1220.00
Indigs

1220.10
Definition: All numbers have their own integrity.


1220.11
The name digit comes from finger. A finger is a digit.
There are five fingers
on each hand. Two sets of five digits give humans a
propensity for counting in increments
of 10.


1220.12
Curiosity and practical necessity have brought humans
to deal with numbers
larger than any familiar quantity immediately available
with which to make matching
comparison. This frequent occurrence induced brainplusmind
capabilities to inaugurate
ingenious human informationapprehending mathematical
stratagems in pure principle. If
you are looking at all the pebbles on the beach or all
the grains of sand, you have no
spontaneous way of immediately quantifying such an experience
with discrete number
magnitude. Quantitative comprehension requires an integrative
strategy with which to
reduce methodically large unknown numbers to known numbers
by use of obviously well
known and spontaneously employed linear, area, volume,
and timemeasuring tools.


1220.13
Indig Table A: Comparative Table of Modular Congruences
of
Cardinal Numbers: This is a comparative table of the
modular congruences of cardinal
number systems as expressed in Arabic numerals with
the individual integer symbols
integrated as indigs, which discloses synergetic wavemodule
behaviors inherent in
nature's a priori, orderly, integrative effects of progressive
powers of interactions of
number:
Visually  Nonintegrated  Indigs  Indig 
1 
1 
1 
11 
11 
2 
1
1
1 
111 
3 
11
11 
1111 
4 
11
1
11 
11111 
5 
111
111 
111111 
6 
111
1
111 
1111111 
7 
11
11
11
11 
11111111 
8 
111
111
111 
111111111 
9 
two hands 
1111111111 
10  1 
too much 
11111111111 
11  2 


1220.14
Man started counting large numbers which he did not
recognize as a discrete
and frequently experienced pattern by modularly rhythmic
repetitive measuring, or
matching, with discrete patterns which he did recognize^{__}as,
for instance, by matching the
items to be counted one for one with the successive
fingers of his two hands. This gave
him the number of separate items being considered. Heeltotoe
stepping off of the
number; or footafterfoot length dimensions; or progressively
and methodically covering
areas with square woven floor mats of standard sizes,
as the Japanese tatami and tsubo; or
by successive mouthfuls or handfuls or bowls full, counted
on the fingers of his hands,
then in multiples of hands (i.e., multiples of ten),
gave him commonly satisfactory volume
measurements.


1220.15
Most readily humans recognized and trusted one and
one making two, or
one and two making three, or two and two making four.
But an unbounded loose set of 10
irregular and dissimilar somethings was not recognizable
by numbers in one glance: it was
a lot. Nor are five loose, irregular, and dissimilar
somethings recognizable in one glance as
a number: they are a bunch. But a human hand is boundaried
and finitely recognizable at a
single glance as a hand, but not as a discrete number
except by repetitively acquired
confirmation and reflexive conditioning. Five is more
recognizable as four fingers and a
thumb, or even more readily recognizable as two end
fingers (the little and the index), two
fingers in the middle, and the thumb (2 + 2 + 1 = 5).

Fig. 1220.16

1220.16
Symmetrical arrays of identically shaped and sized,
integrally symmetric
objects evoke spontaneous number identification from
one to six, but not beyond. Paired
sets of identities to six are also spontaneously recognized;
hence we have dice and
dominoes.


1220.17
Thus humans learned that collections of very large
numbers consist of
multiples of recognizable numbers, which recognition
always goes back sensorially to
spontaneously and frequently proven matching correspondence
with experientially
integrated pattern simplexes. One orange is a point
(of focus). Two oranges define a line.
Three oranges define an area (a triangle). And four
oranges, the fourth nested atop the
triangled first three, define a multidimensional volume,
a tetrahedron, a scoop, a cup.^{1}
(Footnote 1: This may have been the genesis of the cube^{__}
where all the trouble began. Why? Because man's tetrahedron
scoop wpuld not stand on its point, spilled, frustrated counting,
and wasted valuable substances. So humans devised the squarebased
volume: the cube, which itself became an allspacefilling
multiple cube building block easily appraised by "cubing" arithmetic.)


1220.20
Numerological Correspondence: Numerologists do not
pretend to be
scientific. They are just fascinated with a game of
correspondence of their "key" digits^{__}
finger counts, ergo, 10 digits^{__}with various happenstances
of existence. They have great
fun identifying the number "seven" or the number "two"
types of people with their own
ingeniously classified types of humans and types of
events, and thereafter imaginatively
developing significant insights which from time to time
seem justified by subsequent
coincidences with reality. What intrigues them is that
the numbers themselves are
integratable in a methodically reliable way which, though
quite mysterious, gives them
faithfully predictable results. They feel intuitively
confident and powerful because they
know vaguely that scientists also have found number
integrity exactly manifest in physical
laws.


1220.21
The numerologists have also assigned serial numbers
to the letters of the
alphabet: A is one, B is two, C is three, etc. Because
there are many different alphabets of
different languages consisting of various quantities
of letters, the number assignments
would not correspond to the same interpretations in
different languages. Numerologists,
however, preoccupied only in their single language,
wishfully assumed that they could
identify characteristics of people by the residual digits
corresponding to all the letters in
the individual's complete set of names, somewhat as
astrologists identify people by the
correspondences of their birth dates with the creative
picturing constellations of the Milky
Way zoo = Zodiac = Celestial Circus of Animals.


1221.00
Integration of Digits


1221.10
Quantifying by Integration: Early in my life, I became
interested in the
mathematical potentials latent in the methodology of
the numerologists. I found myself
increasingly intrigued and continually experimenting
with digit integrations. What the
numerologist does is to add numbers as expressed horizontally;
for instance:
120 = 1 + 2 + 0 = 3
Or:
32986513 = 3+2+9+8+6+5+1+3 = 37 = 3+7 = 10 = 1+0 = 1,
Numerologically, 32986513= 1
Or:
59865279171 = 5+9 = 14+8 = 22+6 = 28+5 = 33+2 = 35+7 = 42+9 =
51+1 = 52+7 = 59+1 = 60 = 6+0 = 6,
Numerologically, 59865279171 = 6.


1221.11
Though I was familiar with the methods of the calculus^{__}for
instance,
quantifying large, irregularly bound areas^{__}explorations
in numerology had persuaded me
that large numbers themselves, because of the unique
intrinsic properties of individual
numbers, might be logically integratable to disclose
initial simplexes of sensorial
interpatterning apprehendibility.


1221.12
Integrating the symbols of the modular increments of
counting, in the above
case in increments of 10, as expressed in the tencolumnar
arrays of progressive residues
(less than ten^{__}or less than whatever the module employed
may be), until all the columns'
separate residues are reduced to one integral digit,
i.e., an integer that is the ultimate of
the numbers that have been integrated. Unity is plural
and at minimum two. (See Secs.
240.03;
527.52; and
707.01.)


1221.13
As a measure of communications economy, I soon nicknamed
as indigs the
final unitary reduction of the integrated digits. I
use indig rather than integer to remind us
of the process by which ancient mathematicians counting
with their fingers (digits) may
have come in due course to evolve the term integer.


1221.14
I next undertook the indigging of all the successive
modular congruence
systems ranging from onebyone, twobytwo pairs to
"by the dozens," i.e., from zero
through 12. (See modulocongruence tables, Sec.
1221.20.)


1221.15
The modulocongruence tables are expressed in both
decimal and indig
terms. In each of the 13 tables of the chart, the little
superscripts are the indigs of their
adjacently below, decimally expressed, corresponding
integers.


1221.16
The number of separate columns of the systematically
displayed tables
corresponds with the modulocongruence system employed.
Inspection of successive
horizontal lines discloses the orderly indig amplifying
or diminishing effects produced
upon arithmetical integer progression. The result is
startling.


1221.17
Looking at the chart, we see that when we integrate
digits, certain integers
invariably produce discretely amplifying or diminishing
alterative effects upon other
integers.
 One produces a plus oneness; 
 Two produces a plus twoness; 
 Three produces a plus threeness; 
 Four produces a plus fourness. 
Whereafter we reverse,
 Five produces a minus fourness; 
 Six produces a minus threeness; 
 Seven produces a minus twoness; 
 Eight produces a minus oneness. 
 Nine produces zero plusness or minusness. 
One and ten are the same. Ten indigs
(indig = verb intransitive) as a one and produces the
same alterative effects as does one.
Eleven indigs as two and produces the same alterative
effects as a two. All the other
whole numbers of any size indig to 1, 2, 3, 4, S, 6,
7, 8, or 9^{__}ergo, have the plus or
minus oneness to fourness or zeroness alterative effects
on all other integers.


1221.18
Since the Arabic numerals have been employed by the
Western world almost
exclusively as congruence in modulo ten, and the whole
world's scientific, political, and
economic bodies have adopted the metric system, and
the notation emulating the abacus
operation arbitrarily adds an additional symbol column
unilaterally (to the left) for each
power of ten attained by a given operation, it is reasonable
to integrate the separate
integers into one integer for each multisymboled number.
Thus 12, which consists of 1 +
2, = 3; and speaking numerologically, 3925867 = 4.

This provides an octave number system of a plus and minus
octave and an (outsideout) and an (indiseout)
differentiation, for every system has insideness
(concave) and outsideness (convex) as well as two
polar hemisystems.



1221.20
Indig Table B: ModuloCongruence Tables: The effects
of integers: One
is + 1. Two is + 2. Three is + 3. Four is +4. Five is
 4. Six is  3. Seven is  2. Eight is 1. Nine is zero; nine is none.
(The superior figures in
the Table are the Indigs.)

Congruence in Modulo Zero Integrates to Gain or Lose 0:

0  (Like nine)  0 
Congruence in Modulo One Integrates to Gain 1:

1^{1}  (Each row gains 1 

2^{2}  in each column) 

3^{3} 

4^{4} 

5^{5}  +1 

6^{6} 

7^{7} 

8^{8} 

9^{9} 

10^{1} 

11^{2} 

12^{3} 
Congruence in Modulo Two Integrates to Gain 2:

1^{1}  2^{2}  (Each row gains 2 

3^{3}  4^{4}  in each column) 

5^{5}  6^{6} 

7^{7}  8^{8} 

9^{9}  10^{1}  +2 

11^{2}  12^{3} 

13^{4}  14^{5} 

15^{6}  16^{7} 
Congruence in Modulo Two Integrates to Gain 3:

1^{1}  2^{2} 
3^{3} 
(Each row gains 3 

4^{4}  5^{5} 
6^{6} 
in each column 

7^{7}  8^{8} 
9^{9} 


10^{1}  11^{2} 
12^{3} 
+3 

13^{4}  14^{5} 
15^{6} 

16^{7}  17^{8} 
18^{9} 

19^{1}  20^{2} 
21^{3} 
Congruence in Modulo Four Integrates to Gain 4:

1^{1}  2^{2} 
3^{3}  4^{4} 
(Each row gains 4 

5^{5}  6^{6} 
7^{7}  8^{8} 
in each column 

9^{9}  10^{1} 
11^{2}  12^{3} 


13^{4}  14^{5} 
15^{6}  16^{7} 
+4 

17^{8}  18^{9} 
19^{1}  20^{2} 

21^{3}  22^{4} 
23^{5}  24^{6} 
Congruence in Modulo Five Integrates to Lose 4:

1^{1}  2^{2} 
3^{3}  4^{4} 
5^{5} 
(Each row loses 4 

6^{6}  7^{7} 
8^{8}  9^{9} 
10^{1} 
in each column 

11^{2}  12^{3} 
13^{4}  14^{5} 
15^{6} 


16^{7}  17^{8} 
18^{9}  19^{1} 
20^{2} 
4 

21^{3}  22^{4} 
23^{5}  24^{6} 
25^{7} 

26^{8}  27^{9} 
28^{1}  29^{2} 
30^{3} 
Congruence in Modulo Six Integrates to Lose 3:

1^{1}  2^{2} 
3^{3}  4^{4} 
5^{5}  6^{6} 
(Each row loses 3 

7^{7}  8^{8} 
9^{9}  10^{1} 
11^{2}  12^{3} 
in each column 

13^{4}  14^{5} 
15^{6}  16^{7} 
17^{8}  18^{9} 


19^{1}  20^{2} 
21^{3}  22^{4} 
23^{5}  24^{6} 
3 

25^{7}  26^{8} 
27^{9}  28^{1} 
29^{2}  30^{3} 
Congruence in Modulo Seven Integrates to Lose 2:

1^{1}  2^{2} 
3^{3}  4^{4} 
5^{5}  6^{6} 
7^{7} 
(Each row loses 2 

8^{8}  9^{9} 
10^{1}  11^{2} 
12^{3}  13^{4} 
14^{5} 
in each column 

15^{6}  16^{7} 
17^{8}  18^{9} 
19^{1}  20^{2} 
21^{3} 


22^{4}  23^{5} 
24^{6}  25^{7} 
26^{8}  27^{9} 
28^{1} 
2 
Congruence in Modulo Eight Integrates to Lose 1:

1^{1}  2^{2} 
3^{3}  4^{4} 
5^{5}  6^{6} 
7^{7}  8^{8} 
(Each row loses 1 

9^{9}  10^{1} 
11^{2}  12^{3} 
13^{4}  14^{5} 
15^{6}  16^{7} 
in each column) 

17^{8}  18^{9} 
19^{1}  20^{2} 
21^{3}  22^{4} 
23^{5}  24^{6} 


25^{7}  26^{8} 
27^{9}  28^{1} 
29^{2}  30^{3} 
31^{4}  32^{5} 
1 
Congruence in Modulo Nine Integrates to No Lose or Gain:

1^{1}  2^{2} 
3^{3}  4^{4} 
5^{5}  6^{6} 
7^{7}  8^{8} 
9^{9} 
(Each row remains 

10^{1}  11^{2} 
12^{3}  13^{4} 
14^{5}  15^{6} 
16^{7}  17^{8} 
18^{9} 
same value in its 

19^{1}  20^{2} 
21^{3}  22^{4} 
23^{5}  24^{6} 
25^{7}  26^{8} 
27^{9} 
column) 

28^{1}  29^{2} 
30^{3}  31^{4} 
32^{5}  33^{6} 
34^{7}  35^{8} 
36^{9} 
0 
Congruence in Modulo Ten Integrates to Gain 1:

1^{1}  2^{2} 
3^{3}  4^{4} 
5^{5}  6^{6} 
7^{7}  8^{8} 
9^{9}  10^{1} 
(Each row gains 1 

11^{2}  12^{3} 
13^{4}  14^{5} 
15^{6}  16^{7} 
17^{8}  18^{9} 
19^{1}  20^{2} 
in each column) 

21^{3}  22^{4} 
23^{5}  24^{6} 
25^{7}  26^{8} 
27^{9}  28^{1} 
29^{2}  30^{3} 


31^{4}  32^{5} 
33^{6}  34^{7} 
35^{8}  36^{9} 
37^{1}  38^{1} 
39^{3}  40^{4} 
+1 
Congruence in Modulo Eleven Integrates to Gain 2:

1^{1}  2^{2} 
3^{3}  4^{4} 
5^{5}  6^{6} 
7^{7}  8^{8} 
9^{9}  10^{1} 
11^{2} 
(Each row gains 2 

12^{3}  13^{4} 
14^{5}  15^{6} 
16^{7}  17^{8} 
18^{9}  19^{1} 
20^{2}  21^{3} 
22^{4} 
in each column) 

23^{5}  24^{6} 
25^{7}  26^{8} 
27^{9}  28^{1} 
29^{2}  30^{3} 
31^{4}  32^{5} 
33^{6} 


34^{7}  35^{8} 
36^{9}  37^{1} 
38^{2}  39^{3} 
40^{4}  41^{5} 
42^{6}  43^{7} 
44^{8} 
+2 
Congruence in Modulo Eleven Integrates to Gain 3:

1^{1}  2^{2} 
3^{3}  4^{4} 
5^{5}  6^{6} 
7^{7}  8^{8} 
9^{9}  10^{1} 
11^{2}  12^{3} 
(Each row gains 3 

13^{4}  14^{5} 
15^{6}  16^{7} 
17^{8}  18^{9} 
19^{1}  20^{2} 
21^{3}  22^{4} 
23^{5}  24^{6} 
in each column) 

25^{7}  26^{8} 
27^{9}  28^{1} 
29^{2}  30^{3} 
31^{4}  32^{5} 
33^{6}  34^{7} 
35^{8}  36^{9} 


37^{1}  38^{2} 
39^{3}  40^{4} 
41^{5}  42^{6} 
43^{7}  44^{8} 
45^{9}  46^{1} 
47^{2}  48^{3} 
+3 
Copyright © 1997 Estate of R. Buckminster Fuller