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




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 United States was launched on the reasonable  grounds that science should be autonomous  its own field of  biological theory  with respect to organic evolution and to  the teaching  of scientific theories about it.


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 Africa after about 160,000  years ago.


The fully modern era followed, and the Cro-Magnon  peoples had spread across  southern Europe by 30,000 years ago.


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 ( 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


Table 1



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

  Dinosaurs vanish

  Mammals proliferate


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)

  (Out of Africa)                    6 .0 x 105      (6.84)                7.17                    6.06                     14.0


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 I. Prigogine. 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).



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