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CO2 and Climate
Supporting Mathematics
Easter Island

   Supporting Mathematics

              of  the “open space” hypothesis




This website constitutes a mathematical critique that underscores the deeply-erroneous nature of such "open space" hypotheses.

Since marine biologists routinely sample one-liter samples of red-tide outbreaks, the following data provide a starting point for a mathematical portrait:  First, severe and deadly red-tide conditions are common when K. brevis populations reach concentrations of between  10,000  to  1,000,000  or more cells per liter.  Secondly, a one-liter sample of water equals approximately 61.024 cubic inches.  And thirdly, we will utilize the dimensions of a typical cell of K. brevis are also shown below.


Background values: 

A volume of 1 liter = 61.024 cubic inches

Dimensions of a typical cell of K. brevis
                              (Nierenberg, personal communication, 2008; 2008)

L:  ~30 um (= 0.03 mm)  = ~ 0.0012 inches 
W: ~ 0.0014 inches  (“a little wider than it is long" –
D: ~ 10 – 15 um deep  (10 um = .0004; 15 um = .0006), so average = ~ .0005 inches


The data above allow us to make the following calculations:

Volume of a typical cell of K. brevis  =  (L) x (W) x (D)

=  (.0012) x (.0014) x (.0005)
                                           = ~ .000 000 000 840 cubic inches



Therefore, one million K. brevis cells occupy approximately
(1,000,000) x (.000 000 000 840), or an actual physical volume
of approximately .000 84 cubic inches


Since one liter equals 61.024 cubic inches, subtracting 00.000 84 cubic inches occupied by the cells of K. brevis leaves (61.024) – (00.000 84) = approximately 61.023 16 cubic inches still unoccupied


In other words, the dinoflagellates in this liter of water still have approximately 61.023 16 cubic inches
of unoccupied volume (of apparently“ empty space”) still remaining theoretically-available to them.


Percentage unoccupied          
                                    = (61.023 16) divided by (61.024 00)
                                    =  ~ .999 987 2
                                                =  ~  99.998 72 % unoccupied
                                              volume remaining


This means that the above dinoflagellate population manages to routinely visit calamity upon themselves and the aqueous environment in which they live even when they physically-occupy less than two one-thousandths of one percent) of the total volume that seems to remain theoretically-available to them.

(100%) – (99.998 72%) =    .001 28 %

or less than two one-thousandths of one percent

of the volume that seems to remain theoretically-available to them


In other words, despite an apparently enormous amount of open space, and despite the fact that the Karenia  brevis population occupies a "volumetrically-insignificant" portion of the remaining “open space” that seems to be available,

they have, by their combined overpopulation and their
production of unseen, invisible, and calamitous wastes

catastrophically-damaged the watery surroundings in which they live.



Here we depict the physical amount of space that
constitutes two one-thousandths of one percent


Note that the dot in the image denotes two one-thousandths
of one percent of the total area comprising the image.




Red-tides produced by algal blooms of dinoflagellates such
as Karenia brevis occur when the dinoflagellate cells them-
selves cells themselves occupy less than
2/1000ths of 1% of
the total volume of the water sample in which they reside.*

                *and the above 2/1000th calculation assumes K. brevis
                         concentrations of one million or more cells per liter

                            Some K. brevis red-tides occur at much smaller
                       concentrations of as little as ten thousand cells per liter



The step-by-step mathematics outlined below allows us to
prepare a two-dimensional illustration that visually depicts  
the amount of area occupied by 2/1000ths of one percent





(1)  Use software to open a rectangle 500 pixels tall
                                           by 350 pixels wide = 175,000 square pixels

(2)  One percent of this area = (175,000) x (.01) =   1750 square pixels

(3)  1/1000ths of one percent = (1750) x (.001) =     1.750 square pixels

(4)  2/1000ths of one percent = (1750) x (.002) =         3.5 square pixels

(5)  Calculate the square root of 3.5 square pixels = 1.87 pixels
       so that a square of (1.87 pixels) x (1.87 pixels) = 3.5 square pixels




Thus given a starting rectangle of 500 x 350 pixels,
a small square of 1.87 pixels by 1.87 pixels
(length x width)

would visually depict a physical region of
  two one-thousandths of one percent



This example underscores quite clearly that sheer physical
amounts of “open space” available to a population constitute
a fallacious criterion by which to judge overpopulation.










“A continuation of today’s demographic tidal wave may constitute

the greatest single risk that our species has ever undertaken.”


What Every Citizen Should Know About Our Planet






 GPSO 2010

The comments, illustrations, mathematics, and data presented on this site

are offered as a contribution to the Global Population Speak Out dialogue


































Copyright 2010, Randolph Femmer.

All rights reserved.