If we graph humanity's populationthe classical J-curve that characterizes exponentialour schooling, however, we have been taught linear mathematics so thoroughly, so repeatedly,and so well that our minds literally “interpret the world” using the intuitions and elementaryschool arithmetic of our everyday lives.

Thus, in nearly all cases, our early schooling leaves us illthat are behaving in an exponentialtomed to the COUNTERINTUIT

rors that their deceptive behaviors invitehelp us better appreciate the implications

The term dinosaur means "terrible lizard." We are going to examine some tiny, onerine organisms called dinoflagellates. Since some of these species can undergo populsions that cause red-tides and massive fish kills in marine systems, their dflagellates" is appropriate.

Exponential Growth in a Finite Environment

While teaching at the University of Colorado, Albert Bartlett, Professor of Physics and Astrophysics, included his famous lectures on exponential matha population of bacteria growing exponentially within a petri dish (Bartlett,line transcript, 2000).

We are going to employ his classical example here, except our mentalexperiment will feature alates within the finite confines of a bottle of salt water.single dinoflagellate is placed in the bottle at

Suppose that members of this imaginary speciesty secondsing minute.

Since this population grows larger byplying

At the end of the first sixty seconds, there will beThen, one minute later, they will both divide to producegellates.and

Exponential Mathematics - III"Riddles of the Dinoflagellates"

population growth from 8000 B.C. to 2000 A.D., we see that it generatescurve that characterizes exponential number sequences and nuclear detonations. In

our schooling, however, we have been taught linear mathematics so thoroughly, so repeatedly,our minds literally “interpret the world” using the intuitions and elementary

ithmetic of our everyday lives.

Thus, in nearly all cases, our early schooling leaves us ill-prepared to interpret number sequencesexponential or non-linear fashion. As a general rule,

COUNTERINTUITIVE behavior of such progressions, nor do we learn to avoid the errors that their deceptive behaviors invite. There are a series of simple riddleshelp us better appreciate the implications of such exponential patterns.

dinosaur means "terrible lizard." We are going to examine some tiny, oneflagellates. Since some of these species can undergo popul

tides and massive fish kills in marine systems, their designation as "terrible

Exponential Growth in a Finite Environment

While teaching at the University of Colorado, Albert Bartlett, Professor of Physics and Astrophysics, included his famous lectures on exponential mathematics. One of his examples ina population of bacteria growing exponentially within a petri dish (Bartlett,

We are going to employ his classical example here, except our mentalexperiment will feature an imaginary species of onelates within the finite confines of a bottle of salt water.single dinoflagellate is placed in the bottle at ten a.m

Suppose that members of this imaginary species dividety seconds so that their population doubles its numing minute.

Since this population grows larger by repeated multiplicationsplying x 2 each minute), its pattern of growth is exponential.

At the end of the first sixty seconds, there will beThen, one minute later, they will both divide to producegellates. And these will then be followed by eightand sixty-four individuals, etc.

, we see that it generatesnumber sequences and nuclear detonations. In

our schooling, however, we have been taught linear mathematics so thoroughly, so repeatedly,our minds literally “interpret the world” using the intuitions and elementary

prepared to interpret number sequencesAs a general rule, we are not accus-

nor do we learn to avoid the er-are a series of simple riddles, however, that can

dinosaur means "terrible lizard." We are going to examine some tiny, one-celled ma-flagellates. Since some of these species can undergo population explo-

esignation as "terrible

While teaching at the University of Colorado, Albert Bartlett, Professor of Physics and Astro-ematics. One of his examples involves

a population of bacteria growing exponentially within a petri dish (Bartlett, 2005, 1978, and on-

We are going to employ his classical example here, except our mentaln imaginary species of one-celled dinoflagel-

lates within the finite confines of a bottle of salt water. Suppose that aa.m.

divide once every six-its numbers with each pass-

multiplications (multi-2 each minute), its pattern of growth is exponential.

At the end of the first sixty seconds, there will be two dinoflagellates.Then, one minute later, they will both divide to produce four dinofla-

eight, sixteen, thirty-two,

Next suppose that we set the jar aside and return to check the experimentlater. One hour later (eleven a.m.) we notice thatoccurred: First, the bottle is nowand, in addition, they are all

Thus, during the hour that has passed, the disucceeded in .filling their bottlhere is the riddle:

If we base our answer onschooling, our firstmust be

This answerour earliest years.

Employing ordinary arithmetic, if the bottle is completely filled at elevena.m., then itpoint: 10:30. In fact, thnoflagellate population were growing arithmetically.

With a number sequence that is growingour dinoflagellates are doing) the 10:30 answer is incorrect.

Half Full and 50% Empty

Because the dinoflagellates in our experiment are increasingly, their container will be half

If the jar is half-full at 10:59 and its occupants all divide over the nextsixty seconds, then they will double in numberscontainer by 11:00 a.m.

This reveals an unsettling aspect of exponential number sequencescause the dinoflagellate population has onlybefore the disaster that lies just aheadempty container seem to be relatively innocuous, because the dinoflalate numbers are growing exponentially (instead of arithmetically),tainer-wide disaster is just one minute away

Next suppose that we set the jar aside and return to check the experiment(eleven a.m.) we notice that two changes have

First, the bottle is now completely filled with dinoflagellates,dead.

that has passed, the dinoflagellate inhabitants havebottle. completely. Given this background,

When is the bottle half-full?

If we base our answer on the linear math that we learned inschooling, our first-impulse, automatic response is tomust be half-full at 10:30 a.m.

This answer seems logical if we use the math we have been taught sinceour earliest years.

Employing ordinary arithmetic, if the bottle is completely filled at elevena.m., then it seems as though it should logically be halfpoint: 10:30. In fact, this would be the correct answer,noflagellate population were growing arithmetically.

With a number sequence that is growing exponentiallyour dinoflagellates are doing) the 10:30 answer is incorrect.

The patterns that apply to “elementary school arithmetic”DO NOT WORK with numbers that grow exponentially

Half Full and 50% Empty

the dinoflagellates in our experiment are increasing exponential-, their container will be half-full at 10:59.

full at 10:59 and its occupants all divide over the nextthey will double in numbers and completely fill their

an unsettling aspect of exponential number sequences be-dinoflagellate population has only one more minute to exist

before the disaster that lies just ahead. Although conditions in the half-to be relatively innocuous, because the dinoflagel-

late numbers are growing exponentially (instead of arithmetically), con-wide disaster is just one minute away.

math that we learned in our earlyis to imagine that the jar

logical if we use the math we have been taught since

Employing ordinary arithmetic, if the bottle is completely filled at elevenshould logically be half-full at the halfway

the correct answer, but only if the di-

exponentially, however (whichour dinoflagellates are doing) the 10:30 answer is incorrect.

The patterns that apply to “elementary school arithmetic”with numbers that grow exponentially.

If a population attempts an arithmetic interpretationexponential pattern of progressof the catastrophe that is about to overtake them

Since our dinoflagellates comprise an imaginary species anyway, let usconfer upon them an awareness of their surroundings. To the occupants living in a jar that is onevolume is still empty.

If weemen

Thejar is ¼ full, ¼ of the hour must have gone by.”

With so much empty volume available, conditions again seem relatively innocuous.nummisleading.

With only two doublings leftdinoflagellates occupy a largely10:58 a.m. When this 10:58 population doubles, the container will be halfwhen the dinoflagellates double again, their bottle will be completely filled by 11:00 a.m.

Thus, with mostly empty volume availablefind themselves seduced into

Let us then test the exponential pattern in three more instances. Each question belicit from us a first-impulseeryday mathematics that we learned in school. At the same time, however, each question also hasa correct answer that can be determined by analy

At what time is the container At what time is the container At what time is the container

The bottle will be one-eighth full at 10:57 a.m., so that,container-wide disaster, thea.m., only four minutes before disaster, the jar will beof its volume remains unoccupied.

Finally, at 10:55 a.m., with only five minutes left until dibe 98% empty. It is difficult to accept that danger is present when so much empty volumeabounds. Thus, if the chief mathematics that we internumber sequences, then it is completely impossible to see the oncoming disaster

attempts an arithmetic interpretation of events that are beingprogression, the error will blind them to both the degree and the proximity

of the catastrophe that is about to overtake them.

When Is The Bottle One-Quarter Full?

Since our dinoflagellates comprise an imaginary species anyway, let usconfer upon them an awareness of their surroundings. To the occupants living in a jar that is one-quarter full, 75% of their bottle's totalvolume is still empty. With this in mind, we present our second riddle:

When is the container one-quarter full?

If we.try to apply the grocery-store instincts developed by all of our elementary schooling, our first-impulse answer is likely to be 10:15.

The mistaken logic of such reasoning, of course, being that “since thejar is ¼ full, ¼ of the hour must have gone by.”

With so much empty volume available, conditions again seem relatively innocuous. However, since the organisms are innumbers exponentially, what appears be minimal danger is seriouslymisleading.

With only two doublings left before an oncoming catastrophe, thelargely empty environment. As the drawing shows, the bottle is ¼ full at

10:58 a.m. When this 10:58 population doubles, the container will be half-filled at 10:59. Andwhen the dinoflagellates double again, their bottle will be completely filled by 11:00 a.m.

pty volume available, it is easy to understand how the dinoflagellates mightfind themselves seduced into complacency if they interpret their world using linear mathematics.

Let us then test the exponential pattern in three more instances. Each question bimpulse, incorrect response based on our tendency to use the ordinary, ev

eryday mathematics that we learned in school. At the same time, however, each question also hasa correct answer that can be determined by analyzing the data as an exponential progression.

At what time is the container .01/8th full?At what time is the container 1/16th

.full?At what time is the container 1/32nd full?

eighth full at 10:57 a.m., so that, with only three minutes remainingwide disaster, the dinoflagellates inhabit an environment that is 87% empty

a.m., only four minutes before disaster, the jar will be one-sixteenth full while the remainingof its volume remains unoccupied.

Finally, at 10:55 a.m., with only five minutes left until disaster, the jar, which is 1/32cult to accept that danger is present when so much empty volume

Thus, if the chief mathematics that we internalize and use involvesthen it is completely impossible to see the oncoming disaster

eing governed by anto both the degree and the proximity

Quarter Full?

Since our dinoflagellates comprise an imaginary species anyway, let usconfer upon them an awareness of their surroundings. To the occu-

quarter full, 75% of their bottle's totalWith this in mind, we present our second riddle:

quarter full?

store instincts developed by all of our el-impulse answer is likely to be 10:15.

logic of such reasoning, of course, being that “since the

With so much empty volume available, conditions again seem rela-ver, since the organisms are increasing their

mal danger is seriously

before an oncoming catastrophe, theAs the drawing shows, the bottle is ¼ full at

filled at 10:59. Andwhen the dinoflagellates double again, their bottle will be completely filled by 11:00 a.m.

dinoflagellates mightif they interpret their world using linear mathematics.

Let us then test the exponential pattern in three more instances. Each question below tends toresponse based on our tendency to use the ordinary, ev-

eryday mathematics that we learned in school. At the same time, however, each question also haszing the data as an exponential progression.

minutes remaining before87% empty. At 10:56

while the remaining 93%

ter, the jar, which is 1/32nd full, willcult to accept that danger is present when so much empty volume

linear (“arithmetic”)then it is completely impossible to see the oncoming disaster.

If we have a sentient species of dinoflagellates,problem? The answer is: If they try to interpret their world using anization will come TOO LATE

that we humans are much smarter than dinoflagellinvolves demographics, planetary machinery, carrying capacity,tween a million and a billionmathematics.

When Does Realization Occur?

If we have a sentient species of dinoflagellates, when will this population realize that they have aThe answer is: If they try to interpret their world using an arithmetic

TOO LATE.

One can imagine a grocery-store analysis something like this: Sinceit took fifty-nine minutes for the jar to become halflogical" to assume that one has another fifty-nine minutes availablebefore conditions become too severe. And, if that assessment weretrue, then the organisms could address their problem at sothe distant future. This sort of reasoning would beulation under discussion were following a lineartern of growth. The problem is, however, that the organisms exhibitan exponential pattern of growth so that trying to analyze an exponential sequence using an arithmetic interpretation of the datasults in a seriously wrong answer.

Thus, with less than one minute to go before calamitylation interprets its circumstances employing a linear mindset,be blind to the immediacy of its danger.

Thus, having seen how deceptive an exponential prowe can at least be thankful that we are smartertiny, brainless, unicellular dinoflagellates, aren't we

Libraries, Symphonies, and Literature

Actually, when it comes to communications, space travel, libraries,great literature, and symphonies, we are clearly much smarter thandinoflagellates. Dinoflagellates, for example, cannot usenet, memorize Hamlet, or use a cell phone. Nor can they know theirown history, pass legislation, or read a book. It is obvious, therefore,

that we humans are much smarter than dinoflagellates – unless, of course, the subject of the examplanetary machinery, carrying capacity, the enormous

tween a million and a billion, and the powerful and counterintuitive properties of

when will this population realize that they have aarithmetic mindset, their real-

store analysis something like this: Sincenine minutes for the jar to become half-full, it "seems

nine minutes availablevere. And, if that assessment were

address their problem at some time inwould be correct if the pop-

linear or “arithmetic” pat-the organisms exhibit

trying to analyze an expo-nential sequence using an arithmetic interpretation of the data re-

calamity, if this popu-a linear mindset, it will

tial progression can be,than a population of

ren't we?

Libraries, Symphonies, and Literature

Actually, when it comes to communications, space travel, libraries,great literature, and symphonies, we are clearly much smarter than

Dinoflagellates, for example, cannot use the inter-. Nor can they know their

It is obvious, therefore,unless, of course, the subject of the exam

enormous difference be-properties of exponential

If we evaluate our two species only upon the subjectsulation, demographics, and the power and misleading behavior of an exponential sequence,are completely clueless – but, for the most part, so are weour government officials, teachers, journalists, and other opinion leaders do not know the math by which we add one billion additional people to our planet every twelve to fifteenyears; if as individuals we do not know how enormously largea billion really is; if we cannotbirths and deaths; and if we do not understand the power anddeceptive nature of exponential mathematicsblind to the demographic forces that have been shaping, areshaping, and are very possibly wrecking, ou

Numbers making up an exponential sequence are like a firealarm going off in a burning buildingarm can only be heard if our schools, curricula, teachers, andtextbooks, everywhere, teach us the deceptive, misleading,and extraordinarily powerful nature of exponential mathematics.

A continuation of today’s demographic tidal waveconstitute the greatest single

undertaken

What Every Citizen Should Know About Our

Librarians: The book version of Wecskaop is available fromM. Arman Publishing, Fax: 386

evaluate our two species only upon the subjects of pop-ulation, demographics, and the power and misleading beha-vior of an exponential sequence, our dinoflagellate colleagues

but, for the most part, so are we. Ifofficials, teachers, journalists, and other opi-

nion leaders do not know the math by which we add one bill-ion additional people to our planet every twelve to fifteenyears; if as individuals we do not know how enormously large

annot approximately quantify dailyand if we do not understand the power and

deceptive nature of exponential mathematics, then we areblind to the demographic forces that have been shaping, areshaping, and are very possibly wrecking, our future.

Numbers making up an exponential sequence are like a fire-alarm going off in a burning building – but this particular al-arm can only be heard if our schools, curricula, teachers, andtextbooks, everywhere, teach us the deceptive, misleading,

d extraordinarily powerful nature of exponential mathema-

of today’s demographic tidal wave maythe greatest single risk that our species has ever

undertaken.

Excerpted fromWhat Every Citizen Should Know About Our Planet

Used with permission.

Copyright 2009. Randolph Femmer.

Librarians: The book version of Wecskaop is available fromM. Arman Publishing, Fax: 386-951-1101

Copyright 2009. Randolph Femmer.All rights reserved. .

Expanded implications of this excerpt are alsoaddressed in additional PDFs in this collection:

Razor-Thin Films: Earth's Atmosphere and Seas How big is a billion? Conservation planning Population momentum … (… on steroids Eight Assumptions that Invite Calamity Climate - No Other Animals Do This A Critique of Beyond Six Billion Dinoflagellates, 2/1000ths of 1%, and the Open Delayed feedbacks, Limits, and Overshoot Thresholds, Tipping points, and Carrying Capacity and Limiting Factors Humanity's Demographic Journey Ecosystem services and Ecological release J-curves and Exponential progressions One hundred key Biospheric understandings

Sources and Cited References…pending…

Anson, 2009.Anson, 1996Bartlett, 2005.Bartlett, 2000.Bartlett, 1978.Campbell, et al., 1999Cohen, 1995Cohen and Tilman, 1996.Dobson, et al., 1997Duggins, 1980;Estes and Palmisano, 1974Mader, 1996Mill, J.S., 1848.Pimm, 2001.Prescott, et al., 1999.Raven, et al., 1986Wilson, 2002.

Expanded implications of this excerpt are alsoaddressed in additional PDFs in this collection:

Thin Films: Earth's Atmosphere and Seas

rvation planning - Why 10% goals are not enoughPopulation momentum … (… on steroids …)?

Assumptions that Invite CalamityNo Other Animals Do This

ritique of Beyond Six BillionDinoflagellates, 2/1000ths of 1%, and the Open-space delusionDelayed feedbacks, Limits, and OvershootThresholds, Tipping points, and Unintended consequencesCarrying Capacity and Limiting FactorsHumanity's Demographic JourneyEcosystem services and Ecological release

curves and Exponential progressionsOne hundred key Biospheric understandings

Sources and Cited References…pending…

Cohen and Tilman, 1996.

Estes and Palmisano, 1974