I do hate sums, There is no greater mistake than to call arithmetic an exact science. There are Permutations and Aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of Numbers which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then again from the top down, the result is always different.
— Maria Price La Touche, The Letters of a Noble Woman (1908), ed. Margaret Ferrier Young, letter dated July 1878, p. 49 (quoted in Mathematical Gazette, Vol. 12, 1924)
Pendants that are Sums of a set of Contiguous Pendants. This is the most basic of all Ascher sum relationships, and the most heavily studied. One pendant is the sum of two or more contiguous pendants, regardless of cord group index, or color.
There are five basic types of sum relationships:
a.) Pendant Pendant Sums, cords whose value is a sum of contiguous pendants
b.) Pendant Color Sums, cords whose sum is only that of similarly colored pendants
c.) Indexed Pendant Sums, where the pendants are summed, by index, across groups.
d.) Subsidiary Pendant Sums, where subsidiaries of a pendant sum other pendants.
e.) Top cords which are the sums (or double sums) of an adjacent group.
The subset of the 3 pendant summing techniques (Color, Index, Subsidiary), differ from the Pendant Pendant fieldmark, in that they allow Non-Contiguous Sums.
As is often the case with large search spaces, this fieldmark has a delicate balance between precision and recall. The search space is simply pendant cords, without consideration of index to color or position. When unrestrained, search recall is enhanced at the expense of precision, and computation time has exponentional growth. In the never ending struggle to dial it just right, the search is limited to pendants with a value of at least 11, and where the summand cords are contiguous for at least three cords (not including 0 cords in-between). Furthermore, for summation candidates, the maximum moving search window size is set to 250 contiguous (except for 0-value cords) at a time.
Occasionally, a pendant sum can have more than one summand relationship. For example P1 = P2 + P3, but P3, in turn is the sum of P4 and P5. So P1 = P2 + P4 + P5. Although the contiguity constraint eliminates many of these duplicates, when duplicates occur Occam’s Razor is employed, and the summation with the fewest number of summands is chosen.
What to do about off-by-one errors, where for example 483 should match 493? Although off-by-one knot errors occur frequently, in this particular study, unlike top cords, Sums are not matched on an off-by-one digit basis and instead an exact numerical match is required. This reduces the number of found pendant sums about 15% - a considerable amount. Once again, we are trading off recall for precision.
| Measure | Result |
|---|---|
| Number of Khipus That Match | 427 (66%) |
| Number of Significant Khipus | 158 (24%) |
| Five most Significant Khipu | AS069, UR004, UR231, UR113, UR1114 |
| XRay Image Quilt | Khipu by Pendant-Pendant Sum Relationships |
| Database View | Click here |
# Initialize plotly
plotly.offline.init_notebook_mode(connected = False);
# Read in the Fieldmark and its associated dataframe and match dictionary
from fieldmark_ascher_pendant_pendant_sum import Fieldmark_Pendant_Pendant_Sum
aFieldmark = Fieldmark_Pendant_Pendant_Sum()
fieldmark_dataframe = aFieldmark.dataframes[0].dataframe
raw_match_dict = aFieldmark.raw_match_dict()<Figure size 6000x3000 with 0 Axes># Plot Matching khipu
matching_khipus = aFieldmark.matching_khipus()
matching_values = [raw_match_dict[aKhipuName] for aKhipuName in matching_khipus]
matching_df = pd.DataFrame(list(zip(matching_khipus, matching_values)), columns =['KhipuName', 'Value'])
fig = px.bar(matching_df, x='KhipuName', y='Value', labels={"KhipuName": "Khipu Name", "Value": "Number of Pendant Pendant Sum Cords", },
title=f"Matching Khipu ({len(matching_khipus)}) for Number of Pendant Pendant Sum Cords", width=944, height=450).update_layout(showlegend=True).show()# Plot Significant khipu
significant_khipus = aFieldmark.significant_khipus()
significant_values = [raw_match_dict[aKhipuName] for aKhipuName in significant_khipus]
significant_df = pd.DataFrame(list(zip(significant_khipus, significant_values)), columns =['KhipuName', 'Value'])
fig = px.bar(significant_df, x='KhipuName', y='Value', labels={"KhipuName": "Khipu Name", "Value": "Number of Pendant Pendant Sum Cords", },
title=f"Significant Khipu ({len(significant_khipus)}) for Number of Pendant Pendant Sum Cords", width=944, height=450).update_layout(showlegend=True).show()In the case of non-contiguous sum relationships, identification of sum cords appears fairly easy. The sums are usually “right-handed” and positionally based. Identification of pendant-pendant sum cord locations appears to be a much more challenging problem, and is akin to finding where sums occur in a spreadsheet, without being able to look at how the formula was calculated.
Pendant pendant sum cords can be explored along the following axes:
These explorations are lengthy and detailed, and each is excerpted into it’s own page. Conclusions from each of these analyses, is summarized below:
Imagine you’re a khipukamayuq who gets a cord from “lower” 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 “signifies” a pendant sum cord? This study reveals the clues:
HANDEDNESS: - There are a total of 8088 Pendant Pendant Sums
4382 (54%) are Right-handed sums and 3706 (46%) are Left-handed sums
LOCATION: - 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 ⅔ of the time that right handed sum cords occur. - 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. 81% 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 29% of the time may indicate that it belongs to a sum group. - Sum groups often collocate with groups of 0 cords. Especially so in banded groups… - Sum cords often occur at the start of a group or in line with other sum cords in a group
COLOR: - A majority of large-valued pendant sum cords have a seriated color scheme. This provides an additional confirmation of Dr. Clindaniel’s argument in his Ph.D. Thesis that seriated cords are associated with higher cord values. - White is the most common pendant sum cord color (<30%), 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.
HIERARCHY: - 427 khipus have sum relationships of some sort (or stated alternatively, have some relationships of sum sort). - 292 (68%) of those khipus (out of the 427 khipus with pendant pendant sum relationships) had compound sums, with the biggest compound sum depth going 9 levels, but the average being 3.4. - Only 1 khipu, UR196, has a cyclic sum/loop.
OTHER FIELDMARKS: Spatially, most pendant-pendant sum/summand cords occur: - As a majority of the pendants in an overall khipu (70% of the time). - There is a moderate linear trend line (R2=0.46) for khipus, involved in pendant-pendant sum relationships. Roughly 70% of a khipu’s non-zero pendants are involved in being a sum or a summand. - 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 32°
+ Degrees between Pendant Pendant_Sums and Indexed Pendant_Sums is 43°
+ Degrees between Indexed Pendant_Sums and Pendant Pendant_Color_Sums is 46°
NUMBER OF SUMMANDS - 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:
(5165) Right Num_Summands: Range=(2,281), Mean=6.00, Std_dev=11.49
(2927) Left Num_Summands: Range=(2,232), Mean=7.00, Std_dev=12.53AS069:
As an example of what the pendant pendant sum relationship reveals, it’s interesting to study AS069’s pendant pendant sum X-rays. AS069’s pendant-pendant sums appear to be in “incorrect” 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 it’s X-Ray sum view reveals that it’s summation pattern is so cross-linked that it is likely one khipu.
UR1084:
UR1084 has block structure similar to Urton’s Pachacamac khipu UR1104. The number 22 is referenced a lot (a calendar measure?).