Number of groups whose (loosely) left half sum equals the (loosely) right half sum.
Marcia Ascher notes that in AS066 There are a number of groups whose (loosely) left half sum equals the (loosely) right half sum, which in turn equals half of a top cord (the Double Sum Top Cord fieldmark). She notes it as:
This half summing where the left half sum equals the right half sum is a fieldmark worth reviewing.
It is simple to see if there is some split index exists where the left sum equals the right sum. The trick here is to remove the uninteresting false positives, of which there are many:
This leaves 64 interesting khipus sums with 114 group’s which are sum bands.
| Measure | Result |
|---|---|
| Number of Khipus That Match | 87 (13%) |
| Number of Significant Khipus | 87 (13%) |
| Five most Significant Khipu | UR037, AS069, UR004, AS066, UR060 |
| XRay Image Quilt | Khipu by Sum Relationships |
| Database View | Click here |
# Initialize plotly
import plotly
plotly.offline.init_notebook_mode(connected = False);
# Read in the Fieldmark and its associated dataframe and match dictionary
from fieldmark_group_sum_bands import FieldmarkGroupSumBands
aFieldmark = FieldmarkGroupSumBands()
fieldmark_dataframe = aFieldmark.dataframes[0].dataframe
raw_match_dict = aFieldmark.raw_match_dict()# Plot Matching khipu
import pandas as pd
import plotly.express as px
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 Group Sum Bands", },
title=f"Matching Khipu ({len(matching_khipus)}) for Number of Group Sum Bands", 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 Group Sum Bands", },
title=f"Significant Khipu ({len(significant_khipus)}) for Number of Group Sum Bands", width=944, height=450).update_layout(showlegend=True).show()# Khipu Imports
import qollqa_chuspa as qc # A Khipu Maker is known (in Quechua) as a Khipu Kamayuq
import math
# Load khipus
(khipu_dict, all_khipus) = qc.fetch_khipus()
sum_group_df = aFieldmark.dataframes[1].dataframe
sum_group_df['log_num_cords'] = [math.log(num_cords) for num_cords in sum_group_df.num_cords.values]There are 114 group sum matches in the entire Khipu Field Guide. First, let’s review there value (how much they sum to) and where the group sum band splits.
fig = (px.scatter(sum_group_df, x="split_ratio", y="group_sum", log_y=True,
size="num_cords", color="log_num_cords",
labels={"split_ratio": "Where the group is halved - .01 is to the left, .99 is to the right", "group_sum": "group_sum (Log(y) scale)"},
hover_data=['kfg_name', 'num_cords'], title="<b>Sum Paired Groups</b> - <i>Hover Over Circles to View Khipu/Cord Info</i>",
width=944, height=944)
.update_layout(showlegend=True)
.show()
)Investigation of each sum pair is interesting. There is of course, a density at the split at .5 - right down the middle of the group. Additionally, there is a density at .33 - a third of a group. Lesser densities are evident at 0.4(2/5), 0.6(3/5) and at .333(2/6) and .666(4/6). Hovering over the scatter plot, the data shows that the values are significant, involving both large numbers of cords, and cords with large values.
| Unnamed: 0 | kfg_name | group_index | split_index | split_ratio | num_cords | split_sum | group_sum | log_num_cords | |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | AS008 | 16 | 2 | 0.500 | 4 | 18 | 36 | 1.386294 |
| 1 | 1 | AS013 | 4 | 4 | 0.500 | 8 | 18 | 36 | 2.079442 |
| 2 | 2 | AS023 | 0 | 2 | 0.250 | 8 | 20 | 40 | 2.079442 |
| 3 | 3 | AS026A | 1 | 4 | 0.211 | 19 | 15 | 30 | 2.944439 |
| 4 | 4 | AS066 | 36 | 2 | 0.500 | 4 | 25 | 50 | 1.386294 |
def fetch_group(khipu_name, group_index):
return khipu_dict[khipu_name][int(group_index)]
group_tuples = list(zip(list(sum_group_df.kfg_name.values), list(sum_group_df.group_index.values)))
groups = [fetch_group(kfg_name, group_index) for (kfg_name, group_index) in group_tuples]
banded_groups = [aGroup for aGroup in groups if aGroup.is_banded_group()]
seriated_groups = [aGroup for aGroup in groups if not aGroup.is_banded_group()]
print(f"Out of {len(groups)} groups, there are {len(banded_groups)} banded groups and {len(seriated_groups)} seriated_groups")Out of 157 groups, there are 37 banded groups and 120 seriated_groupsimport utils_loom as uloom
import re
from collections import Counter
color_strings = [[aCord.main_color() for aCord in aGroup[:]] for aGroup in banded_groups]
color_strings = uloom.flatten_list(color_strings)
def is_complex_color(anAscherColor): return (len(re.split('[\:\-\%]',anAscherColor)) > 1)
basic_colors = [color for color in color_strings if not is_complex_color(color)]
complex_colors = [color for color in color_strings if is_complex_color(color)]
basic_counter = Counter(basic_colors)
complex_counter = Counter(complex_colors)
print(f"There are {len(set(basic_colors))} basic colors over {len(basic_colors)} cords, and {len(set(complex_colors))} complex colors over {len(complex_colors)} cords.")
print(f"20 Most Common Basic Colors: {basic_counter.most_common(20)}")
print(f"20 Most Common Complex Colors: {complex_counter.most_common(20)}")There are 11 basic colors over 220 cords, and 2 complex colors over 31 cords.
20 Most Common Basic Colors: [('W', 88), ('AB', 30), ('MB', 27), ('B', 25), ('PR', 10), ('YB', 10), ('LB', 10), ('HB', 7), ('KB', 5), ('LG', 4), ('LK', 4)]
20 Most Common Complex Colors: [('W:MB', 22), ('LG:AB', 9)]color_strings = [[aCord.main_color() for aCord in aGroup[:]] for aGroup in seriated_groups]
color_strings = uloom.flatten_list(color_strings)
from collections import Counter
def is_complex_color(anAscherColor): return (len(re.split('[\:\-\%]',anAscherColor)) > 1)
basic_colors = [color for color in color_strings if not is_complex_color(color)]
complex_colors = [color for color in color_strings if is_complex_color(color)]
basic_counter = Counter(basic_colors)
complex_counter = Counter(complex_colors)
print(f"There are {len(set(basic_colors))} basic colors over {len(basic_colors)} cords, and {len(set(complex_colors))} complex colors over {len(complex_colors)} cords.")
print(f"20 Most Common Basic Colors: \n{basic_counter.most_common()}")
print(f"20 Most Common Complex Colors: {complex_counter.most_common()}")There are 39 basic colors over 959 cords, and 67 complex colors over 284 cords.
20 Most Common Basic Colors:
[('MB', 215), ('AB', 186), ('W', 180), ('YB', 105), ('B', 62), ('KB', 45), ('LB', 22), ('HB', 20), ('RB', 13), ('BS', 12), ('PB', 12), ('GG', 12), ('RL', 6), ('YG', 6), ('DG', 6), ('BG', 5), ('G0', 5), ('NB', 5), ('DB', 4), ('BB', 4), ('0G', 3), ('R', 3), ('PR', 3), ('G', 2), ('GB', 2), ('LG', 2), ('MG', 2), ('PG', 2), ('LK', 2), ('VR', 2), ('VG', 2), ('RG', 2), ('GY', 1), ('EB', 1), ('FR', 1), ('KG', 1), ('LC', 1), ('RD', 1), ('PK', 1)]
20 Most Common Complex Colors: [('W:KB', 34), ('W:AB', 32), ('W-AB', 18), ('AB:MB', 16), ('W:MB', 15), ('GG:MB', 14), ('MB:KB', 12), ('W-KB', 11), ('BB:KB', 10), ('YB:MB', 8), ('W:NB', 7), ('AB:GG', 6), ('AB:KB', 5), ('W:FB', 4), ('AB:RB', 4), ('YB:PB', 4), ('W:GG', 4), ('W-MB', 4), ('W%MB', 3), ('AB:FR', 3), ('W:YB', 3), ('MB:CB', 3), ('YG-DB', 3), ('0G:DG', 2), ('B:CB', 2), ('W:B', 2), ('MB-HB', 2), ('AB-MB', 2), ('BY:MB', 2), ('GB-HB', 2), ('RL:AB', 2), ('YB:PG', 2), ('YB:AB', 2), ('AB:G', 2), ('MG:RB', 2), ('W:AB:KB', 2), ('W:GG:MB', 2), ('W-BG', 2), ('W:DB', 2), ('DB-CB', 2), ('W%AB', 1), ('LG:RL', 1), ('HB:KB', 1), ('LG-BY', 1), ('YB-KB', 1), ('PK-R', 1), ('YB:KB', 1), ('YB:LK', 1), ('W:BL', 1), ('W:LG', 1), ('W:LG:AB', 1), ('LG-AB:W', 1), ('W-LA', 1), ('W:GL', 1), ('AB:GG:MB', 1), ('AB:GB:MB', 1), ('MB:HB', 1), ('W:HB', 1), ('W:RL', 1), ('RL-LC', 1), ('BG:MB', 1), ('YB-0Y', 1), ('W:W:GG', 1), ('W:GL:KB', 1), ('GL:MB', 1), ('AB:NB', 1), ('YG:DB', 1)]%%capture --no-display
def khipu_moment(aKhipu):
""" If the moment-weight is '1', then the moment-arm is simply the average of the position indices """
khipu_df = sum_group_df[sum_group_df.kfg_name == aKhipu.name()]
sum_band_groups = [aKhipu[int(group_Pos)] for group_Pos in khipu_df.group_index.values]
mid_point = aKhipu.num_cord_groups()/2.0
moment_pt = mean([aGroup.position()+1 for aGroup in sum_band_groups]) if len(sum_band_groups) > 0 else mid_point
#moment_pt = sum([aGroup.position()+1 for aGroup in aKhipu.cord_groups() if aGroup.is_roughly_linearly_decreasing()])/aKhipu.num_linear_decreasing_groups()
return moment_pt - mid_point
def khipu_mean_group_sum(aKhipu):
""" If the moment-weight is '1', then the moment-arm is simply the average of the position indices """
khipu_df = sum_group_df[sum_group_df.kfg_name == aKhipu.kfg_name()]
sum_band_groups = [aKhipu[int(group_Pos)] for group_Pos in khipu_df.group_index.values]
mean_sum = mean([aGroup.group_cord_sum()/2 for aGroup in sum_band_groups]) if len(sum_band_groups) > 0 else 0
return mean_sum
def khipu_sum_bands(aKhipu):
khipu_df = sum_group_df[sum_group_df.kfg_name == aKhipu.kfg_name()]
return len(khipu_df)
group_sum_khipus = [aKhipu for aKhipu in all_khipus if aKhipu.kfg_name() in matching_khipus]
group_sum_moments = [khipu_moment(aKhipu) for aKhipu in group_sum_khipus]
mean_moment_rep = mean(group_sum_moments)
mean_moment_rep_text = f'{mean_moment_rep:2.1f}'
print(f"Average moment arm is: {mean_moment_rep}")
csb_df = pd.DataFrame(columns = ['khipu_id', 'name', 'num_groups', 'mean_group_sum', 'num_sum_bands', 'moment'])
for i, aKhipu in enumerate(group_sum_khipus):
record = {'kfg_name': aKhipu.kfg_name(),
'num_groups': aKhipu.num_cord_groups(),
'mean_group_sum':khipu_mean_group_sum(aKhipu),
'num_sum_bands': khipu_sum_bands(aKhipu),
'moment': khipu_moment(aKhipu),
}
csb_df = pd.concat([csb_df, pd.DataFrame([record])], ignore_index=True)
fig = (px.scatter(csb_df, x="moment", y="num_groups",
size=list(csb_df.num_sum_bands.values),
color=list(csb_df.mean_group_sum.values),
labels={"moment": "Khipu Moment Arm", "num_groups": "Number of Overall Groups in Khipu"},
hover_data=['kfg_name'],
title="<b>Khipus with Group Sum Bands</b> - Marker Size is Number of Group Sum Bands",
width=944, height=944)\
.add_shape(type="line", x0=mean_moment_rep, y0=0, x1=mean_moment_rep, y1=140)\
.add_annotation(x=8, y=150,
text=f"Mean moment arm is ~{mean_moment_rep_text} (Vertical Line)", showarrow=False)\
.update_layout(showlegend=True).show())