One of the classical gems of mathematicsâand to my way of thinking, a pinnacle of human achievement â is the ancient discovery of incommensurable numbers, quantities that cannot be expressed as the ratio of integers.

â2

The Pythagoreans discovered in the fifth century BC that the side and diagonal of a square have no common unit of measure; there is no smaller unit length of which they are both integral multiples; the quantities are incommensurable. If you divide the side of a square into ten units, then the diagonal will be a little more than fourteen of them. If you divide the side into one hundred units, then the diagonal will be a little more than 141; if one thousand, then a little more than 1414. It will never come out exactly. One sees those approximation numbers as the initial digits of the decimal expansion:

The discovery shocked the Pythagoreans. It was downright heretical, in light of their quasi-religious number-mysticism beliefs, which aimed to comprehend all through proportion and ratio, taking numbers as a foundational substance.

According to legend, the man who made the discovery was drowned at sea, perhaps punished by the gods for impiously divulging the irrational.

Excerpt From: Hamkins, Joel David. âProof and the Art of Mathematicsâ Apple Books.

Who was Marcia Ascher?

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

Marcia Ascherfrom Wikipedia Marcia Alper Ascher (April 1935 â 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 masterâs 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 book Code of the Quipu: A Study in Media, mathematics, and Culture (University of Michigan Press, 1981); it was republished in 1997 by Dover Books as Mathematics 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) and Mathematics 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 Elsewhere won an honorable mention in the 2002 PROSE Awards in the mathematics and statistics category.

1. Ascher Summation Relationships

Numerous summation relationships can be harvested from Marcia Ascherâs notes. Examining these summation pattern using the lens of âis it possible to identify this algorithmicallyâ, and then âhow likely does it occur?â, I investigated the following relationships:

Difference relation between the first pendant, itâs subsidiary, and the pendantâs neighbor, where the neighborâs value equals the original pendant minus its subsidiary.

Subsidiary cords that are Sums of Similarly Indexed (by Color Index, not Position Index) subsidiary cords in contiguous groups to the right or left of the cord.

Sum (+)

Ascher

22 (4%)

164

~71% of the KFG khipus exhibit some type of Ascher âsummationâ relationship. Of these 11 Ascher relationships, the following have been selected for analysis.

One additional statistical relationships is explored - white cords that are the first cords of a pendant sum group/group. These are markers of summand cords, and of sum âgroupsâ, respectively. Together, along with cord position, they help the khipu reader to identify where summation relationships occur:

The X-Ray Ascher Sum Relationships Table is a useful visual tool for understanding and cross-correlating all of these relationships. Review the image table of X-ray sum relationships. Sort by Group Group Sums or some other # column (the number column not the image column!). Iâm sure you will find the X-rays reveal fascinating layers of summing.

2. Process:

Step 1 - Transcription of the Ascher Databooks

The first step was to transcribe and typeset the microfilm of Marcia Ascher databooks for the khipus that I was able to recover from the Harvard KDB. Using her notes, I have annotated 220 khipu, slightly less than 40% of the recovered Harvard KDB khipus.

Step 2 - Extracting the âAscher Fieldmarksâ

Marcia Ascher has noted many interesting fieldmarks in cord groups (she succinctly calls them âgroupsâ). From her transcribed notes, Iâve extracted and summarized any interesting equations that she had discovered and written about. There are 61 khipus that contain some âAscher relationshipâ, regarding a khipuâs cords and/or adjacent groups. Many of these notes can be translated into a Fieldmark. As you can see, many of the summation Fieldmarks noted by Marcia Ascher generalize, and distribute over the entire Khipu Field Guide.

Step 3 - Examining the Ascher Fieldmarks

Potential Fieldmarks are extracted from the list of Ascher relationships noted in the 61 khipu. Some khipus have multiple fieldmarks/relationships. In examining the fieldmarks over the khipus the questions that are asked include:

Does investigating the Fieldmark make sense? Does it seem to occur enough times?

Can I write a reasonable fieldmark search test for each important note exhibited in the khipu.

Like any search algorithm, there is a tradeoff between recall (all groups that vaguely specify the search test) and precision (groups that are clearly interesting to us.) For example, two tradeoffs need to be evaluated - one which falsely calls sums such as 1 <= 1+0 which probably isnât a real sum cord (high recall/false positives), and the combinatorial explosion of considering all cords (high precision/false negatives) which can lead to intractable compute times.

What does graphical investigation reveal. What are a fieldmark distributions for qualifying cords and groups.

Are there any tentative conclusions?

3. Evaluating Ascher Sum Relationships:

The lines of investigation include:

How often does this fieldmark occur?

What is there relationship to other fieldmarks, such as banded or seriated groups?

How are the Ascher colors of this fieldmark distributed?

How is the fieldmark distributed across the khipu? Is it right-handed, left-handed, or all over the khipu?

How much of a benford match signature is there for this relationship?

Which khipus bear closer inspection?

How many khipu have some sort of Sum Khipu Relationship?

Of the seven key Ascher Sum Relationships (pendant-pendant sums, indexed-pendant-sums, colored-pendant-sums, subsidiary-pendant-sums, group-sum-bands, group-group-sums, and ascher-decreasing-groups), how many khipus have at least one sum.

Code

import pandas as pdimport utils_loom as uloomimport utils_khipu as ukhipusum_relations_df = pd.read_csv(f"{ukhipu.khipu_fieldmarks_data_directory()}/ascher_sums_overview.csv")num_khipus_with_sums = sum_relations_df[sum_relations_df['num_ascher_sums']>0].shape[0]print(f"Out of {ukhipu.num_kfg_khipus()} khipus, there are {ukhipu.pct_kfg_khipus(num_khipus_with_sums)} khipus with some sort of Ascher relationship")

Out of 654 khipus, there are 464 (70.9%) khipus with some sort of Ascher relationship

So approximately ~71% of khipus in the Khipu Fieldguide have some sort of Ascher Summation relationship. This is not a good indicator for the idea that khipus are âlinguisticâ in nature.

In almost all cases Ascher equations have a sum cord on the left equal to the sum of summand cords on the right. There are exceptions, for example AS164. However, when the search for such sums becomes open-ended over the entire range of khipu two observations present themselves:

3.1 Handedness

The summand cords are both on the right like Ascher suggests, and on the left! The surprising discovery in the exploration of these relationships is that these types of sum have a âhandednessâ with sums pointing to the right (right-handed sums have sum on the left, summands on the right), or sums pointing to the left (left-handed sums have sum on the right, summands on the left).

Look at the back of your hands, with the thumbs sticking out:

A Right-Handed Sum has 10 = 1+2+3+4 with the sum on the right-thumb, and the fingers representing 1,2,3,4. Right-handed sums are conventionally colored as Red (using the R mnemonic - Red/Right-Handed)

A Left-Handed Sum has 1+2+3+4 = 10 with the sum on the left-thumb, and the fingers representing 1,2,3,4. Left-handed sums are conventionally colored as Blue.

3.2 Contiguity

In almost all cases in the ascher equations, sums are contiguous, ie. the sum of cords x through y. To reduce to reduce the search space to tractable compute times, this contiguity constraint is used in finding cords for various ascher sum relationship fieldmarks.

4. Pendant Cord Sum Relationships:

Pendant Cord Sum Relations are the most common sum relationships in khipus, with one or more types of sum relationships occuring in about ÂŒ of all khipus in the Khipu Field Guide.

What is most intriguing is the distribution of these sums. For each of the four types of pendant sum relations, the distribution of right to left handed sums is asymmetrical, and roughly breaks down to a 60/40 ratio. Furthermore, these handed sums gravitate, as you would expect, to the left for right handed sums, and to the right for left-handed sums.

5. Pendant Cord Sum Drawings:

For each of the summation relationships the following drawings are typically offered:

Sum Maps: For pendant sum relationships, a sum map is generated. The sum map shows sum cords on top, as columns, with the column contents being the summand cords. The coloring obeys the same rules as the X-Ray Diagram, so for example in pendant pendant sums, groups of sum cords will show up as blocks of color. The farther from the diagonal a color sum is, the more handed and distant it is. Summand cords that are close to their sum cord are close to the diagonal.

Summation X-Rays: For most sum relationships, a Summation X-Ray is generated. This shows a more detailed view than a sum map, and shows the colors that are being summed, their values, their group/band relationships, etc. Additionally, when useful, the Summation X-Rays show the longest sum path, i.e. sums of sums of sums etc. for the entire khipu.

Detailed Schemas: For each relationship a detailed relation schema is shown.

Pendant cords that are Sums of a set of Contiguous Pendant cords, regardless of color, or parent cord group.

Imagine youâre a khipukamayuq who gets a cord from âlowerâ in the hierarchy. How do you know which cords are sum cords? In the case of indexed pendant groups, it makes some sense how to find them. But in the case of pendant pendant sums, what âsignifiesâ a pendant sum cord? This study reveals the clues:

Right-handed sums occur on the left side of the khipu

Left-handed sums occur on the right side of the khipu

As expected the largest set of Right-handed sum cords are in the first group, first cord position. Many of these come from smaller khipus.

Left handed sum cords occur about â of the time that right handed sum cords occur.

The most common group for a right handed cord is NOT the first group (from the right-end), but instead, the 5th group, and then the 12th. This is unexpected.

Pendant pendant sum cords group together ~50% of the time. This grouping is called a sum group.

Sum groups also group together with other sum groups. 75% of Sum Groups have a NON-ZERO sum cord group on BOTH their Left and Right.

In highly banded khipus, sum groups are often bordered by zero-valued cord groups.

The use of white, as a first cord in a group, which occurs ~41% of the time (higher than its overall average of 27%) may indicate the presence of a sum group.

Groups of sum groups are often surrounded by groups of 0 cords. (20% of the time). Especially in banded khipusâŠ

A majority of large-valued pendant sum cords have a seriated color scheme. This provides an additional confirmation of Dr. Clindanielâs argument in his Ph.D. Thesis the majority of the large-valued ascher cord colors in sums are barberpole or mottled - few are solid.

White is the most common pendant sum cord color (<27%), followed by AB(Light Brown), MB(Moderate Brown), and B(Moderate Yellowish Brown) and (YB)Light Yellowish Brown.

White does not appear to play a role as a grammatical marker for pendant sum summation, but it does appear to serve as a marker for cord sum groups. More examination of this can be found on the White Cord EDA page.

Color does not appear to play a role in summation patterns, although location does.

Out of the 427 (66%) of KFG khipus with pendant pendant sum relationships, 290(68%) had compound sums.

The biggest compound sum depth was 9 levels, but the average was 3.4 Â±1.5

Only 1 khipu, UR196, has a cyclic sum loop.

OTHER FIELDMARKS: Spatially, most pendant-pendant sum/summand cords occur:

Roughly 70% of a khipuâs non-zero pendants are involved in being a sum or a summand. There is a strong linear trend line (R^{2}=0.78) for the relationship between number of pendant cords, and those involved as sums or summands.

There is little to no pattern synchronicity in cords between the three sum relationships.

Degrees between Pendant Pendant_Sums and Pendant Pendant_Color_Sums is 32Â° Degrees between Pendant Pendant_Sums and Indexed Pendant_Sums is 43Â° Degrees between Indexed Pendant_Sums and Pendant Pendant_Color_Sums is 46Â°

The statistics show the first evidence of the predominance of right-handed sums. While the general distribution of num_summands is similar, the maximum range is two-thirds narrower for the number of summands in left-handed sums:

(4382) Right Num_Summands: Range=(2,281), Mean=7.00, Std_dev=12.85 (3706) Left Num_Summands: Range=(2,127), Mean=6.00, Std_dev=10.30

CASE STUDIES

AS069: As an example of what the pendant pendant sum relationship reveals, itâs interesting to study AS069âs pendant pendant sum X-rays. AS069âs pendant-pendant sums appear to be in âincorrectâ places. The majority of the right-handed sums, which are usually on the left of the khipu, are instead located on the right. The majority of the left-handed sums, which are usually on the right of the khipu, appear in the middle, or on the left. Something appears odd about this gigantic khipu.

UR113: Cord group 2 has lots of unique values for summands that are referenced a lot.

UR231: A similar visual inspection of UR231 is intriguing. UR231 was suspected of being two separate khipus, but a look at itâs X-Ray sum view reveals that itâs summation pattern is so cross-linked that it is likely one khipu.

UR1084: UR1084 has block structure similar to Urtonâs Pachacamac khipu UR1104. The number 22 is referenced a lot (a calendar measure?).

Pendant cords that are Sums of a set of Pendant cords, with the Same Color, ignoring parent cord group boundaries.

Pendant Sums by Color have characteristics that are very similar to Pendant-Pendant Sums:

There is a moderate relationship between Color Pendant Sums and Pendant Pendant Sums (R^{2} = 0.70).

The ratio of color pendant sums to normal to pendant pendant sums is 37%.

As expected the largest set of Right-handed sum cords are in the first or second group, first cord position. Many of these come from smaller khipus.

Left handed sum cords occur about â of the time that right handed sum cords occur.

The most common group for a right handed cord is NOT the first group, but instead, the 5th group, the 7th group and then the 13th. Like pendant-pendant-sums, this is also unexpected.

Pendant color sum cords group together 60% of the time. This grouping is called a sum group.

Sum groups also group together with other sum groups. 70% of Sum Groups have a NON-ZERO sum cord group on BOTH their Left and Right.

Sum groups are often bordered by zero-valued cord groups.

The use of White, as a first cord color in a group, which occurs 41.5% of the time (higher than its overall average of 27%) may indicate that it belongs to a sum group.

Pendant cords that are Sums of Similarly Indexed cords in Contiguous Groups to the right or left of the cord.

Once again, we see the classic 60/40 split (here 57%/43%) between right and left-handed sums. This consistent 60/40 split across all three sum relations is astonishing.

Dropping AS069 gives us more insight into the distance between sums and summands. Itâs not that large. When the outlier AS069 is dropped, the distance between summands and sum cord follows a roughly triangular distribution - Large sum cord values are close to their summands, and small sum cord values can be farther away (1-40 cords) from their summands.

Indexed pendant sums follow a similar overall location distribution as pendant-pendant sums and pendant-color sums.

Oddly, with indexed pendant sums the cord values decrease, somewhat, with an increasing number of summands.

Subsidiary cords that are Sums of Pendant cords right or left (inclusive) of the subsidiary cord.

We could cut and paste the conclusions from indexed pendant sums, and it would almost be correct :-) - Once again, we see the classic 60/40 split between right and left-handed sums. Here it is also (57%)/(43%). This consistent 60/40 split across all four sum relations is astonishing. - Dropping AS069 gives us more insight into the distance between sums and summands. Itâs not that large. When the outlier AS069 is dropped, the distance between summands and sum cord follows a roughly triangular distribution - Large sum cord values are close to their summands, and small sum cord values can be farther away (1-30 cords) from their summands. - Subsidiary pendant sums follow the same overall location distribution that pendant-pendant sums, pendant-color sums, and color pendant sums follow.

The pendant_subsidiary_neighbor relationship seems likely to be a fluke. Occuring 1.23% of the time, and even less, when the pendant-subsidiary difference is >= 3 (0.6%), Iâm 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 ânever.â They say âmaybe.â Maybe thereâs something here, but I canât see it.

Along with pendant index subsidiaries, this is an interestingly different fieldmark. Like Pendant-Color-Sum indexing, this fieldmark uses color indexing instead of position indexing. Is the use of color-indexing a user-interface issue - itâs just harder to pick a cord and run your thumbs on it, if itâs a subsidiary? Is itâs use a result of an âoptionsâ grammar (only include the color if the condition occurs)?

These relations occur in only 8% of the khipu. Additionally only 173 such relations occur - compare that to the 8,091 pendant pendant sum relationships (just 2% of those). So is this just a statistical fluke? The presence of a subsidiary at all renders that argument unlikely. This is a known technique, not a statistical fluke - the bar chart for matching/significant khipu adds to that argument, but it seems a sign/signature of some sort, rather than a common technique. AS038 pendant 19 seems to be the poster child for this relationship.

Due to the scarcity of occurrence, as well as the fact that itâs a subsidiary relationship, Iâm going to pass on further investigation of this fieldmark at this time.

Groups whose left âhalfâ sum equals the right âhalfâ sum. Off-by-one loose matching is allowed.

Group sum bands split mostly at the middle 1/2, and occasionally at 2/5 and 3/5 of the way through the group.

Of those groups that are group sum bands, group sum bands occur on seriated groups twice as often as banded groups (2:1) (compared to the overall KFG prior distribution of 4 seriated groups for every 3 banded groups (1.33:1).

Large group sum bands occur mostly on the right.

It is not necessary for the group sum bands to have equal numbers of cords, or to be multiples of 2 for each side.

It is not evident when a group sum band occurs, other than by doing the arithmetic. It seems this is a convenient way of pre-organizing something. Sometimes a top cord occurs. Sometimes the cord attachment is different. But overall, there appears to be little in the way of a signifier.

Groups whose cord values decrease from left to right.

The Ascher relationship - Decreasing Group Cord Values provides the simplest introduction to the analysis of Ascher sum relationships. A brief review of the conclusions from that analysis:

Decreasing cord value groups occur in roughly 3% of the khipus 2 or more times (17 of 592 khipus).

Decreasing cord value groups occur in roughly 1/7th (14%) of the khipus (67 of 592 khipus).

Of those groups that are decreasing cord groups, 4 seriated groups for every banded group (4:1) (compared to the overall KFG prior distribution of 4 seriated groups for every 3 banded groups (1.33:1).

Other than the fact that white predominates as a first cord color, there does not appear to be any apparent ordering significance on seriated groups with respect to Ascher cord colors.

Of the khipus that they are on, they occur roughly on 1/3 of the groups.

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 groups.

Their mean benford match signature (.707) indicates that they are largely of an âaccountingâ nature.

A few khipu bear closer inspection - these include UR208, UR113, UR231, UR068, and AS212

86 sum top cords exist. Less than 1% of pendant-pendant sum relationships. Clearly although top cord sums were the first sum relationship to be found (by Locke in 1926), they are rare.

Sum top cords strongly associate with banded groups.

Sum top cords strongly associate with 6 cord groups (70%)

Most sum top cords are less than 1000. Waranka scale.

Only 9 (1.3%) of the khipu in the Khipu Fieldguide exhibit this fieldmark, and only 3 share the fieldmark with group sum bands. Although this is a valid identifiable fieldmark, itâs statistical occurrence makes it difficult to make much of use of it as a fieldmark.

All the local caĂ§iques come to these towns along the road to serve [âŠ] [the accountants] have a store of firewood and maize and everything else, and they count by some knots in ropes what each caĂ§ique has brought. And when they brought us a few loads of firewood or sheep or maize or chicha, they removed the knots from those who had it in charge, and tied it elsewhere: so that in everything they have a very great account and reason.

Pizarro, Hernando. (1533) 1920. âA los seĂ±ores oydores de la audiencia real de su magestad.â In Informaciones sobre el antiguo PerĂș, edited by Horacio H. Urteaga, 16â180. Vol. 3. Lima: SanmartĂ.

Now that we have taken a look at Ascher sum relationships, we can synthesize what we know about khipus and sums.

An astonishing number â ~71% (464/654) â of khipus in the Khipu Fieldguide have 1 or more summation relationships. This is not a good indicator for the idea that khipus are âlinguisticâ in nature.

Summations come in two-types, right-handed and left-handed, with right-handed sums having their sum cords on the left of the khipu with their summand cords to the right of the sum cord, generally within 20 to 50 cords of the sum cord. Similarly left-handed sums, generally have their sum cords on the right side of the khipu.

Right-handed/left-handed sums generally split along 60/40 lines, for all types of pendant sum relations.

More than one type of sum relationship can exist in a khipu. In fact, khipus often consist of multiple layers of summation relationships.

âX-rayâ pictures of the network of summation relationships quickly reveal the overall structure of a khipu, what cords are important, where âthe actionâ is, etc. As an example, the pictures reveal that the largest khipu in the fieldguide, AS069, is probably incorrectly spliced. As another example, look at UR006, the two-year calendrical khipu first analyzed by Gary Urton. The X-ray drawings clarify and confirm the two year organization of the khipu.

Other than actually KNOWING, using arithmetic, that a particular cord is a sum cord, how does one identify it? As shown in the pendant pendant sum study, there are a significant number of âcluesâ

Seriated vs Banded Sums. Summation patterns on banded khipus are different from seriated khipus. Banded khipus frequently have a set of zero valued cord groups with no knots, interspersed between sum groups - groups of cords where each cord sums a section of the khipu

The biggest right-handed sums, are generally in the first four or five cords of a khipu

The biggest left-handed sums, are generally in the last four or five cords of a khipu

The use of white as a first cord in a group - see here and here, which occurs ~41% of the time (higher than its overall average of 27%) may indicate the presence of a sum group.

Sum Top Cords strongly associate (>95%) with banded groups, and 70% of the time with 6 cord groups. They are also quite rare, occuring in less 4.5% of the khipus.

Sums of sums exists. The maximum compound summing depth is 9, but the usual sum depth is around 2-5.