This notebook is based on the article Information Control in the Palace of Puruchuco - An Accounting Hierarchy in a Khipu Archive from Coastal Peru by Gary Urton and Carrie J. Brezine.
import numpy as np
import pandas as pd
import khipu_kamayuq as kamayuq # A Khipu Maker is known (in Quechua) as a Khipu Kamayuq
import khipu_qollqa as kq
from pandas import Series, DataFrame
# Plotly
import plotly
from plotly.offline import iplot, init_notebook_mode
import plotly.graph_objs as go
import plotly.express as px
import plotly.figure_factory as ff
plotly.offline.init_notebook_mode(connected = False)
(khipu_dict, all_khipus) = kamayuq.fetch_khipus()The article references seven khipus:
CM009 had to be recreated from Carol Mackey’s notes and drawings of her 1970 doctoral thesis.
An examination of Ascher Sum Relationships for UR063 reveals:
The authors note that UR073 is broken.
We note first that khipu UR73 has been broken; it bears only 69 of what we surmise were originally about 111 pendant strings. The similarity between UR63 and UR73 in color patterning and numeric values is very close; thus we feel confident in assuming that these were originally a matching pair.
An examination of Ascher Sum Relationships for UR073 reveals:
The paper describes UR063 as having six subunits, labeled A-F
UR63 is organized by spacing and color seriation into six pendant string groupings, labeled a–f. The number of strings in each group is shown in brackets at the bottom of the columns. The six columns comprise: a) three sets of (5 x 4 =) 20 strings organized into five groups of four color- seriated strings; b) two sets of (3 x 4 + 2 x 3 =) 18 color-seriated strings; and c) one set of (3 x 4 + 3 =) 15 color-seriated strings.
subunit_a = [aCord for aCord in UR063.pendant_cords()[0:20]]
subunit_b = [aCord for aCord in UR063.pendant_cords()[20:38]]
subunit_c = [aCord for aCord in UR063.pendant_cords()[38:56]]
subunit_d = [aCord for aCord in UR063.pendant_cords()[56:76]]
subunit_e = [aCord for aCord in UR063.pendant_cords()[76:96]]
subunit_f = [aCord for aCord in UR063.pendant_cords()[96:112]]
def cord_sum(cord_list): return sum([aCord.knotted_value() for aCord in cord_list])
def describe_cord_list(list_of_cord_lists, titlelist=None):
description = ""
for i, aCordList in enumerate(list_of_cord_lists):
title = f"{titlelist[i]} " if titlelist else ''
list_len = len(aCordList)
contents = f"{[aCord.knotted_value() for aCord in aCordList]}"[1:-1]
description += f"{title}({list_len}) {contents}\n"
if len(list_of_cord_lists) > 1: description += "\n"
for i, aCordList in enumerate(list_of_cord_lists):
title = f"{titlelist[i]} " if titlelist else ''
list_len = len(aCordList)
contents = f"{[(aCord.level_name, aCord.longest_ascher_color()) for aCord in aCordList]}"[1:-1].replace("\'", "")
description += f"{title}({list_len}) {contents}\n"
print(description)
describe_cord_list([subunit_f, subunit_e, subunit_d, subunit_c, subunit_b, subunit_a],
['subunit_f', 'subunit_e', 'subunit_d', 'subunit_c', 'subunit_b', 'subunit_a'])subunit_f (15) 0, 0, 0, 0, 1, 0, 1, 0, 5, 0, 1, 0, 219, 6, 0
subunit_e (20) 0, 0, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 10, 1, 0, 1, 157, 7, 12, 3
subunit_d (20) 2, 0, 0, 0, 3, 1, 0, 0, 1, 1, 0, 0, 12, 1, 0, 0, 367, 8, 8, 2
subunit_c (18) 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 9, 2, 1, 0, 148, 7, 2
subunit_b (18) 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 12, 1, 1, 229, 6, 13
subunit_a (20) 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 1, 9, 1, 1, 0, 127, 12, 9, 1
subunit_f (15) (p97, W), (p98, AB), (p99, GG), (p100, W:KB), (p101, W), (p102, AB), (p103, GG), (p104, W:KB), (p105, W), (p106, AB), (p107, GG), (p108, W:KB), (p109, W), (p110, AB), (p111, GG)
subunit_e (20) (p77, W), (p78, AB), (p79, GG), (p80, W:KB), (p81, W), (p82, AB), (p83, GG), (p84, W:KB), (p85, W), (p86, AB), (p87, GG), (p88, W:KB), (p89, W), (p90, AB), (p91, GG), (p92, W:KB), (p93, W), (p94, AB), (p95, GG), (p96, W:KB)
subunit_d (20) (p57, W), (p58, MB), (p59, GG), (p60, W:KB), (p61, W), (p62, AB), (p63, GG), (p64, W:KB), (p65, W), (p66, AB), (p67, GG), (p68, W:KB), (p69, W), (p70, AB), (p71, GG), (p72, W:KB), (p73, W), (p74, AB), (p75, GG), (p76, W:KB)
subunit_c (18) (p39, W), (p40, AB), (p41, GG), (p42, W), (p43, AB), (p44, GG), (p45, W:KB), (p46, W), (p47, AB), (p48, GG), (p49, W:KB), (p50, W), (p51, MB), (p52, GG), (p53, W:KB), (p54, W), (p55, MB), (p56, GG)
subunit_b (18) (p21, W), (p22, AB), (p23, GG), (p24, W:KB), (p25, W), (p26, AB), (p27, GG), (p28, W:KB), (p29, W), (p30, AB), (p31, GG), (p32, W:KB), (p33, W), (p34, W:AB), (p35, GG), (p36, W), (p37, AB), (p38, GG)
subunit_a (20) (p1, W), (p2, AB), (p3, GG), (p4, W:GG), (p5, W), (p6, MB), (p7, GG), (p8, W:GG), (p9, W), (p10, MB), (p11, GG), (p12, W:GG), (p13, W), (p14, MB), (p15, GG), (p16, W:GG), (p17, W), (p18, MB), (p19, GG), (p20, W:GG)
There are some discrepancies, between the data and the article.
article_subunit_a=[1, 2, 1, 9, 1, 127, 12, 9, 1]
article_subunit_b=[1, 2, 1, 12, 1, 1, 229, 6, 13]
article_subunit_c=[1, 1, 9, 2, 1, 148, 7, 2]
article_subunit_d=[2, 3, 1, 1, 1, 12, 1, 367, 8, 8, 2]
article_subunit_e=[2, 1, 2, 1, 10, 1, 1, 157, 7, 12, 3]
article_subunit_f=[1, 1, 5, 1, 219, 6]
print(f"{cord_sum(subunit_a)=} - {sum(article_subunit_a)=} = {cord_sum(subunit_a)-sum(article_subunit_a)}")
print(f"{cord_sum(subunit_b)=} - {sum(article_subunit_b)=} = {cord_sum(subunit_b)-sum(article_subunit_b)}")
print(f"{cord_sum(subunit_c)=} - {sum(article_subunit_c)=} = {cord_sum(subunit_c)-sum(article_subunit_c)}")
print(f"{cord_sum(subunit_d)=} - {sum(article_subunit_d)=} = {cord_sum(subunit_d)-sum(article_subunit_d)}")
print(f"{cord_sum(subunit_e)=} - {sum(article_subunit_e)=} = {cord_sum(subunit_e)-sum(article_subunit_e)}")
print(f"{cord_sum(subunit_f)=} - {sum(article_subunit_f)=} = {cord_sum(subunit_f)-sum(article_subunit_f)}")cord_sum(subunit_a)=165 - sum(article_subunit_a)=163 = 2
cord_sum(subunit_b)=266 - sum(article_subunit_b)=266 = 0
cord_sum(subunit_c)=172 - sum(article_subunit_c)=171 = 1
cord_sum(subunit_d)=406 - sum(article_subunit_d)=406 = 0
cord_sum(subunit_e)=197 - sum(article_subunit_e)=197 = 0
cord_sum(subunit_f)=233 - sum(article_subunit_f)=233 = 0So we are off by 3.
Let’s sort subunits by color sum:
def sum_cord_values(aCordList):
sums_by_color = {aColor:0 for aColor in sorted(set([aCord.longest_ascher_color() for aCord in aCordList]))}
for aCord in aCordList: sums_by_color[aCord.longest_ascher_color()] += aCord.knotted_value()
return sums_by_color
print(f"{sum_cord_values(subunit_a)=}")
print(f"{sum_cord_values(subunit_b)=}")
print(f"{sum_cord_values(subunit_c)=}")
print(f"{sum_cord_values(subunit_d)=}")
print(f"{sum_cord_values(subunit_e)=}")
print(f"{sum_cord_values(subunit_f)=}")sum_cord_values(subunit_a)={'AB': 0, 'GG': 11, 'MB': 13, 'W': 139, 'W:GG': 2}
sum_cord_values(subunit_b)={'AB': 6, 'GG': 15, 'W': 244, 'W:AB': 1, 'W:KB': 0}
sum_cord_values(subunit_c)={'AB': 1, 'GG': 3, 'MB': 9, 'W': 159, 'W:KB': 0}
sum_cord_values(subunit_d)={'AB': 11, 'GG': 8, 'MB': 0, 'W': 385, 'W:KB': 2}
sum_cord_values(subunit_e)={'AB': 8, 'GG': 14, 'W': 171, 'W:KB': 4}
sum_cord_values(subunit_f)={'AB': 6, 'GG': 2, 'W': 225, 'W:KB': 0}The sum of these subunits, in turn is supposed to roughly equal UR068 cords 34-53
subunit_UR068 = [aCord for aCord in UR068.pendant_cords()[33:53]]
describe_cord_list([subunit_UR068])
sum_UR063 = cord_sum(subunit_a) + cord_sum(subunit_b) + cord_sum(subunit_c) + cord_sum(subunit_d) + cord_sum(subunit_e) + cord_sum(subunit_f)
print(f"{sum_UR063=}")
partial_sum_UR068 = cord_sum(subunit_UR068)
print(f"{partial_sum_UR068=}")(20) 2, 0, 1, 0, 8, 1, 1, 0, 8, 1, 3, 1, 57, 5, 6, 2, 1236, 46, 59, 10
(20) W, AB, GG, W:GG, W, AB, AB:GG, W:KB, W, AB, GG, W:KB, W, AB, GG, W:KB, W, AB, GG, W:KB
sum_UR063=1439
partial_sum_UR068=1447It takes some finessing to make the Puruchuco scheme work for aligning the sums of UR068 to match a section of UR063. If you align the sections by color, and insert a “null cord” holder, when needed, it aligns well.
Here’s a tabular “excel” view similar to Figure 6 of the article:
| KHIPU | # CORDS | ||||||||||||||||||||
| UR068 34-53 | p34 | p35 | p36 | p37 | p38 | p39 | p40 | p41 | p42 | p43 | p44 | p45 | p46 | p47 | p48 | p49 | p50 | p51 | p52 | p53 | |
| 20 | W | AB | GG | W:GG | W | AB | AB:GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | |
| 20 | 2 | 0 | 1 | 0 | 8 | 1 | 1 | 0 | 8 | 1 | 3 | 1 | 57 | 5 | 6 | 2 | 1236 | 46 | 59 | 10 | |
| UR063 (Sum) | 2 | 0 | 1 | 0 | 9 | 1 | 1 | 0 | 8 | 2 | 3 | 1 | 57 | 6 | 4 | 1 | 1247 | 46 | 44 | 6 | |
| UR063 | |||||||||||||||||||||
| subunit_f | 15 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 5 | 0 | 1 | 0 | 219 | 6 | 0 | |||||
| subunit_e | 20 | 0 | 0 | 0 | 0 | 2 | 0 | 1 | 0 | 2 | 0 | 1 | 0 | 10 | 1 | 0 | 1 | 157 | 7 | 12 | 3 |
| subunit_d | 20 | 2 | 0 | 0 | 0 | 3 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 12 | 1 | 0 | 0 | 367 | 8 | 8 | 2 |
| subunit_c | 18 | 0 | 0 | 0 | null cord | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 9 | 2 | 1 | 0 | 148 | 7 | 2 | |
| subunit_b | 18 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 2 | 0 | 1 | 0 | 12 | 1 | 1 | null cord | 229 | 6 | 13 | |
| subunit_a | 20 | 0 | 0 | 1 | 0 | 2 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 9 | 1 | 1 | 0 | 127 | 12 | 9 | 1 |
| subunit_f | 15 | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | |||||
| subunit_e | 20 | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB |
| subunit_d | 20 | W | MB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB |
| subunit_c | 18 | W | AB | GG | null cord | W | AB | GG | W:KB | W | AB | GG | W:KB | W | MB | GG | W:KB | W | MB | GG | |
| subunit_b | 18 | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | W:AB | GG | null cord | W | AB | GG | |
| subunit_a | 20 | W | AB | GG | W:GG | W | MB | GG | W:GG | W | MB | GG | W:GG | W | MB | GG | W:GG | W | MB | GG | W:GG |
An examination of URO64’s pendant-pendant sums reveals intentional pendant summing on top of the summing of the Puruchuco hierarchy. This is especially evident in the last few cords of the last few clusters - a hallmark of left-handed pendant-pendant sums.
An examination of CM009’s sum relationships reveals intentional pendant summing on top of the summing of the Puruchuco hierarchy, as well as evidence of Ascher decreasing clusters.
An examination of URO68’s pendant-pendant sums reveals intentional pendant summing on top of the summing of the Puruchuco hierarchy.
Additionally 4 of the clusters are Ascher decreasing groups.
As befitting a “high-level khipu”, UR067 has only two pendant-pendant sum relationships, and those seem largely unintentional and of small value.
As befitting a “high-level khipu”, UR066 has only two pendant-pendant sum relationships, and those seem largely unintentional and of small value. However, on visual examination it last four (of six) clusters appear to be rough ascher decreasing groups.
Continuing the replication of the article, we examine the interaction between the next two levels.
subunit_c (20) 1, 0, 0, 0, 3, 1, 1, 0, 6, 1, 3, 0, 43, 3, 2, 2, 459, 44, 39, 13
subunit_b (20) 2, 0, 1, 0, 8, 1, 1, 0, 8, 1, 3, 1, 57, 5, 6, 2, 1236, 46, 59, 10
subunit_a (20) 3, 0, 0, 0, 8, 2, 1, 0, 8, 1, 3, 1, 56, 5, 4, 1, 1213, 43, 64, 17
subunit_c (20) W, AB, GG, W:KB, W, AB, GG, W:KB, W, AB, GG, W:KB, W, AB, GG, W:KB, W, AB, GG, W:KB
subunit_b (20) W, AB, GG, W:GG, W, AB, AB:GG, W:KB, W, AB, GG, W:KB, W, AB, GG, W:KB, W, AB, GG, W:KB
subunit_a (20) W, AB, GG, W:KB, W, AB, GG, W:KB, W, AB, GG, W:KB, W, AB, GG, W:KB, W, AB, GG, W:KB
(20) 0, 0, 1, 0, 19, 4, 3, 0, 22, 3, 9, 2, 156, 13, 12, 0, 2904, 133, 161, 40
(20) W, AB, GG, W:KB, W, MB, YG, W:KB, W, AB, AB:GG, W:KB, W, AB, GG, AB:KB, W, AB, AB, W:KB
sum_UR068=3498
partial_sum_UR067=3482A similar “excel” table view of the sums for UR068 to UR067
| KHIPU | # CORDS | ||||||||||||||||||||
| UR067 p13-p32 | p13 | p14 | p15 | p16 | p17 | p18 | p19 | p20 | p21 | p22 | p23 | p24 | p25 | p26 | p27 | p28 | p29 | p30 | p31 | p32 | |
| 20 | W | AB | GG | W:KB | W | MB | YG | W:KB | W | AB | AB:GG | W:KB | W | AB | GG | AB:KB | W | AB | AB | W:KB | |
| 20 | 0 | 0 | 1 | 0 | 19 | 4 | 3 | 0 | 22 | 3 | 9 | 2 | 156 | 13 | 12 | 0 | 2904 | 133 | 161 | 40 | |
| UR068 p14-p73 | 6 | 0 | 1 | 0 | 19 | 4 | 3 | 0 | 22 | 3 | 9 | 2 | 156 | 13 | 12 | 5 | 2908 | 133 | 162 | 40 | |
| subunit_c | 20 | 1 | 0 | 0 | 0 | 3 | 1 | 1 | 0 | 6 | 1 | 3 | 0 | 43 | 3 | 2 | 2 | 459 | 44 | 39 | 13 |
| subunit_b | 20 | 2 | 0 | 1 | 0 | 8 | 1 | 1 | 0 | 8 | 1 | 3 | 1 | 57 | 5 | 6 | 2 | 1236 | 46 | 59 | 10 |
| subunit_a | 20 | 3 | 0 | 0 | 0 | 8 | 2 | 1 | 0 | 8 | 1 | 3 | 1 | 56 | 5 | 4 | 1 | 1213 | 43 | 64 | 17 |
| subunit_c | 20 | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB |
| subunit_b | 20 | W | AB | GG | W:GG | W | AB | AB:GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB |
| subunit_a | 20 | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB | W | AB | GG | W:KB |
The article mentions the following hypothesis:
We suggest that khipus may have contrasting number qualities depending on whether they represented instructions coming from the state administration to a local accounting center, or records produced within a local accounting center with regard to existing community resources. In the first circumstance, we suspect that khipu values would have tended to be even decimal values or calculations of values in standard proportional shares.12 If a khipu account was compiled from within some local administrative center to be sent upward to higher-level officials, counts of resources could be expected to have reflected the vagaries of the natural distribution of items in society. Such numbers are less likely to be whole and rounded or perfectly proportional. This is partly because there are many more “interdecimal” values (e.g., 41, 89, 53) than there are full decimal ones (e.g., 10, 20, 100).
Let’s briefly review the Benford Match for the 7 khipus
# Read in the Fieldmark and its associated dataframe and match dictionary
from fieldmark_khipu_summary import Fieldmark_Benford_Match
aFieldmark = Fieldmark_Benford_Match()
fieldmark_df() = aFieldmark.fieldmark_df()
def benford_match(aKhipuName):
return fieldmark_df()[fieldmark_df().kfg_name== aKhipuName].iloc[0].benford_match
# Remember:
level_3 = ['UR067', 'UR066']
level_2 = ['UR064', 'CM009', 'UR068']
level_1 = ['UR063', 'UR073']
for khipu_name in level_3:
print(f"III - {khipu_name} {benford_match(khipu_name):2.2f}")
print()
for khipu_name in level_2:
print(f"II - {khipu_name} {benford_match(khipu_name):2.2f}")
print()
for khipu_name in level_1:
print(f"I - {khipu_name} {benford_match(khipu_name):2.2f}")III - UR067 0.60
III - UR066 0.63
II - UR064 0.66
II - CM009 0.80
II - UR068 0.71
I - UR063 0.49
I - UR073 0.45These differences are large enough to question the hypothesis proposed in the article.