A Logarithmic Scaling of the Time Series of Cosmic Evolutionary Events to the Base Seven
Bernard A. Power
Consulting Meteorologist (ret.’d)
J. F. Power Ph.D
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
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 109 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  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 ( log7 N = log7 ( 7n) = 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
Event Time of Emergence ( Nt ) yrs Log7 Nt Log10 Nt Log e
Big Bang 13.7 x 109 11.99 10.14 23.3
Primitive life 3.5 x 109 11.29 9.5 21.9
Cambrian explosion 500 x 106 10.29 8.7 20
K/T extinction 65 x 106 9.24 7.8 18
Great Apes Era
Gibbons 18 x 106 (8.59)
Orangutans 14 x106 (8.46) 8.31 7.06 16.3
Gorillas 7.0 x 106 (8.10)
Chimpanzees 7.0 x 106 (8.10)
Ergast 1.8 x 106 (7.40)
Archaics 1.4 x 106 (7.27)
Near Modern Era
Hetro group 1.6 x 105 (6.16)
“Eve”group 1.4 x 105 (6.10)
“Adam” group 6.0 x 104 (5.66)
(Neanderthals) 1.5 x 105 (6.12) 6.01 5.08 11.7
Fully Modern Era
Cro- Magnon Man 3 x 104 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 = log10 N / log10 b.
For example, the logarithm to the base 2 of 8 ( i.e. b = 8) is: 0.903 /0.30103 = 3; ( 23 = 8). The log to the base 7 of 8 is 0.903/0.845 = 1.069 ( 71.0689 = 8). The log to the base e (e = 2.71828) of 8 is 0.903/ 0 .4343 = 2.079 ; ( e2.079 = 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 109 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 7n, 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
3. D. R. Brooks and E. O. Wiley. Evolution as Entropy. (Univ. of Chicago Press, Chicago and London, 1986).
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