Benedikt, Appendix 3 Illustrations.
Page 1
Figures for Appendix 3:
"On Omega at Different Scales"
D
+
=
medium links only
D/4 l < D/2
short links only
0 l < D/4
Fully connected:
all link-lengths from
0 to D
+
+
medium-long links only
D/2 l < 3D/4
Figure A3.1. Decomposition of a fully connected system of diameter D into four
subsystems, each using similar-length links: "short," "medium," "medium-long," and
"long."
50
40
10
medium
short
20
0
long
30
medium-long
% links of
length l
Total number of links = N(N-1)/2,
where N = number of nodes.
D
l
Figure A3.2 The percentage of all links in a fully connected graph that fall into
the four scale categories, "short," "medium," "medium-long" and "long"
as defined in Figure A3.1.
long links only
3D/4 l D
Benedikt, Appendix 3 Illustrations.
4
Signaly
strength
upper limit of sensitivity for short links
lower limit of sensitivity for medium links
2
upper limit of sensitivity for medium links
lower limit of sensitivity for med-long links....
.25D
2
x
D
.75D
.5D
4
6
8
Distance of receiver from transmitter
Figure A3.3. Showing how signal strength combined with receiver sensivity-thresholds
can serve as a scaling device, courtesy of the inverse square law.
Cpot
Cpot
and
(bits)
0
small
Scale, l
large
D
Figure A3.4. Potential complexity and Omega (hypothetical) at different scales in
a single system (M = 15)
Page 2
Benedikt, Appendix 3 Illustrations.
cumulative Cpot
2
Cpot
1
Cpot at each scale
("alpha-distribution")
0.2
0.4
0.6
0.8
1.0
Communication range, g, as a fraction of D
Figure A3.5 How cumulative Cpot varies with communication range, g (M » 0).
10
12
14
medium
15
6
Organization,
10
R
20
contours
of equal
16
8
contours of equal Cpot
small/short
4
5
long/large
medium-long/large
2
5
10
15
Complexity, Cact
Figure A3.6 Representing at multiple scales in a single hypothetical system on the
-surface; one of indefinitely many possible 4-dot patterns describing various systems with
M=4
Page 3
Benedikt, Appendix 3 Illustrations.
10
12
14
20
system A
16
system B
15
6
8
as projected
into cave
projection lines
(to origin)
Organization,
R
10
"cave of
consciousness"
4
5
2
5
10
15
Complexity, C
Figure A3.7 Two systems, each represented at four scales, showing scales outside
the cave of consciousness projected back in.
10
12
14
20
contours
of equal
16
15
6
8
average position
average , projected
onto -surface
Organization,
10
R
4
X
X'
5
(11.5, 6.5)
= 7.5
2
5
10
15
Complexity, C
Figure A3.8 Representing the average magnitude of
appropriately on the
-surface.
Page 4
Benedikt, Appendix 3 Illustrations.
Page 5
R
R
Cpot= logM
C
C
Figure A 3.9 An array of dots on the -surface (contours not shown) representing increasing
magnitudes of Cpot = log22M (M being the group-length analyzed) of a single linear string of N symb
The gray dots are impossible because of the Bohr-Pauli rule. The black dots are impossible because
of the subsumption rule.
Cpot
R
smallest scale
C
largest scale
Figure A 3.10 The black dot represents a scale larger than the whole, which thus can break
the subsumption rule. It too is projected back into the "cave of consciousness" (shaded area).