 feet, according to the above pi value 1601/338.

