Geometric Numbers

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- Curvature of Rectangular Numbers, Part I, by Jonathan
Tennenbaum

- Curvature of Rectangular Numbers, Part II, by Jonathan
Tennenbaum

- Polygonal Numbers (The Prisoner and the Polygon), by Bruce
Director

- Geometric Numbers (Prisoner and Professor), by Bruce Director

Curvature of Rectangular Numbers -- Part 1

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?

A pedagogical discussion Part II

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]

The Prisoner and the Polygon

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.

The Prisoner and the Professor

"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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