The discovery by Leland Locke in 1926 of top cord values matching the sum of their adjacent group cords was a major breakthrough in khipu studies and confirmed the use of base 10 in Inkan-era khipus. UR166 set the stage for modern khipu studies.
Interestingly, the sums often don’t quite match, and are off by one digit, not just in the ones place, but sometimes in the tens, hundreds, or thousands place. The summer/yupana user, or the reader of the knots, erred - I suspect the latter.
Since the combinatorial search space is pretty small, (adjacent groups), I can check for off-by-one digit errors in the ones, tens, etc. count, indicating that the sum is approximate (~) on the rendered khipus - see our old friend UR166 for an example. That check and a review of the database shows that 21 khipus match, although UR117C probably does not count. That’s ~4% of the overall database. A significant number and clearly a useful fieldmark. Their distribution is as follows:
# Initialize plotly
import plotly
plotly.offline.init_notebook_mode(connected = False);
# Read in the Fieldmark and its associated dataframe and match dictionary
from fieldmark_sum_top_cords import FieldmarkSumTopCords
aFieldmark = FieldmarkSumTopCords()
fieldmark_dataframe = aFieldmark.dataframes[0].dataframe
raw_match_dict = aFieldmark.raw_match_dict()# Bookkeeping to produce table at top of this page
from statistics import mean
import pandas as pd
import qollqa_chuspa as qc
# Load khipus
(khipu_dict, all_khipus) = qc.fetch_khipus()
cord_value_and_location = []
for khipu_name in aFieldmark.matching_khipus():
aKhipu = khipu_dict[khipu_name]
top_cord_sum_groups = aKhipu.top_cord_sum_groups()
for group in top_cord_sum_groups:
top_cord_value = mean([aCord.knotted_value for aCord in group.pendant_cords()])
top_cord_location = mean([aCord.pendant_index() for aCord in group.pendant_cords()])
cord_value_and_location.append((khipu_name, top_cord_value, top_cord_location))
top_cord_df = pd.DataFrame(cord_value_and_location, columns=['kfg_name', 'cord_value', 'location'])
top_cord_dict = {}
for row_index in range(len(top_cord_df)):
khipu_name = top_cord_df.kfg_name.values[row_index]
cord_value = top_cord_df.cord_value.values[row_index]
if khipu_name in top_cord_dict:
top_cord_dict[khipu_name].append(cord_value)
else:
top_cord_dict[khipu_name]= [cord_value]
top_cord_dict = dict(sorted(top_cord_dict.items(),key= lambda x:len(x[1])))
top_cord_table = sorted([[kfg_name, str(len(top_cord_dict[kfg_name])), str(mean(top_cord_dict[kfg_name]))] for kfg_name in top_cord_dict],
key = lambda x:(x[1], x[2]),
reverse = True)
top_cord_khipus = [top_cord_table[i][0] for i in range(len(top_cord_dict))]
top_cord_count = [len(top_cord_dict[khipu_name]) for khipu_name in top_cord_khipus]
top_cord_mean = [mean(top_cord_dict[khipu_name]) for khipu_name in top_cord_khipus]
top_cord_min = [min(top_cord_dict[khipu_name]) for khipu_name in top_cord_khipus]
top_cord_max = [max(top_cord_dict[khipu_name]) for khipu_name in top_cord_khipus]
top_cord_khipu_mean = [khipu_dict[khipu_name].mean_pendant_cord_value() for khipu_name in top_cord_khipus]
top_cord_khipu_min = [khipu_dict[khipu_name].min_pendant_cord_value() for khipu_name in top_cord_khipus]
top_cord_khipu_max = [khipu_dict[khipu_name].max_pendant_cord_value() for khipu_name in top_cord_khipus]
top_cord_data = zip(top_cord_khipus, top_cord_count, top_cord_mean, top_cord_min, top_cord_max, top_cord_khipu_mean, top_cord_khipu_min, top_cord_khipu_max)
colnames = ['Khipu Name', 'Num of Top Cords', 'Mean Top Cord', 'Min Top Cord', 'Max Top Cord', 'Mean Khipu Cord', 'Min Khipu Cord', 'Max Khipu Cord']
top_cord_display_df = pd.DataFrame(top_cord_data, columns = colnames)| Khipu Name | Num Top Cords | Mean Top Cord | Min Top Cord | Max Top Cord | Mean Khipu Cord | Min Khipu Cord | Max Khipu Cord |
|---|---|---|---|---|---|---|---|
| AS044 | 9 | 42 | 30 | 76 | 15 | 0 | 160 |
| UR118 | 7 | 460 | 369 | 581 | 122 | 0 | 1957 |
| UR1096 | 7 | 27 | 12 | 60 | 5 | 0 | 60 |
| AS066 | 7 | 113 | 32 | 237 | 24 | 0 | 300 |
| AS199 | 6 | 376 | 50 | 587 | 104 | 0 | 922 |
| UR166 | 5 | 701 | 17 | 1417 | 244 | 0 | 1417 |
| AS007 | 5 | 126 | 80 | 152 | 16 | 0 | 170 |
| QU009 | 4 | 78 | 50 | 98 | 10 | 0 | 239 |
| UR119 | 4 | 423 | 390 | 481 | 85 | 0 | 643 |
| AS006 | 4 | 125 | 105 | 140 | 34 | 4 | 140 |
| UR186 | 3 | 250 | 75 | 596 | 38 | 0 | 596 |
| AS072 | 3 | 200 | 30 | 530 | 69 | 0 | 530 |
| AS001 | 2 | 97 | 93 | 101 | 27 | 3 | 101 |
| UR047 | 2 | 80 | 60 | 100 | 16 | 4 | 120 |
| QU014 | 2 | 738 | 541 | 935 | 45 | 0 | 935 |
| UR246 | 2 | 302 | 288 | 316 | 67 | 5 | 316 |
| UR087 | 2 | 15 | 15 | 16 | 3 | 0 | 26 |
| HP047 | 1 | 842 | 842 | 842 | 191 | 0 | 2643 |
| AS215 | 1 | 76 | 76 | 76 | 7 | 1 | 76 |
| AS003 | 1 | 76 | 76 | 76 | 21 | 4 | 76 |
| UR149 | 1 | 601 | 601 | 601 | 48 | 0 | 601 |
| QU015 | 1 | 512 | 512 | 512 | 248 | 0 | 2792 |
| AS004 | 1 | 312 | 312 | 312 | 246 | 10 | 932 |
| UR048 | 1 | 25 | 25 | 25 | 7 | 0 | 158 |
| UR1057 | 1 | 2 | 2 | 2 | 1 | 0 | 104 |
| UR241 | 1 | 18 | 18 | 18 | 6 | 0 | 61 |
| QU019 | 1 | 173 | 173 | 173 | 30 | 0 | 601 |
| QU016 | 1 | 129 | 129 | 129 | 15 | 0 | 129 |
| AS002 | 1 | 117 | 117 | 117 | 33 | 8 | 117 |
In this case, it’s pretty clear what we are searching for. The only wrinkle is that they khipu-kamayoq who made the khipu frequently was off by one in one digit - getting 468 instead of 469 or 569 instead of 469. That case has to be checked.
# Code excerpts:
def tag_sum_groups(self):
if len(self.cord_groups) >= 2:
for i in range(len(self.cord_groups)-1):
(sum_matches, exact_match) = self.cord_group_sums_match(self.cord_groups[i], self.cord_groups[i+1])
if sum_matches:
if self.cord_groups[i].num_cords() == 1:
self.cord_groups[i].tag_sum_group(self.cord_groups[i+1], is_exact_sum= exact_match)
elif self.cord_groups[i+1].num_cords() == 1:
self.cord_groups[i+1].tag_sum_group(self.cord_groups[i], is_exact_sum= exact_match)
def cord_group_sums_match(self, left_group, right_group):
left_sum = left_group.group_cord_sum()
right_sum = right_group.group_cord_sum()
if left_sum > 0: # A lot of uninformatively tagged 0 sum cord groups. Much ado about nothing.
if left_sum == right_sum:
return (True, True)
elif left_sum in self.off_by_one(right_sum) or right_sum in self.off_by_one(left_sum):
return (True,False)
else:
return (False, False)
else:
return (False, False)
def near_digits(self, aDigit):
min_digit = max(0, aDigit-1)
max_digit = min(9,aDigit+1)
return list(range(min_digit, max_digit+1))
def off_by_one(self, aNumber):
digit_chars = list(f"{aNumber}")
int_chars = [int(aChar) for aChar in digit_chars]
near_combos = [self.near_digits(x) for x in int_chars]
final_possibilities = []
for i, aCombo in enumerate(near_combos):
for possible_digit in aCombo:
final_possibilities.append("".join(digit_chars[0:i]) + f"{possible_digit}" + "".join(digit_chars[i+1:]))
off_by_one_possibilities = [int(aString) for aString in final_possibilities]
return off_by_one_possibilitiesThis is such a rare (and unambigous) fieldmark that any occurrence is deemed significant
| Measure | Result |
|---|---|
| Number of Khipus That Match | 22 (3%) |
| Number of Significant Khipus | 22 (3%) |
| Five most Significant Khipu | AS044, AS066, UR1096, UR118, AS199 |
| Image Quilt | Click here |
| Database View | Click here |
from plotly import express as px
# 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 Top Cord Sum Cords", },
title=f"Matching Khipu ({len(matching_khipus)}) for Number of Top Cord 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 Top Cord Sum Cords", },
title=f"Significant Khipu ({len(significant_khipus)}) for Number of Top Cord Sum Cords", width=944, height=450).update_layout(showlegend=True).show()Note that all double sum top cord khipus exist on khipus that also have “regular” top sum cords.
A total of 86 sum top cords exist over 22 khipus with sum top cordsfig = (px.scatter(top_cord_df,
x="location", y="cord_value",
#color="handed_color", color_continuous_scale=['#3c3fff', '#ff3030',],
labels={"location":"Top Cord Location From Left",
"cord_value":"Top Cord Sum Value"},
hover_name="kfg_name", hover_data=['location', 'cord_value'],
title=f"<b>Top Cord Sums - Position vs. Sum Value</b>",
width=944, height=944)
.update_layout(showlegend=False)
.update(layout_coloraxis_showscale=False)
.show()
)An examination of the sum top cord data reveals that many of the sum top cords are associated with a group of six pendant cords. How much? An astonishing 70%.
import utils_loom as uloom
sum_top_cord_df = aFieldmark.dataframes[1].dataframe
num_summand_cords = []
for kfg_name in sum_top_cord_df['kfg_name'].tolist():
kfg_top_cord_df = sum_top_cord_df[sum_top_cord_df.kfg_name == kfg_name]
summand_string = kfg_top_cord_df.summand_string.values[0]
num_summand_cords.append(len(aFieldmark.cords_from_sum_string(khipu_dict[kfg_name], summand_string, one_based=False)))
sum_top_cord_df['num_summand_cords'] = num_summand_cords
num_6_summands = sum_top_cord_df[sum_top_cord_df.num_summand_cords == 6].shape[0]
num_8_summands = sum_top_cord_df[sum_top_cord_df.num_summand_cords == 8].shape[0]
total_sum_cords = sum_top_cord_df[sum_top_cord_df.num_summand_cords > 0].shape[0]
from collections import Counter
num_summands_counter = Counter(num_summand_cords)
bar_plots_tuples = [(key, num_summands_counter[key]) for key in num_summands_counter if key > 0]
# Plot Matching khipu
matching_df = pd.DataFrame(bar_plots_tuples, columns =['KhipuName', 'Value'])
fig = px.bar(matching_df, x='KhipuName', y='Value', labels={"KhipuName": "Khipu Name", "Value": "Number of Top Cord Sum Cords", },
title=f"Matching Khipu ({len(matching_khipus)}) for Number of Top Cord Sum Cords", width=944, height=450).update_layout(showlegend=True).show()
print(f"{uloom.percent_info(num_6_summands, total_sum_cords, rounddigits=0):} Sum Top Cords have 6 Summands")
print(f"{uloom.percent_info(num_8_summands, total_sum_cords, rounddigits=0):} Sum Top Cords have 8 Summands")47% (40 of 86) Sum Top Cords have 6 Summands
15% (13 of 86) Sum Top Cords have 8 Summands