Mind Over Mathematics:
Geometric Numbers


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Pedagogical Discussion

Curvature of Rectangular Numbers -- Part 1

by Jonathan Tennenbaum

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        Our pedagogical discussions concerning the problem
"incommensurability" in Euclidean geometry demonstrated, among
other things, that the shift from linear to plane, or from plane
to solid geometry cannot be made without introducing new
principles of measure, not reducible to those of the lower
domain. Thus, the relationship of the diagonal to the side of a
square can only be constructed in plane geometry, and is
inaccessible -- except in the sense of mere approximations -- to
the mode of measurement characteristic of the simple linear
domain (i.e., that embodied in "Euclid's algorithm"). In the
following discussion, we propose to explore that change from a
somewhat different standpoint.
        I choose, as a point of departure for this exploration, the
issues posed by any attempt to compare the areas of various plane
figures. The famous problem of "squaring the circle" falls under
this domain. But I propose, before looking at that, to start with
something much simpler. For example: How can we compare the areas
of arbitrary polygons, by geometrical construction? Or, to start
with, take the seemingly very simple case of rectangles. Let's
forget what we were taught -- but do not know! -- namely the
proposition that the area of a rectangle is equal to the product
of the sides. (Actually, even if the assumptions of Euclidean
geometry were perfectly true, the proposition in that form is
either false or highly misleading: an AREA is a different species
of magnitude, distinct from all linear magnitudes.) In the
interest of making discoveries of principle, let us resolve to
use nothing but geometrical construction.
        Experimenting and reflecting on this problem, the insightful
reader might come to the conviction, that the problem of the
relationships of area among rectangles of different shapes and
sizes, pivots on the following special case: Given an arbitrary
rectangle, how to construct "many" other rectangles having the
equivalent area. Or perhaps even to characterize the entire
manifold of rectangles of area equivalent to the given one.
        The first line of attack, which might occur to us, were to
find a way to cut up the given rectangle into parts, and
rearrange them somehow to form other rectangles. Should we admit
any limitation to the shapes and numbers of the parts? To avoid a
bewildering bad infinity of options, let us focus first on what
would appear to be the "minimum" hypothesis, namely to divide the
given rectangle into congruent squares (i.e., squares of equal
size). A bit of reflection shows us, that such a division is only
possible for the special case, that the sides of the given
rectangle are linearly commensurable (i.e., are multiples of a
common unit of length). So, for example, if the sides of the
given rectangle are 3 and 4 units long, respectively, then by
cutting the rectangle lengthwise and crosswise in accordance with
divisions of the sides into 3 and 4 congruent lengths,
respectively, we obtain a neatly packed array of 12 congruent
squares. We discover, that it is possible to rearrange those
squares to obtain five other rectangles: 4 by 3 (instead of 3 by
4), 2 by 6, 6 by 2, 1 by 12, and 12 by 1 (i.e., six in all counting
the original one, or three if we ignore the order of the sides).
        Experiment further. If we start, for example, with a square,
and divide the sides into five congruent segments, we obtain 25
congruent squares. The "harvest" of rectangular rearrangements is
disappointingly small: all we find is the long, skinny 1 by 25!
        Carrying out such simple experiments, the attentive reader
might detect a number of potential pathways of further inquiry.
One of these would be to ask, for a given total number of
congruent squares, how many different rectangles can be formed as
arrangements of exactly that number of squares? We can then
organize the number into species or classes, according to the
resulting number of rectangular arrangements (or "rectangular
numbers" as the Greek geometers called them). The class of
numbers for which only one rectangular arrangement is possible
(disregarding the order of the sides) are known as "prime
numbers." After these, we have a class of numbers with exactly
two rectangular arrangements, such as 6, 10, 14, 15, 21, etc. (The
otherwise mind-destroying game of "Scrabble" might be put to good
use, by employing the wood squares for experiments.)
        For the present purposes, however, we would like to
construct as many different rectangles as possible out of the
original one. We note, that the number of rectangles generated
from any given division of the rectangle is very narrowly
bounded, and certainly does not include all geometrically
constructible ones. How to obtain more? If we stick to the method
of division into squares, the only option is to increase the
number of divisions. So, for example, we can bisect the unit
length in our 3 by 4 rectangle, obtaining a division into 6 times
8, or 48 squares. This raises the total number of rectangles
obtained by rearrangement to 10 (5 not counting the order of the
sides). By repeated such subdivisions, we might hope to increase
the density of population of rectangles so generated, whose areas
are all equivalent to the area of the original rectangle. It
might be interesting to see how the population grows, as we add
new divisions.
        But, should we be satisfied with this approach? Aren't we
plunging into a "bad infinity" of particulars? Is there no way to
obtain an overview of the whole domain? And remember, our
geometrical domain is not limited to linear commensurability of
sides. Indeed, a bit of reflections suggests, that for EVERY
given segment, there must exist a rectangle, whose area is
equivalent to the given one, and one of whose sides is that
length. How might we construct such a rectangle?
        For a glimpse at a higher bounding of our problem, try the
following construction: Take the rectangles constructed from any
given rectangle by divisions into squares and rearrangement, as
above, and superimpose them by bringing the lower left-hand
corners into coincidence and aligning the sides along the
vertical and horizontal directions. What do you see?



The Curvature of Rectangular Numbers

A pedagogical discussion Part II

by Jonathan Tennenbaum

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        The general task, posed in last week's discussion, was to
generate the manifold of rectangles whose areas are equivalent to
a given rectangle.  The initial tactic chosen, was to divide the
given rectangle into an array of congruent squares, and rearrange
them into rectangles of different dimensions, but equivalent
area.  It became clear, that this tactic only yields a discrete
``population'' of rectangles (``rectangular numbers''), whose
number depends on some characteristic of the number of divisions
chosen.  On the other hand, if we arrange the resulting
rectangles in such a way, that their lower left-hand corners
coincide, and their sides are lined up along the horizontal and
vertical axes, then a hidden harmony springs into view: the upper
right- hand corners of the rectangles, so arranged, appear to
describe a HYPERBOLA, or at least a hyperbola-like curve.  The
idea suggests itself, that the discreteness of dividing and
rearranging parts to form individual rectangles, is bounded from
the outside by a higher continuity (ordering), whose presence
reveals itself in the hyperbolic ``envelope'' of the rectangles.
        To proceed further, let us change our tactic, concentrating
on the idea, that there must exist a PROCESS of TRANSFORMATION
which generates the entire manifold of equivalent-area rectangles
and hyperbolic ``envelope'' at the same time.  We might adopt the
attitude, that any pair of rectangles of equivalent area
expresses a kind of INTERVAL within the implied ``hyperbolic''
ordering of the whole.
        With this in mind, start with any given rectangle, and
consider the following approach.  If we triple the length of the
rectangle, keeping the width the same, then we obtain a rectangle
whose area is clearly equivalent to three times that of the
original one.  If we then reduce the width of the new rectangle
to one-third of its original value, while keeping the length
unchanged, then the area of the resulting rectangle (with three
times, the length, but one-third the width of the original) will
clearly be equivalent to the original rectangle's area.  In fact,
we might verify that equivalence in the former, discrete manner,
namely by dividing the original rectangle lengthwise into three
congruent rectangles, and then rearranging them to obtain the new
one.  In the same way, we could quadruple the length of the
original rectangle and reduce its width to one-fourth, and so on.
Obviously, nothing prevents us from applying the same procedure
with ANY factor (i.e. not only 3 or 4), or from reversing the
roles of ``length'' and ``width'' in this procedure.
        At this point, something might occur to us, which allows us
to ``jump'' the gap between the discreteness of our former
procedure, and the underlying ordering of the problem.  Up to
now, we have considered as primary a process of multiplying or
dividing lengths or widths by some integral number.  But now we
realize, that the crux of the matter, lies not in this
duplicating or dividing up, but rather in the relationship of
``INVERSION'' between the transformation applied to the length
and the transformation applied to the width.  This suggests a new
approach, which does not depend upon whole-number relationships
at all.
        Thus, take any rectangle with length A and width B.  Now
imagine A prolonged to ANY ARBITRARY LENGTH X.  Those two
lengths, A and X, define an interval.  Evidently, what we must
do, is to ``invert'' that interval with respect to B! In other
words, construct a length Y, for which the interval (proportion)
``Y to B'' is (in relative terms) congruent to the interval ``A
to X''.
        The required construction can be approached in many
different ways.  For example, generate a horizontal line, and
erect a perpendicular line at some point P.  Starting from P, lay
off a vertical line segment PQ, whose length is equivalent to X,
and determine a point R between P and Q, such that PR is
equivalent to the length A.  Next, chose an arbitrary point S,
lying to the left of P on the horizontal line, and construct a
vertical line segment ST whose length is equivalent to B.  Now,
generate a straight line through the points T and Q.  Leaving
aside the case, where that line happens to be parallel to the
horizontal axis, the line through TQ will intersect the
horizontal axis at some point O.  Finally, generate a straight
line through O and R.  That straight line will intersect the
vertical line ST at some point U.  Reflect on the relationship
formed, relative to ``projection'' from O, between the line
segments on the two vertical lines from P and S.  Evidently, the
interval of PR to PQ (i.e. A to X) is congruent to the interval
of SU to ST, the latter being equivalent to B.  Thus, SU gives us
the value Y  for the required  ``inversion'' of the
transformation from A to X.  In other words, the transformation
of A to X, and the transformation from B to Y are inversions of
each other, and the rectangle with sides X, Y will have the
equivalent area to the rectangle with sides X and Y.
        Consider the case, in which the value of X is changing, and
observe the manner in which the positions of O and U vary in
relation to X.  The hyperbolic envelope is already implicit.
        Those skillful in geometry will be able to devise
essentially equivalent constructions, which make it possible to
generate the hyperbolic envelope and the entire array of
equi-area rectangles at the same time.  Just to give a brief
indication: Start with a rectangle, whose sides A and B lie on
vertical and horizontal axes.  Let O and M denote the lower
left-hand and lower right-hand corner-points of the rectangle.
Generate any ray from O, with variable angle, which intersects
the upper horizontal side of the rectangle, at a point P.
Prolonging the right vertical side of the rectangle upward, the
same ray will intersect that vertical line at some point, Q.  Now
draw the vertical line at P and the horizontal line at Q.  Those
two lines intersect at a point R.  Now examine the relationship
of the rectangle with upper right-hand corner R and lower
left-hand corner O, to the original rectangle.  Examine the
motion of R as a function of the angle of the ray from O.
        For those who feel the compulsion to scribble algebraic
equations,  now is the time to kick the habit! The whole point
here is to think GEOMETRICALLY.  The notion of ``geometrical
interval'' supercedes that of discrete arithmetic relationship...
[jbt]

- PEDAGOGICAL DISCUSSION -

The Prisoner and the Polygon

by Bruce Director

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        Back from lunch, our prisoner eagerly digs deeper into his
investigations of the nature of number, fueled by enthusiasm from
his recently demonstrated capacity to discover truth by his own
powers of reason. He's determined to avoid the various textbooks
lying around (not wanting to fraternize with the enemy), relying
instead on a well-worn copy of Euclid's Elements, whose text
contains the footprints of some classical Greek discoveries. The
more profound nature of these discoveries are not explicitly
stated in Euclid's Elements, but the profound nature of these
ancient thoughts are, neverthelss, reconstructible in the mind.
Combining centuries of discoveries in his mind simultaneously, he
turns to Book 9, Propositions 21-34 to reconstruct for himself,
the indicated discoveries concerning even and odd numbers,
pondering these Propositions, in dialogue with the more advanced
standpoint of Nicholas of Cusa's "On Conjectures."  (As noted
last week, "The odd number appears to have more of unity than the
even number, because the former cannot be divided into equal
parts and the latter can be. Therefore, since every number is
one out of unity and otherness, so there will be numbers in which
the unity prevails over the otherness, and others in which the
otherness appears to absorb the unity.")
        From Cusa's standpoint, the indicated principles of Euclid
can be stated as follows: When two even numbers are added, the
otherness still prevails over unity, producing an even number.
When two odd numbers are added, the unities from each one are
combined, making otherness prevail over the unity, and producing
an even number. When an even and an odd number are added, the
unity of the odd number remains, producing an odd number. In sum
when like numbers are added, the otherness prevails over unity,
and an even number is produced. When unlike numbers are added,
unity prevails over otherness, and an odd number is produced.
        When an even number is added an even number of times (i.e.,
multiplied by an even number), otherness continues to prevail,
resulting in an even number. When an odd number is added an even
number of times (i.e., multiplied by an even number) otherness
prevails and an even number results. When an odd number is added
an odd number of times (i.e., multiplied by an odd number), unity
still prevails over otherness, and an odd number results.
        Different from addition, unlike numbers, when mulitplied,
produce even numbers, and like numbers preserve their type. (The
reader will find it liberating to demonstrate for yourself, that
this is true in all cases; also the reader should discover the
similar principle for subtraction and division, and for the
second order types of even-even, even-odd, odd-even, and odd-odd,
with respect to addition, multiplication, subtraction and
division.)
        Having discovered so much from the construction of linear
numbers, the prisoner extends his experiments into a new domain
and now investigates the construction of polygonal numbers. He
begins with the polygon with the smallest number of sides, the
triangle.

                                                  *
                                     *           * *
                          *         * *         * * *
                 *       * *       * * *       * * * *
          *     * *     * * *     * * * *     * * * * *
     *   @ @   @ @ @   @ @ @ @   @ @ @ @ @   @ @ @ @ @ @

and so on.
        Unlike linear numbers, triangular numbers are constructed
not by adding one, but by adding the linear numbers themselves.
Each successive triangle, contains within it, all previous
triangles, plus the next linear number. The added part is called
a Gnomon, (denoted in the above figures by the symbol @) which in
Greek geometry, means a shape, which when added to a figure,
yields a figure similar to the original one. The word Gnomon is
derived from the Greek word to know. (The triangular pillar on a
sun-dial, which casts the shadow that marks the time, is also
called a Gnomon.) In the above representation, each Gnomon is
represented by a different symbol.
        (The reader is again urged to make your own hand drawings of
the construction of triangular numbers, instead of relying on
these computer generated representations. Hand drawings are an
efficient means of unfolding the cognitive process. When you
make these drawings, locate for yourself, the preceeding
triangles, in the successive one.)
        Triangular numbers are constructed by adding all previous
linear numbers together. 1; 1+2; 1+2+3; 1+2+3+4; ...; resulting
in the series of triangular numbers, 1, 3, 6, 10, .... The
differences (intervals) between each triangular number forms the
series, 2, 3, 4, 5, .... The difference between the differences
is always 1. Here, unity is found, not in the construction of the
numbers, but in the differences of the differences.
        Intrigued by this discovery, he extends the experiment to
the next polygon, the square. Square numbers are constructed
thusly.

                                             * @ & $ # %
                                 * * * * @   @ @ & $ # %
                       * * * @   * * * * @   & & & $ # %
               * * @   * * * @   * * * * @   $ $ $ $ # %
         * @   * * @   * * * @   * * * * @   # # # # # %
     *   @ @   @ @ @   @ @ @ @   @ @ @ @ @   % % % % % %

and so on.
        Again, each square contains within it all previous squares,
plus the addition of a Gnomon. (The Gnomon with respect to each
square is denoted by the symbol @. The last figure represents
each Gnomon with a different symbol.) The square numbers increase
by adding every second linear number, to the previous square
number, 1+3; 1+3+5; 1+3+5+7; resulting in the series of square
numbers, 1, 4, 9, 16, 25, 36,... The differences (intervals)
between each square number, forms the series of odd numbers, 1,
3, 5, 7, 9, ..., and the difference between any two odd numbers
is always 2, or is always divisible by 2.
        The prisoner can now prove, why these difference are always
odd, by looking at the nature of each Gnomon, from the standpoint
of his previous discoveries about the nature of even and odd
numbers. (When making your hand drawings, distinguish each
successive Gnomon and see how each square contains, nested within
it, all previous squares. Then look at each Gnomon from the
standpoint of the nature of adding even and odd numbers.)
        The prisoner now thinks, "Under what conception can I bring
the generating principle of the square numbers into a One." The
square numbers are obviously not equal to one another, so
equality is not the right conception. But, congruence is not
self-evident, as no modulus can be found, relative to which all
square numbers are congruent. But the differences (intervals)
between the square numbers, (i.e., the odd numbers) while not
equal, are all congruent to unity relative to modulus 2. Here the
unity is found, not in the formation of the square numbers, nor
in the differences between the square numbers, or even in the
difference between the differences. Unity is found, as that to
which all the differences between the square numbers, are
congruent, relative to the modulus of the difference of the
differences. (In this case, modulus 2.)
        Excited by the ability of his mind to increase its cognitive
power, by discovering a congruence, not on the surface, but in
the underlying generating principle, he drives the process
further. By extending his experiments to polygons of increasing
number of sides, the prisoner seeks to force new anomalies to
emerge, so he can find what new ordering principles he can
discover.
        So on to pentagonal numbers. Which he constructs thusly:

                                           *
                                          * *
                              *          * * *
                             * *        * * * *
                    *       * * *      * * * * *
                   * *     * * * *     * * * * *
           *      * * *    * * * *     * * * * *
          * *     * * *    * * * *     * * * * *
     *    * *     * * *    * * * *     * * * * *

and so on.
        Here again, each pentagonal number contains nested within
it, all previous pentagonal numbers. (Here the reader must make
his own hand drawings, as this computer is utterly incapable of
doing the work for you, let alone the thinking.) Each pentagonal
number increases over the previous pentagonal number by the
addition of every third linear number. 1+4; 1+4+7; 1+4+7+10;
1+4+7+10+13; resulting in the series of pentagonal numbers, 1, 5,
12, 22, 35, .... The differences (intervals) between the
pentagonal numbers forms the series, 4, 7, 10, 13.... The
difference between any two differences is always 3, or is
divisible by 3.
        Like the square numbers, and the triangular numbers, the
pentagonal numbers are not equal, and no modulus can be found,
relative to which all pentagonal numbers are congruent. But, when
the prisoner looks to the generating principle of pentagonal
numbers, a modulus can be found under which the ordering
principle can be thought of as a One. The differences between the
pentagonal numbers are all congruent to unity relative to modulus
3. Again, unity is found, as that to which all the differences
between the pentagonal numbers are congruent, relative to the
modulus of the difference of the differences. (In this case,
modulus 3.)
        This process can be extended to polygons of ever-increasing
numbers of sides, forming hexagonal numbers, heptagonal numbers,
octagonal numbers and so on. The prisoner spends some time,
carefully drawing each series of polygons, so as to bring the
generating principle of each polygonal series into a One in his
mind. (The reader is well advised to do the same.)
        Having done this, a new, more profound question comes before
the prisoner's mind. "What is the generating principle, under
which the generating principle of all polygonal numbers can be
brought into a One?"
        With each new polygon, a new series of numbers is
constructed. Unlike linear numbers, which increase by adding one,
the polygonal numbers, increase by an increasing amount each
time. Each polygonal series, is unified, not with respect to each
number of the series, but by the differences between those
numbers, which are all congruent to unity relative to a modulus
formed by the differences of the differences. (The reader will
see that the modulus is always two less than the number of sides
of the polygon.)
        The prisoner has discovered a generating principle, of a
generating principle.
        (These discoveries, some of which were embodied in classical
Greek knowldedge, were subsequently investigated by Pascal and
Fermat, formed a basis for Leibniz' discovery of the differential
calculus, and were reworked by Gauss' in the development of the
complex domain.)
        The prisoner steps back and looks at his work, taking a deep
breath of fresh air. He feels as though he's climbed a high peak,
on a path whose direction and steepness has changed along the
way. The path began with the simple step of adding one, to
construct the linear numbers. The path became more curved and the
angle of ascent changed, as the concept of numbers was extended
into the domain of polygons. Now, at the summit, the change in
curvature, and changing angle of ascent, are thought of as a One,
under a principle that is congruent with the principle which he
started, thought of in an entirely new way. Now the addition of
unity, is found, not in the generation of the numbers themselves,
but in the generation of the moduli, under which the differences
between each polygonal number series are themselves made
congruent to unity.
        In seeing, with his mind, this whole process from the
summit, he asks himself, "What curvature is all this a reflection
of?"
        His free-thinking is suddenly interrupted by the sound of
footsteps. He looks up to see a well-dressed, slightly paunchy
baby boomer, with an air of self-importance about him. The man is
clutching a very large heavy textbook. As he comes close, the
prisoner looks quizzically at the stranger, who sticks out his
hand, saying, "Dr. Crumbbucket here. Glad to meet you. I'm a
visiting professor, of applied and theoretical bullshit. I
understand you're in need of instruction."
        The prisoner stares for a moment, as the fresh air seems to
rush out of his head. His lunch gurgles in his stomach. He
prepares to defend his mind.

- PEDAGOGICAL DISCUSSION -

The Prisoner and the Professor

by Bruce Director

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        "I have information that you've been playing around with
numbers," Dr. Crumbbucket inquired of our prisoner." Perhaps I
could help you to learn the ropes."
        "Well," our prisoner says slowly, trying to buy some time to
collect his thoughts, "I was just sort of making some
experiments."
        "Experiments!" Crumbbucket shrieks. "With numbers? No one
experiments with numbers. There are well-established rules for
the proper manipulations of the figures. Rules which have been
handed down from professor to professor, generation to
generation. Complicated rules, intricate rules. These take
years to learn. Either you can learn these rules, or we give you
an electronic calculator with pictures on it. No one can learn
by experiments with numbers. There's nothing to experiment with.
Besides, you can't do experiments in the mind."
        "Not {in} the mind," the prisoner corrects, "{About} the
mind. These experiments are to discover how my own mind thinks."
        "Whatever," the professor mumbles, after a short pause.
        "Do you know all the rules?" The prisoner is still trying to
collect his thoughts.
        "Virtually all of them. And as soon as a new one is
invented, I learn that one, too."
        "Is this what you had to do to get your PhD.?" the prisoner
asks.
        "Yes. I had to memorize, aggrandize, temporize, fantasize,
eulogize, surmise, bastardize, etymologize, generalize,
syllogize, tautologize, ventriloquize, analyze, brutalize,
formalize, legalize, socialize, symbolize, agonize, fraternize,
tyrannize, plagiarize, Anglicize, summarize, and vulgarize, but,
I haven't, at least not yet, had to hypothesize. If you want to
learn, we can begin the lessons immediately."
        Crumbbucket's face is getting redder as he speaks, and small
beads of sweat are forming on his forehead and on his chin. The
prisoner has a sinking feeling that his whole day is about to be
wasted. With no place to go, he has to think fast. Suddenly, a
discovery, that, until now was only half-formed in his mind,
comes into view. He decides to put the Professor to a test.
        "Let me first show you what I've discovered by experiment,"
the prisoner says.
        "Okay, but don't take long. We have a lot of work to do, if
you want to learn what I have learned."
        The prisoner quickly reviews his experiments and discoveries
with even, odd and polygonal numbers, to set the professor up for
the test.
        "That's no big deal. We have rules for all those things.
If you knew the rules, you wouldn't have had to go through all
those manipulations with lines, and dots, and all those drawings.
Let's get on with it."
        "Before we go on, dear Professor, let me put to you a series
of questions, so you can better understand the results of my
experiments. Are you agreeable to this?"
        "If it doesn't take too long," the professor answers,
shifting his weight from side to side, while one of his knees
vibrates quickly back and forth.
        "Okay," the prisoner begins, "Since I've already discovered
some things about linear and polygonal numbers, I now ask what
happens when I add areas?"
        "Areas?"
        "Yes. Areas. If I have an area whose magnitude is one, and
I add another area whose magnitude is one, what is created?"
        "Well, that's obvious. 1 + 1 = 2."
        "And, if I add an area whose magnitude is two to an area
whose magnitude is two, what is created?"
        "The same: 2 + 2 = 4."
        "And, if I add an area whose magnitude is four and I double
it, what happens?"
        "The same thing. 4 + 4 = 8. Of course, 2 x 4 = 8 is the
same thing. As with 2 + 2 + 2 + 2 = 8. Likewise the same with 2
x 2 x 2 = 8. Or 2^3=8."
        "Okay. Well, let's try drawing these areas and see what
happens?"
        "Why do you waste time with drawings?" growled the
Professor. "I just showed you how you can add, multiply, or take
the powers to get the answer. Why in the devil's name do you
want to waste time with drawings?"
        "Just try it. It won't take long. Here," the prisoner
gently hands the professer his pencil and paper.
        "Me? Draw?"
        "Yeah, please. Just try it."
        "Whatever," grumbles the professor, as he reluctantly takes
the pencil and paper.
        "Now, draw a square whose area is one," instructs the
prisoner. The professor complies, drawing a small square in the
middle of the paper. (As usual, the reader is urged to make your
own drawings.)
        "Now draw another square of the same size, attached to the
previous square," comes the next instruction.
        "What has been created?" the prisoner asks.
        "A one by two rectangle," replies the professor.
        "And what is the area of the rectangle?"
        "Two."
        "See, we've added two squares, and we've gotten a
rectangle," the prisoner says proudly.
        "What's the difference?" says Crumbbucket dismissively, "I
got the same answer following the rules: 1 + 1 = 2. And that was
much quicker."
        "Keep going," says the prisoner, ignoring the professor's
insolence.
        "Please, draw another one by two rectangle attached to the
one you've already drawn. Now what have you created?"
        "A two by two square."
        "And what is the area of that square?"
        "Four," the professer responds. "But so what, I already
figured the answer. 2 + 2 = 4. Also 2 x 2 = 4."
        "Please. Can we continue?" The prisoner coaxes the
professor to continue the drawing. Dr. Crumbbucket draws another
four by four square attached to the previous one, making a two by
four rectangle whose area is eight. And continuing, drawing
another two by four rectangle attached to the previous one,
making a four by four square whose area is sixteen.
        "See," the prisoner says excitedly, "First you had a square,
and you added a like square, making a rectangle whose area was
double the square. Then you added a like rectangle, and you got
a square whose area was double the rectangle. Then you added a
like rectangle, and you got a square whose area was double the
area of that rectangle. As you proceeded, you got another square,
then a rectangle. The first addition made a rectangle, the
second addition made a square, the third addition made a
rectangle, the fourth addition made a square, and so on."
        "But I got the same answer this way, 1x2=2x2=4x2=8x2=16...,
or alternatively, 2^0=1, 2^1=2, 2^2=4, 2^3=8, 2^4=16..." answers
the professor.
        Grinning from ear to ear, the prisoner rejoins, "But from
your way, you didn't discover the series of alternating squares
and rectangles. Now you've discovered that the odd-numbered
additions of areas make rectangles and the even-numbered
additions make squares."
        The professor snorts, shrugs his shoulders and says, "Are
you ready to learn the rules?"
        "Can we try one more series of questions?" asks the
prisoner.
        "Just one more," agrees the professor hesitantly, his
curiosity getting the better of him.
        "Try this," instructs the prisoner. "Draw a square whose
area is the same as the first square, one. Next to that, draw a
square whose area is the same as the one by two rectangle, two.
And next to that, draw the two by two square, and next to that a
square whose area is the same as the two by four rectangle. Do
this for all the areas you created by the first series of
drawings."
        The professor makes a neat drawing of squares, one next to
the other with areas one, two, four, eight, sixteen, and so
forth.
        "Now, Dr. Professor. What is the length of the side of the
first square whose area is one?"
        "One, of course," the professor answers.
        "And what is the length of the side of the second square
whose area is two?"
        "The square root of two," the professor states matter of
factly.
        "And what is the square root of two?"
        "It's the length of the side of the square whose area is
two, and is denoted with a symbol thusly," the professor responds
without blinking, tracing a radical sign in the air with his
finger.
        "But," replies the prisoner, "I already know the area of the
square is two. You are simply repeating yourself, to tell me
that the length of the side, is `The length of the side of the
square whose area is two.'"
        "The square root of two," the professor repeats, more
emphatically than before.
        "But that doesn't say anything. What's the square root of
two?" the prisoner asks again. "Can we continue? What is the
length of the side of the next square, the one whose area is
four?"
        "Two," answers the professor.
        "Very fine. And what is the length of the side of the next
square whose area is eight?" asks the prisoner.
        "The square root of eight." This time the professor's pride
in his ability to answer is tinged with trepidation, anticipating
the prisoner's response.
        "There you go again. You have only repeated the question as
the answer. I ask, `What is the length of the side of a square
whose area is eight?' and you answer, `The length of the side
whose area is eight.' That is not an answer. From that, we have
discovered nothing."
        Perceiving the professor's obvious distress, the prisoner
tries to be gentle, hoping that his prodding will liberate the
professor's mind.
        The professor stares for a moment in disbelief at the
resistance of the prisoner to accept his answer.
        The prisoner asks again, "What is the length of the side of
the square whose area is two or eight? Or in your words, what is
the square root of two, or eight?"
        "Here, hold this," the professor hands back the pencil and
paper after the briefest moment's pause, and picks up his heavy
textbook, wildly flipping the pages. "I know it's in here
somewhere," he says as he balances the book in one hand, turning
the pages with the other. The prisoner stands mute with a wry
smile on his face.
        "Just a minute. I'll find it," begs the professor. "Damn
it! Wrong book. Hang on a minute. Don't go away, I'll be right
back. I have to get my other book."
        "I shall return," the professor calls, his voice trailing
off. The prisoner watches the professor scurry down the hall,
half of his shirt-tail hanging out of his pants, the sound of
clanging chains diminishing as he gets further away.
        Free from the immediate encounter with the professor, the
prisoner turns his thoughts back to the drawings just created.
He spends some time making similar drawings, in which he
increases the area by three each time, then by four, then by
five. Each time he creates an alternating series of squares and
rectangles, with the first addition being a rectangle, the second
a square, the third a rectangle, and so forth. The rate at which
the areas grow, changes, but the type of change in each case is
the same; the odd-numbered additions (powers) make rectangles,
the even-numbered additions (powers) make squares. He has
discovered even and odd, in a new domain.
        When he added linear numbers, thus forming polygonal
numbers, the rate of growth changed for each type of number, but
remained the same within each series. Among linear numbers, he
discovered congruences, between even and odd, between even/even,
even/odd, odd/even, and odd/odd. Among the polygonal numbers, he
discovered congruences with respect to the change between each
number. Areas (geometric numbers), reflect an entirely
different type of change, as the numbers are increased.
        These two-dimensional geometric numbers, reflect a new
domain. Congruence with respect to even and odd remains, but in
an entirely different way. Here, evenness reflects squares and
oddness reflects rectangles. When these magnitudes are
transformed into only squares, the sides of the even ones are
commensurable with the area, while the sides of the odd ones are
incommensurable with the area. The concept of number cannot be
seperated from the content of number, which is a reflection of
the domain in which that number is situated. Even something as
seemingly simple as even and odd, is different in different
domains. The poor professor didn't even suspect, that from the
method he used, he really didn't know the area of half the
squares he drew, even though he seemed to be able to draw them.
"There's probably some hope for him," the prisoner thinks to
himself, "if he'll only try to discover, rather than just learn."
        The prisoner asks himself again, "What curvature do these
processes reflect?" In his mind's eye, he sees, with respect to
each type, different principles of growth, which are reflected as
a series of nested curves; a circle, an Archimedean spiral, and
an equiangular spiral. Each type of curvature, is reflected
simultaneously, yet distinctly, in each process.
        Now he thinks of a new, most important project: "What is the
nature of the curvature, which bounds these curves?"


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The preceding article is a rough version of the article that appeared in The American Almanac. It is made available here with the permission of The New Federalist Newspaper. Any use of, or quotations from, this article must attribute them to The New Federalist, and The American Almanac.


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