The Biotic Pattern of Prime
Numbers Supports the Bios Theory of Creative Evolution from Radiation to
Complexity
H. Sabelli,
Chicago Center for Creative Development. Chicago, IL 60614
Mathematical analyses reveal life-like (biotic) creativity and
harmonic features in the prime number series. Bios is a pattern characterized
by causal novelty (measurable as reduction in recurrence by randomization).
Biotic features of novelty, causation, diversification, non-random complexity,
and episodic patterning are observed (after detrending) in the series of prime
numbers, the differences between consecutive primes, and the distribution of
primes. Plots of the sine versus cosine transforms of the primes show circular
symmetry as in electromagnetic radiation, which is also demonstrable in
heartbeats and in the Schrödinger equation. Recursions of sinusoidal functions
generate bios. Bios has been demonstrated in quantum, cosmological,
meteorological, cardiac, population, and economic series. These results support
the Bios Hypothesis that natural and human processes and their evolution are
causal creative processes, largely driven by cyclic engines homologous to
electromagnetic radiation which are capable of generating diversity, novelty
and complexity.
1 Introduction
Here I study the prime numbers with methods developed to analyze
empirical creative processes. In addition
to the intrinsic mathematical interest of prime numbers, they also are the exemplar
of novelty, in the sense that they are not multiples of any other, and they are
generated mathematically, not by random chance. Natural
and human processes are creative, i.e. they generate diversity, novelty and
complexity. Yet, they are described as deterministic or indeterministic. From
quantum mechanics and the origin of cosmological non-uniformities to biological
evolution and socioeconomic processes, “chance” is explicitly assumed to
account for novelty. This fails to provide methods to foster healthy creative
processes and to hinder the emergence of destructive ones. The goal of our
research program is to contribute to the development of a science of creative
processes and its application to human endeavors.
Table 1. Non-Periodic
Processes
|
Stationary Stable standard deviation. Recurrence. Stable degree of complexity. Flat power spectrum. Uniform, stable pattern. |
Creative Diversification. Novelty. Complexification. 1/f power spectrum. Continual transformation. |
Stochastic Random difference between consecutive terms. |
RANDOM |
RANDOM WALK |
Causal (“deterministic”) Patterned series of differences between consecutive
terms. Partial auto-correlation. |
CHAOS |
BIOS |
Taking biological processes as a model for creativity,
we have developed a series of methods that allow one to measure
diversification, novelty and life-like complexity in time series [Sabelli and
Abouzeid, 2003]. These methods identify Bios as a distinct pattern [Kauffman
and Sabelli, 1998]. Biotic patterns are generated by the recursion of
trigonometric functions such as A(t+1) = A(t)
+ g * sin(A(t)) that involve action
(recursion), harmonic opposition (sine function) and conservation (the A(t)
term). Table 1 summarizes the similarities and differences between biotic
processes and random, chaotic and stochastic processes as measured by time
series analyses.
Biotic patterns are found at
all levels of organization:
(1) Quantum
physics: Schrödinger wave function [Sabelli and
Kovacevic, 2006] and its relativistic version, the Klein-Gordon-Fock equation [Thomas et al.,
2006].
(2) Cosmology: temporal distribution of
galaxies [Sabelli and Kovacevic, 2006] and quasars [Thomas et al., 2006].
(3) Planetary:
air and ocean temperature, river levels, river and shore fractal forms [Sabelli, 2005].
(4) Molecular
biology: sequences of bases in DNA [Sabelli, 2005].
(5) Physiology: heartbeat series and
respiration [Carlson-Sabelli et al, 1994; Sabelli
and Carlson-Sabelli, 2005].
(6) Biology: population size of several animal species [Sabelli
and Kovacevic, accepted for publication].
(7) Economic series [Patel and Sabelli, 2003; Sugerman and Sabelli, 2003; Sabelli,
2005].
(8) Music [Levy et al, 2006].
2. Methods
Following
the notion that physical action, bipolar opposition and causal creativity are
basic features of fundamental natural and human processes, Bios Analysis
involves three steps:
1. Measuring Action:
Everything is a process, i.e. a sequence of actions (energy change in time).
Processes thus have numerical properties, cardinal and ordinal. Data must then
be collected and interpreted as time series, and their analysis must include
measures of quantity (changes of intensity in time) and of order, such as
directionality (temporal asymmetry) and transitivity (causality) between
consecutive terms. As linear action often predominates, detrending by subtracting from each term the local average may
be required to detect periodic and creative patterns. Samples of prime numbers
up to two million primes are studied. Each series is compared with surrogate
copies randomized by shuffling, and with random integer series sorted in
ascending order, random data, brown noise, 1/f noise, and chaotic and biotic
series generated with the process equation A(t+1) = A(t) + g * sin(A(t)); these
series are chaotic for g between 3.9 and 4.6035…, and biotic for larger g.
2. Measuring
Opposition: Every process includes opposite actions and sub-processes;
fundamental processes such as radiation and natural macroscopic cycles embody
simple harmonic opposites, i.e. bipolar, bidimensional and diverse. When only
one time series is available, the opposite processes that produce it can
presumably be brought to light by generating and examining the time series of
the sine and the cosine of the data. Complement
plots [Sabelli, 2000) are constructed by plotting the sine and the cosine
of each term in orthogonal axes, and connecting successive terms with lines;
presumably, as sine and cosine are complementary opposites, the pattern
observed in complement plots of a time series reveals the opposites contained
in the process that generates it. The chords form various patterns that can be
readily identified.
3. Measuring Creativity:
Natural processes are creative, i.e. generate diversity, novelty, and complex
patterns. Regarding the pattern of heartbeat intervals as a model, we
characterize biotic complexity by (1) diversification, (2) episodic
complexes, (3) novelty, and (4) 1/f power spectrum [Patel and Sabelli, 1993],
four fundamental properties of bios absent in chaotic attractors, (5)
fractality, present also in chaos, and (6) lower entropy (Schrödinger, 1945;
Prigogine, 1980]; entropy decreases in the transition from chaos to bios in
mathematical recursions (Sabelli, 2005).. These time series properties are
quantified with the Bios Data Analyzer (Sabelli et al. 2005] and the
Bios Analyzer [http://www.inverudio.com].
Diversification is the increase in variance with increasingly larger
samples (global diversification) or embedding (local diversification); one
constructs vectors of increasing length starting with each term of the series,
averages the S.D. of vectors of equal length, and plots these results as a
function of the embedding, Sabelli and
Abouzeid, 2003]. Diversification is observed in biotic series and in
Brownian noise but not in random, periodic or chaotic series. Recurrence plots
demonstrate transformations of pattern, and portray scale-free fractal
structure, with distinct clusters of recurrences (complexes). Novelty in time series can be detected and
measured by the quantification of recurrences of isometric vectors of N
consecutive terms (N =2, 3,… 200). In biotic series and in Brownian noise,
there are less recurrences than in their shuffled copies, in contrast to
periodic and chaotic series that have a larger number of recurrences than
randomized copies [Sabelli, 2001]. As
repetition characterizes static order, I define novelty as lower than random
recurrence. Novelty is the defining feature of bios that differentiates
it from chaos. Statistical entropy
measures diversity and symmetry (not disorder, as a series and its shuffled
copy have the same entropy). These two components of entropy can be separately
quantified by calculating entropy with a range of bins (2 to 100) and plotting
it as a function of the logarithm of the number of bins; the slope measures
diversity (slope) and the intercept at two bins measures symmetry (1 for
symmetric data) [Sabelli et al, 2005].
3.
Results
3.1 Recurrence Plots and Quantification
Recurrence plots of detrended prime series (figure 1) reveal distinct clusters of isometries
(complexes) separated by recurrence-free intervals. These
complexes display complex internal structure regardless of scale, as observed
with fractal series. Shuffling largely obliterates
complexes.
Figure 1. Recurrence plots of the detrended first 10000
primes,
50 embeddings, cutoff radius 1. 300,000 comparisons
Recurrence plots of the
series of differences ΔP(n) = P(n+1) - P(n) between consecutive primes,
the series of differences between consecutive primes Δ P(n) after
detrending, and of the series of differences of differences, also show marked
structure. Shuffling blurs and even erases completely
such organization.
The
demonstration of these multifarious patterns points to continually creative
processes; complexes are
observed in biotic series and in brown noise, while random and chaotic series
show more uniform recurrence plots, similar to those of their shuffled copies
[Sabelli, 2005]. Pattern in the series of differences implies that the changes
that generate the process are causal actions, not random events. The creative
process is causal, i.e. biotic not stochastic.
The number of isometric
recurrences (figure 2) is lower in detrended prime series, the series of
differences between consecutive primes, the series of differences between
consecutive primes after detrending, and the series of differences of
differences, than in their shuffled copies.
As action is the integral
of changes of energy in time, also the series generated by the sum of
differences between consecutive primes A(t+1) = A(t) + ΔP(n) and the sum
of the differences of differences [after detrending the prime series P(n) or
the series of differences ΔP(n)] have been examined, and found to show
marked structure (figure 3). Novelty is clearly demonstrated in all these series.
Figure 2. Embedding plots of the detrended first 3000
primes.
Thick lines show the results obtained with the original
series. Thin blue lines show results after randomization.
Figure 3. Recurrence
plots of the series A(t+1) = A(t) + ΔP(n) (left and center) and the series
A(t+1) = A(t) + ΔP(n+1) - ΔP(n) (right) after detrending the prime
series (left and right), or the series of differences (center).
3.2 Biotic statistics
Obviously variance increases in the ascending series of primes,
but also detrended series display global and local diversification (figure 4),
while a shuffled copy of the data has a uniform S.D. as observed with chaotic
and random data.
Figure 4. Increase variance with increase
number of data points (left) or with embedding (right)
Prime number series have much lower entropy than random data. The
entropy-bin regression lines (figure 5) show that the series of detrended
primes and of the series of the difference between consecutive terms are
asymmetric (intercept less than 1) as observed with creative processes (bios, Brownian, 1/f “noise”) while
periodic, chaotic, and random series are symmetric.
Distributing the series of primes in equal bins generates an
irregular series. Recurrence plots of this series show complexes with
intricate forms.
Figure 5. Statistical entropy calculated with an
increasing number of equal bins.
Figure 6.
3.3 Trigonometric
Analysis
Plotting the series of prime numbers modulo 2
π, as well as trigonometric transformations of the primes (figure 6), show
periodic features, indicating that harmonic processes may contribute to the
generation of prime numbers. In particular, plotting opposite trigonometric
transforms, sine versus cosine, reveals a remarkable pattern of multiple
circles that many persons spontaneously describe as a Mandala (figure 7).
|
|
|
Initial 1500 prime
numbers |
Random integers
ascending order |
Random integers,
equidistant |
|
|
|
A(t+1) = A(t) + 4.64 *
sin(A(t)) |
Schrödinger (*10) |
Heartbeat intervals |
Figure 7. Complement plots of rounded series. Random
series do not show Mandala patterns. |
4.
Discussion
4.1 Biotic pattern of the
prime series
The
combination of novelty, complexes, diversification,
entropic asymmetry and Mandala symmetry observed in the prime series is
characteristic of biotic patterns generated by the recursion of trigonometric
functions, including the Schrodinger equation. The prime series also
demonstrates 1/f power spectrum, fractality and anti-persistence [Wolf, 1989,
1997, 1998; Shanker, 2006], features also evident in biotic series generated by
the recursion of trigonometric functions [Patel and Sabelli, 2003]. These
features can also be observed in stochastic noise generated by random changes
but prime numbers are generated mathematically and the demonstration of pattern
in the series of differences and differences of differences also implies causal
rather than aleatory change. We have also found biotic behavior in series
generated by the Riemann’s zeta function [Kauffman and Sabelli, This Meeting].
The results obtained not only reveals biotic features
in the pattern of the prime number series but they also provide evidence for a
biotic process in their generation. Prime numbers appear to be generated by a
cumulative harmonic processes such as those that generate bios in mathematical
recursions. Mathematical experiments with recursions indicate the need for
bipolar and bidimensional opposition and for the accumulation of change to
generate bios; e.g. A(t+1) = A(t) + g*sin(A(t)) produces bios while A(t+1) =
g*sin(A(t)) generates only chaos [Sabelli and Carlson-Sabelli, 2005].
4.2 Number, Physics and Novelty
The biotic series of primes is generated within the
cumulative process of adding 1 to each successive member of the number series.
A number of studies connect number theory, and in particular the prime series,
with fundamental physical processes, as illustrated by the work of Hugh
Montgomery, Andrew Odlyzko, and Alain Connes. Physical processes by necessity
“obey” the universal laws of mathematics. Numbers are not only the product of
human minds but they abstract basic properties of physical action. Action is
the integral of changes of energy in time. In the prime series, the homolog of
temporal order is sequence, and the homolog of action is the change in the
difference between consecutive primes and the differences of differences. As
illustrated by recursions that accumulate these changes (figure 3), the
generation of primes shows a biotic pattern (once the linear component of
change in the prime series is removed by detrending). As discussed in the
introduction, also physical, biological and human processes display bios.
Action is cardinal (intensity of energy), ordinal
(time) and arithmetic (sum or integral). Change is always a multiple of action
units (quanta). Numbers have three aspects: ordinal, cardinal, and formal (as
highlighted by Pythagoras, Pierce, and Gödel). Ordinal, cardinal, and formal
are not three series of numbers, but three aspects of each number. Order and
quantity, ordinals and cardinals, grow together in the linear sequence of
numbers. By necessity accumulation (cardinality) requires increase in time
(order). The accumulation of change generates novelty and complexity; the sum
of random change generates complex Brownian noise, and the sum of chaotic
events generates Bios. The mere sequence of numbers generates patterns,
starting with the alternation of odd and even numbers (period 2); period 2 is a
universal dialectic of nature. It also generates the prime numbers, each
representing a new form. Primes are examples of novelty, complexity and
nonlinear changes in quality associated with changes in linear quantity.
The law of quantity and quality posits a general relation
between linear and nonlinear processes: changes in linear quantity generate
nonlinear changes in quality, which may be gradual and linear or abrupt and
nonlinear (e.g. biological thresholds and physical critical points). Linear and
non-linear changes are associated. as complementary aspects of processes, not
two different classes of processes. Pythagoras was perhaps the first scientist
to show the relation between quantity and quality, in his empirical study of
the relation between the length of the chords and harmony between their musical
sound. Galileo famously stated that the book of
nature is written in the language of mathematics, and highlighted the
importance of scaling in physical processes. Scaling is fundamental in biology
[Brown and West, 2000]. Hegel advanced the notion that quantitative
change becomes qualitative change (e.g. water into ice) that Engels generalized
as a law of nature. Expanding this concept, primes show that linear increases in quantity create novelty,
quality and complexity. The law of
quantity and quality provides a method to analyze and promote creative
processes.
The prime number theorem relates the three aspects of
number: the frequency of novelty (density of primes) decreases with the
increase in their ordinal and cardinal value. On the other hand, the prime
number series is fractal. This coexistence between numerical dependent and
independent quality exemplifies a complementarity of opposites such as observed
in harmonic processes.
The
sequence of integers may be regarded as period 1. Each prime number is the
origin of the infinite series of its multiple harmonics. In turn, the process
that generates primes appears to involve harmonic functions [Shlesinger, 1986;
Ares and Castro, 2006]. The zeros of the Riemann zeta-function can be regarded
as harmonic frequencies in the distribution
of primes. Here we observe harmonic features in the prime series itself. In
particular, plotting opposite trigonometric transforms, sine versus cosine,
reveals a pattern of multiple circles, a
two-dimensional view of the helical motion of the trajectories of the prime
generating process in the third, here collapsed, dimension of time. Prime
generation involves rotation. Such pattern of
multiple circles is found in few but significant processes: heartbeat
intervals [Sabelli, 2000] and in integer transforms of series generated by the
Schrödinger and the Klein-Gordon-Fock
equations [unpublished]. The pattern is also displayed by integer
transforms of sine waves and of biotic series generated by recursions involving
sine waves but not in other similar series.
Many persons spontaneously describe this pattern as a
Mandala, an ancient religious symbol found in almost all civilizations. Young
children spontaneously draw Mandala forms. They also appears in dreams and
doodles of adults [Jung, 1967]. Mandala forms occur also in E8 [http://aimath.org/E8/mcmullen.html].
The appearance of the same form in physical and biological processes,
psychosocial symbols and mathematical series points to deep homologies in the
universe.
4.4 Bios as a primal process
From harmonic motion and electromagnetic radiation to proteins and
DNA, helical patterns are fundamental. The sinusoidal pattern of electromagnetic radiation embodies a mutual
transformation of opposites that may impart this dialectic to more complex
physical and mental processes. The widespread
occurrence of Bios is to be expected as there are multiple recursive processes
in nature that involve cycling starting with electromagnetic radiation that
construct atoms, molecules, and organisms, and transmits information across
cosmological distances as well as in atoms, computers and brains. There
are likewise cyclical processes at higher levels of complexity, including biological, economic, and social where bipolarity
would imply synergic and conflictual interactions, as contrasted to
one-sided notions of competition and struggle as the motor of biological
evolution and economic development. According to Bios Theory, these synergic and
conflictual interactions are cyclic
engines that drive motion and qualitative change.
As radiation embodies harmonic opposites,
recursions of harmonic functions generate biotic novelty, diversity and
complexity, and bios is ubiquitous in nature, I conjecture that homologous
processes of harmonic feedback causally generate complexity at all levels of
organization. The analysis of primes extends previous observations on physical,
biological and socioeconomic processes to the most basic mathematical structure
of reality.
Physics embodies number, and human minds create number,
pointing to a profound homology between levels of organization. Such homology
is illustrated, for instance, by the Mandala created by sinusoidal waves as
radiation, by the prime number series, and by personal and religious mental
processes. As number series abstract fundamental
properties of action, and the generation of primes epitomizes causal novelty,
the results support the notion that causality is creative, not deterministic.
Creation is causal, not aleatory. I conjecture that causality and creativity in
nature are mathematically determined [Sabelli, 2005].
The biotic pattern of prime number series points to the
role of cumulative changes in the generation of novelty, and the patterns
observed in their trigonometric transformation to harmonic processes. These results are consistent with the Bios Hypotheses:
1. Cumulative increases in quantity and order create novelty, quality and complexity.
This process is universal, because physical action is cumulative (action is the
integral of energy changes) and asymmetric (action involves temporal order).
2. Fundamental natural and mental processes
are harmonic, i.e. involve bipolar, diverse, bidimensional, and mutually
transforming (not mutually excluding) opposites. In other words, they contain cyclic engines that drive motion and qualitative
change. Such harmonic opposite processes are not portrayed appropriately by
traditional, dialectic or Boolean logic, but can be represented by trigonometric functions. This bios model of
opposition may then be useful to model logical negation in quantum computation
[Sabelli and Thomas, in press].
3. Biotic processes combining cyclic
engines and accumulation are ubiquitous in
natural and human processes, and may play a significant
role in physical and biological evolution [Sabelli,
2007].
The analysis of primes thus grounds Bios in Number
Theory, and in turn points to dialectic features in the logic of mathematics.
These notions are pursued in a companion paper [Kauffman and Sabelli, This
Meeting].
Acknowledgements: This research was supported by generous gifts from Ms. Maria
McCormick to the Society for the Advancement of Clinical Philosophy.
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