**A Logarithmic Scaling of the
Time Series of Cosmic Evolutionary Events to the Base Seven **

Bernard A. Power

Consulting Meteorologist (ret.’d)

and

J. F. Power Ph.D

**1. Introduction**

**2. The Cosmic Time
Scale**

**3. A Logarithmic
Scaling **

**4. Basic Events
of the Cosmic Time Series**

**6. Graphs of the
Results**

**7. Discussion of
Results**

**8. Evidence that logarithmic mechanisms to the base seven may play some physical role
**

**References and Notes**

__1. Introduction __

The current controversy
of evolution versus Intelligent Design in the

Inevitably the dispute has
widened to some extraneous matters. One
of these is the tendency of some to use
evolution against “creationism” and, by implication, against the idea of
creation in general. This inevitably drags into the dispute the venerable
account of the seven days of creation
( six days of creation and one of rest) in the first chapter of
Genesis This may be shortsighted as we shall now explore.

__2. The Cosmic Time Scale for the Emergence of Complexity in the Universe__

Our cosmic beginning is
the Big Bang which is now widely
accepted as occurring about 13.7 billion
years ago ( 13.7 x 10^{9} yrs in convenient notation). The next 9
billion years saw the proliferation of stars, galaxies and nebulae. The planet
earth solidified from the solar system around 5
billion years ago.

The next emergent step in
complexity is life in the form of the first primitive bacteria and algae emerged on earth around 3.5 billion years ago
(1).

On present consensus these
primitive life forms appear to have stayed pretty much the same until about 500
million years ago when what is called
the Cambrian explosion occurred and
higher plants and animals emerged abundantly.

Mammals flourished after
about 65 million years ago, emerging around the so-called Cretaceous
Catastrophe which is presumed to have abruptly ended the dinosaur age when the
earth was hit by an asteroid.

The great apes came in with
the gibbons at 18 million years ago, the orangutans at 14 million, gorillas and
chimpanzees at 7 million years.

The various hominids, walking erect and using tools, as * Homo erectus etc. *appear around 1.8 to 1.4 million years ago and persisted
until around 250,000 years ago. The
Archaic tool-using hominids appear at around 1.4 million years ago. The
Neanderthals lasted from say 150,000 years down to 28,000.years ago.

The near-modern Hetro group,
and ‘Mitochondrial Eve and Adam ’ descendants, spread out of

The fully modern era
followed, and the Cro-Magnon peoples had
spread across southern

Clearly, we have seven __highly
unequal__ time episodes of emergent or
evolutionary complexity on the linear
time scale of terrestrial years, the
first cosmic episode being around 9 billion years long and the latest
fully modern episode being only a hundred thousand years long. Table 1.

__3. A
logarithmic scale of emergent complexity__

Such an immensely long, disparate
time- series has, of course, been recognized as logarithmic, and so it has occasionally been
graphed logarithmically, usually using
common logs to the base 10.

Dawkins [1] gives several
examples of the occurrence of
logarithmic relationships in biology, such as Kleber’s Law which links
the common logarithm of body mass to the log of metabolic rate for a wide variety of organisms. In population probability applications the logs used are often to the base 2. Some
organic growth phenomena are logarithmic to the base of natural logs which
is 2.71828. Entropy
in the Boltzman formulation is natural logarithmic. All logarithms are interrelated by simple numerical factors and
are therefore convertible. The choice of a particular log base may often be a matter of numerical
convenience, but it may in some cases be dictated by the nature of an
underlying physical process.

When we convert the cosmic
times listed above to logarithms ( using any log base) the data all plot along a quasi- straight
line with a correlation coefficient of around 0.99. Also, each log plot gives a
line with a different number of equal logarithmic intervals over the time
series of seven very unequal episodes or
eras of physical complexity. Thus, for
example, a logarithmic plot to the base ten has 5 equal logarithmic intervals
(slope 0.84), natural logs to the base 2.71828 give 12 equal logarithmic
intervals ( slope 1.93).

After some further study we
have found that logarithms the base seven ( log_{7} N = log_{7}
( 7^{n}) = n) , where N is years, are
of particular interest since they provide a uniquely equal relationship
of seven equal logarithmic time
intervals now matching the seven unequal
cosmic time intervals ( Slope 0.998).
Table 1 and Figure 1.

In other words, on this one
particular logarithmic scale to the base
7, the scientific account of seven
unequal time eras in the observed
emergence of complexity becomes
transformed to a series of seven
mathematically equal episodes, stretching from the time of the Big Bang
down to the current era of the emergence of mankind**. **The logarithmic series is relatively insensitive to quite large
variations in the time estimates for the events.

Of course, one log scale is still
no more to be preferred than any other
in the absence of some applicable physical theory of emergent complexity or
evolutionary process. But neither can the base 7 logarithmic scale be ruled out
as arbitrary since it uniquely gives the best mathematical relationship to the complexity series (slope
0.998).

It should now be clear what
we meant when we said above that a careless excursion out of science proper and
into the philosophical and theological
field of creation accounts might, on polemical grounds at least, be
shortsighted as well as unjustified.

__4. Basic Events of the Cosmic Time Series__

__Table 1__

**Event Time
of Emergence ( N _{t} ) yrs Log_{7}
N_{t }Log_{10}
N_{t }Log _{e}**

__Big Bang__ 13.7
x 10^{9} 11.99 10.14 23.3

__Primitive life__ 3.5
x 10^{9} 11.29** **9.5** **21.9** **

__Cambrian explosion__ 500 x 10^{6} 10.29 8.7 20

__K/T extinction__ 65
x 10^{6} 9.24
7.8 18

Dinosaurs vanish

Mammals proliferate

__Great Apes Era__

Gibbons 18 x 10^{6} (8.59)

Orangutans 14 x10^{6} (8.46) 8.31 7.06 16.3

Gorillas
7.0 x 10^{6} (8.10)

Chimpanzees 7.0 x 10^{6} (8.10)

__Hominids __

Ergast 1.8 x 10^{6 } ^{ }(7.40)

Archaics 1.4 x 10^{6} (7.27)

(Out of ^{5} (6.84) 7.17 6.06 14.0

__Near Modern Era__

Hetro group 1.6 x 10^{5} (6.16)

“Eve”group 1.4 x 10^{5} (6.10)

“Adam” group 6.0 x 10^{4} (5.66)

(Neanderthals) 1.5 x 10^{5} (6.12) 6.01 5.08 11.7

__Fully Modern Era__

Cro- Magnon Man 3 x 10^{4} 5.30 4.48 10.3

Number of logarithmic intervals:** 7 **5** **13

__Note.__** **Logarithms
of a given number N to any desired base , say base b, can easily be computed
from the common logarithms by the
following formula: log _{b} N = log_{10} N / log_{10 }b.

For
example, the logarithm to the base 2 of 8 ( i.e. b = 8) is: 0.903 /0.30103 = 3; ( 2^{3} = 8). The log to the base 7 of 8 is 0.903/0.845 = 1.069 ( 7^{1.0689} = 8). The log to the base e (e = 2.71828) of 8 is
0.903/ 0 .4343 = 2.079 ; ( e^{2.079} = 8).

__5. Plot of Log _{7}
Data__

**Figure 1. Logarithmic plot to base 7 of (A) all cosmic
data, and (B) biological data only**

For Curve A the least squares regression equation is: (a)
negative slope: y = − 0.994 x +
12.18

(b) positive slope: y = +
0.994 x + 5.22

(c) correlation coefficient:
0.998

For curve B the least squares
regression equation is: (a) negative slope:
y = − 1.02 x + 11.30

(b) positive slope: y = + 1.02
x + 5.17

(c) correlation coefficient:
0.9988

__6. A Plot of Biological Elements Only __

From the standpoint of the
processes involved in the cosmic series of events one might wonder why purely physical events of the Big Bang, the emergence of radiation, the formation of
galaxies, etc and the formation of the
earth in the first cosmic
‘interval’ are lumped in with the biological events which follow so much later. If then, we plot only the purely biological
events from the appearance of life on earth at 3.5 x 10^{9} yrs to the
arrival of Man at 100,000 to 30,000
years, we have a slightly better
relationship than before ( Curve B of
Figure 1 above ).

__7. Discussion of Results__

The main point which stands
out is that when the time series since the Big Bang of seven main cosmic
complexity eras is plotted
logarithmically to the base 7 all seven highly unequal cosmic intervals or eras fit
into seven equal logarithmic
intervals. (When only the biological events are considered the fit is a bit better, and
the number of intervals drops).

The scientific meaning of
this would seem to depend on whether some relevant physical law can be found which is
logarithmic to the base seven. If none
exists then the correlation is probably scientifically meaningless, since any
log series to any base will encompass the data in some different number of equal intervals. If, however, some physical relationship does exist
involving groupings of 7^{n}, then the logarithmic correlation to the
base 7 might point to some underlying evolutionary mechanism of importance.

A critical scientific review
of the findings, and their possible
scientific meaning seems needed.

[It should also perhaps be
pointed out that our present analysis and purpose is scientific. There is no question of any comparison with the details of the Genesis
accounts and events, which are not only completely at variance time-wise,
but are also in several instances at variance with the sequencing of the physical events of the
scientific data. The purposes and meanings of Genesis are not for science to
consider. From the polemical viewpoint of the present debate,
that is of evolution versus Intelligent
Design, the demonstrated possibility of
grouping the data scientifically into the same number of equal intervals as the
Genesis account of Creation would seem point to a need to prudently keep in
mind the proper boundaries of scientific
debate and avoid straying into philosophy
or ultimate meaning.]

__9. Evidence that the logarithmic base 7 might possibly
have a
role in the physical and/or
biological evolution of the universe__

One possibility is that the
evolution of complexity, both physically and biologically, is conditioned by the laws of **compressible fluid flow.** The physical evolution
of the cosmos involves the laws of gas dynamics and thermodynamics which
are basic to modern cosmology and
astrophysics. These same laws are
involved in biophysics and biochemistry. They have various integral
physical parameters, and their thermodynamic processes are logarithmic since
the Boltzman formulation of entropy is logarithmic. In emerging biological complexity, entropy is always involved
[2,3]. It is interesting also that
the periodic table of the chemical
elements has seven periods.

__References and Notes__

1.The dates and events in the
evolutionary event time-line used here have been abstracted from Richard Dawkins’ *The Ancestor’s Tale*. Weidenfeld and Nicholson, (2004).

**2.** G. Nicolis and *Exploring
Complexity*. (W. H. Freeman and Company, New York, 1989).

**3. ** D. R. Brooks and E. O. Wiley. *Evolution as Entropy*. (Univ. of Chicago
Press, Chicago and London, 1986).