          Home Red-Tides CO2 and Climate Supporting Mathematics Links Easter Island Summary  Supporting Mathematics A MATHEMATICAL CRITIQUE              of  the “open space” hypothesis   PART ONE   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; floridamarine.org 2008) L:  ~30 um (= 0.03 mm)  = ~ 0.0012 inches  W: ~ 0.0014 inches  (“a little wider than it is long" – floridamarine.org)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 volumeof 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 inchesof 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 percentof 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 theirproduction of unseen, invisible, and calamitous wastes catastrophically-damaged the watery surroundings in which they live.     PART TWO   Here we depict the physical amount of space thatconstitutes 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.     Background:  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 physicalamounts of “open space” available to a population constitutea 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.  