Page 93 - J. C. Turner "History and Science of Knots"

The Peruvian Quipu 83

group, these were divided into three unequal parts which then were encoded
in the first three groups. As the exact result of the calculation is not always
an integer, the numbers had to be rounded in order to be encoded on the
quipu . Therefore we get an error which we state in percent of the number
on the quipu. For the first group this error is about 0.6% for j 0 3 (i.e.
0.33796 < (ply/pry ) < 0.34204) and 1 .4% for j = 3, in the second group 0.7%
for j : 1, and in the third group 0.9%, again for j # 1. That means that the
actual numbers are close to those we must expect when the quipu really is a
division table.
Multiplication of integers by integers and fractions : AS55 and AS56
These are two small quipus with seven and three pendants respectively which
were found together. Though the pendants are not grouped by spacing or
colour, an implicit grouping appears when examining the values. We name
the pendants on the first quipu p, q1, q2i q3, rl, r2, r3 and those on the second
quipu s1, s2, s3. With this notation, we find several attractive relations:

glg2q3 rlr2r3 = SlS283
glr283 = r1s2g3 = s1g2r3
818283 = g1r2s3
qq ' ri+l = sj • sj+l for j = 1, 2, 3

where the addition in the indices is mod 3, e.g. 3+1=1. We even find a more
general pattern, namely

ql = B3Cx r1 = B6C2x s1 = B4Cx
q2 = x r2 = BCx 82 = Cx
q3 = y r3 = B2C2y S3 = B2Cy

where B = 7/8 and C = 34/33. Hence the numbers are multiples of the values
on the third and fourth cord on the larger quipu by fractions. Of course, we
cannot be sure that this is indeed the pattern behind the quipu. But it fits very
well, as the error we make is remarkably small, namely in all cases smaller than
0.4 %, in ten out of the twelve cases even smaller than 0.2 %. For a further
examination of these quipus including the first cord and the three subsidiaries
on the larger quipu, see [6], pp. 149-151.
Geometric interpretation: AS120, AS143 and AS149 The quipus
AS143 and AS149 have the same underlying structure as AS120, which we
considered when dealing with division into unequal parts. AS143 has four and
AS149 five pendant groups. The ratios pig/pry are the numbers bl, b2, c, a and
b, c1, c2, c3, a in the following table, thus they stand in the same order as on