By **Ashok Khosla**

THE ASCHER PROPERTIES PAGE

IS A WORK IN PROGRESS

AS ARE ITS CHILDREN

I have mentioned briefly mention the Aschers, the husband and wife team of Robert and Marcia, who wrote **The Code of the Quipu**, in the Introduction page. Manuel Medrano suggested to me that Marcia Ascher's observations be investigated in depth - This is the "simplest" (only mathematicians would say that LOL) kind of khipu analysis - being mostly analysis of sums and differences. Consequently it's a great place to start in reviewing the literature and learning more about khipu.

A little about Marcia Ascher, the mathematician is due. An excerpt from Wikipedia:

Marcia AscherFrom Wikipedia

Marcia Alper Ascher (April 1935 – August 10, 2013) was an American mathematician, and a leader and pioneer in ethnomathematics.[1] She was a professor emerita of mathematics at Ithaca College.

Ascher was born in New York City, the daughter of a glazier and a secretary. She graduated from Queens College, City University of New York in 1956, and married Robert Ascher, an anthropologist graduating from Queens College in the same year. They both became graduate students at the University of California, Los Angeles; she completed a master's degree in 1960, and moved with her husband to Ithaca, New York, where he had found a faculty position at Cornell University. She joined the mathematics department at Ithaca college in 1960, as one of the founders of the department. She retired as full professor emerita in 1995. She died on August 10, 2013.

BOOKS

With her husband, Ascher co-authored the bookCode of the Quipu: A Study in Media, mathematics, and Culture(University of Michigan Press, 1981); it was republished in 1997 by Dover Books asMathematics of the Incas: Code of the Quipu. She was also the sole author of two more books on ethnomathematics,Ethnomathematics: A Multicultural View of Mathematical Ideas(Brooks/Cole, 1991) andMathematics Elsewhere: An Exploration of Ideas across Cultures(Princeton University Press, 2002). The Basic Library List Committee of the Mathematical Association of America has recommended the inclusion of all three books in undergraduate mathematics libraries.Mathematics Elsewherewon an honorable mention in the 2002 PROSE Awards in the mathematics and statistics category.

The first step was to transcribe and typeset the microfilm of the Ascher databooks as notes on each khipu page, for each khipu that has Marcia Ascher's notes, for the khipus that I was able to recover from the Harvard KDB. Using her notes, I have annotated 220 khipu, slightly less than forty percent of the recovered khipus.

Marcia Ascher has noted many interesting relationships in cord clusters (she succinctly calls them "groups"). From her transcribed notes, I've extracted and summarized any interesting relationship equations that she has discovered and written about.</a> There are 61 such "Ascher relationships" regarding a khipu's cords and clusters. We can review each of them and see how they generalize, distribute, and manifest themselves over the entire Khipu Database.

For each of the 61 relationships, the strategy will be as follows:

- Write a suitable search test for the given relationship. Like any search algorithm, there is a tradeoff between
**recall**(all clusters that vaguely specify the search test) and**precision**(clusters that are clearly interesting to us.) - Does investigating the relationship make sense? Does it seem to occur enough times?
- Build an image-quilt of the khipus sorted by some criteria to do a visual inspection of the resulting search.
- Graphically investigate cords and clusters as appropriate for qualifying cords and clusters.
- Try to arrive at some brief conclusions.

Let's go through each of her properties/relationships, suggested by Marcia Ascher, and see where they occur and what we can learn from their distribution.

AS020 has an intriguing property. Its two cord clusters are arranged in descending sorted order. That is: P_{i} > P_{i+1} for all the cords in a cluster. You can see the obvious mnemonic help here. By organizing cords by largest to smallest, it makes it easier for the reader to recite the khipu.

There is an occasional hiccup, where one cord is slightly less than it's neighbor, but the concept still holds, in general. We can do a statistical test for this - Do a least-squares line-fit to the data. If the line's slope is negative, it's decreasing. If the residuals of the fit are within a given tolerance, it's decreasing linearly.

The detailed analysis is here, and an image-quilt of qualifying khipu is here.

- Decreasing cord value clusters occur in roughly 1/7th (14%) of the khipus (67 of 511 khipus).
- Of the khipus that they are on, they occur roughly on 1/3 of the clusters.
- They occur slightly more frequently on the right half of the khipu (analyzed by a moment arm analogy).
- They frequently occur in groups of 2 or 3 clusters.
- Their mean benford match signature (.707) indicates that they are largely of an "accounting" nature.
- A few khipu bear closer inspection - these include UR208, UR113, UR231, UR068, and AS212

This image summarizes the analysis.

View an interactive image here.

AS029 has a property that relates adjacent pendants and their subsidiary to their neighbor.

In group 1, wherever the pendant value is larger than its subsidiary's value, the difference is the value of another pendant in the group. That is:

\[P_{i}-P_{i,s1}=P_{j}\;\;for\;(i,j)\;=\;(5,4),\;(7,8),\;(9,10),\;(10,4)\]

By contrast, in group 8, whenever the pendant value is smaller than its subsidiary' s value, the difference is the value of another pendant in the group. Namely,

\[P_{i,s1}-P_{i}=P_{j}\;\;for\;(i,j)\;=\;(1,2),\;(2,1),\;(4,1)\]

The detailed analysis is here, and an image-quilt of qualifying khipu is here.

The pendant_subsidiary_neighbor relationship seems to be statistically "accidental". Occuring 1.23% of the time, and even less, when the pendant-subsidiary difference is >= 3 (0.6%), I'm inclined to write off this relationship as a statistical fluke, masquerading as a relationship, and not an intentional writing/recitation design strategy. A good decipherer never says "never." They say "maybe."
**Maybe** there's something here, but it's light is shining pretty dimly.

Next on the Ascher relationsips list is the fact that some clusters appear to be sums of other clusters in the khipu. The excerpt from Marcia Ascher describes her notes for AS038:

Groups 1 and 2 are sum-groups of groups 3-6.

Values in group 2 are the sums of values in the corresponding positions in groups 4, 5, 6., for the first 12 positions.

Then for one position, one value is not included and so the next 5 sums involve values in 2 corresponding positions and one neighboring position. That is:\[P_{2i}=P_{4i}+P_{5i}+P_{6i}\;for\;i=(1,2,3...,12) \]\[P_{2,13}=P_{5,13}+P_{6,13} \]\[P_{2,i}=P_{4,i-1}+P_{5i}+P_{6i}\;for\;i=(14,15,...,18) \]

Values in group 1 are the sums of values in corresponding positions in groups 2 and 3:\[P_{1i}=P_{2i}+P_{3i}\;for\;i=(1,2,3...,18) \]Since group 2 sums groups 4, 5, 6, these could alternately be interpreted as the sums of values in groups 3, 4, 5, 6.

Subsidiaries on pendants in group 2 sum the subsidiaries, color by color, on the corresponding pendants in groups 3, 4, 5, 6.

Subsidiaries with color RL: GG, B:W, and BD:W are not summed. Thus:\[P_{2,i°sub_j}=P_{3,i°sub_j}+P_{4,i°sub_j}+P_{5,i°sub_j}+P_{6,i°sub_j}\;for\;i=(1,2,3...,18)\;\;\;j=(B,W,B,W,GG,RL,BD,BS) \]

The detailed analysis is here, and an image-quilt of qualifying khipu is here.

TBD...