This thesis is dedicated to the Yellow Pig and to yellow pig followers everywhere, with thanks to all who have encouraged my mathematical pursuits. I would like to thank my upstairs neighbors in Redwood City for waking me up early every morning so I could write for hours before work, those few individuals who have made useful LATEX documentation available on the Internet, and all who pointed me in the direction of relevant and interesting sources. Finally, I would like to thank my parents for their diligent proofreading and editing, my thesis committee for their suggestions and support, my other thesis readers who caught egregious mistakes that had somehow been overlooked, and those who provided feedback after finding my thesis on the web at http://www.simons-rock.edu/~sara/thesis/.
``Both a proof and a novel must tell an interesting story.''
Ian Stewart, from Nature's Numbers
Introduction
Mathematical Literature
In September I proposed to write a thesis which combined mathematics
and creative writing, a thesis which would make math accessible to the
general reader and present math as fun. I began by considering both
what math I wanted to include and how I wanted to present that
material. As part of this process, I looked at examples of popular
mathematical literature, a somewhat unknown genre.
Popular mathematical literature, in contrast to math textbooks, is designed to appeal to a general audience. I have subdivided this genre into mathematical fiction and recreational mathematics, two categories of literature whose boundaries are sometimes blurred. Mathematical fiction is based on a fictional story model, whereas recreational mathematics is similar to an instructive essay. In other words, mathematical fiction concentrates on the story, using a mathematical premise, while recreational mathematics presents mathematics in an entertaining and accessible way. Normally, the narrator of mathematical fiction is omniscient or a character within the story, whereas in recreational mathematics the narrator is likely the author.
I consider Edwin Abbott's Flatland and the stories in Clifton Fadiman's anthologies to be examples of mathematical fiction, and Douglas Hofstadter's Gödel, Escher, Bach to be an example of recreational mathematics. Hans Magnus Enzensberger's The Number Devil seems to be a bit of both, though I would place it on the side of recreational mathematics because it is instructive (in tone). I feel that my thesis, like The Number Devil, predominantly falls into the category of recreational mathematics. Both are fictional stories, but they use the story primarily as a vehicle for mathematical instruction.
As part of my thesis, I have read many examples of popular mathematical literature. I will now briefly summarize some of these books and stories. Edwin Abbott's Flatland is the classic example of mathematical fiction, written 120 years ago. It is a somewhat satirical story of a flat land inhabited by two-dimensional shapes. In 1960, Dionys Burger wrote Sphereland, a sequel in which Flatland is visited by a three-dimensional being. By way of analogy, he leads the reader to consider the meaning of the fourth dimension. Enzensberger's The Number Devil is a children's story about a boy who dislikes mathematics until a magical number devil visits him in his dreams. It is written on a pretty basic level, and there are a lot of gaps and imprecise terminology, but it is an enjoyable and informative story. Just as I was finishing my thesis I stumbled upon another batch of books. These included Erik Rosenthal's The Calculus of Murder, an entertaining detective story in which a part-time private investigator, who also happens to be a mathematician, investigates the murder of a well-known San Francisco businessman, and Marta Sved's Journey into Geometries, a story intended for readers with little mathematical background that explores hyperbolic geometry as it is taught to a girl named Alice.
I also found dozens of mathematical short stories. Clifton Fadiman edited Fantasia Mathematica and The Mathematical Magpie, two anthologies of math fiction. The stories in these anthologies are examples of science fiction that are largely based on the mathematical. Aldous Huxley's ``Young Archimedes'' is a tragic story of a young Italian mathematical and musical prodigy. ``The Devil and Simon Flagg'' by Arthur Porges is an amusing tale of the devil and a mathematician working together to solve Fermat's Last Theorem. Robert A. Heinlein's ``And He Built a Crooked House'' explores the possibilities of four-dimensional geometry in real estate. In Martin Gardner's ``No-Sided Professor,'' some members of the Moebius Society become nonlateral surfaces. In ``The Island of Five Colors,'' also by Gardner, a mathematician attempts to paint an island off the coast of Liberia with four colors, only to discover that five colors are needed. Edward Page Mitchell's ``The Tachypomp'' is about a struggling math student who loves the professor's daughter and must discover infinite speed to win her hand in marriage. ``The Feeling of Power'' by Isaac Asimov is about a futuristic society in which computers are relied on for computation to such an extent that no one knows how to perform arithmetic. Mark Clifton's ``Star, Bright'' is the story of an exceptionally gifted young girl who uses the fourth dimension to travel.
Rudy Rucker also edited an anthology of math fiction, entitled Mathenauts. Like Fadiman's anthologies, Mathenauts is a collection of science fiction short stories. Greg Bear's ``Tangents'' is about a small boy who has a gift for understanding the fourth dimension. Rucker's own ``A New Golden Age'' is about the consequences of popularizing pure mathematics. His ``Message Found in a Copy of Flatland'' locates Abbott's Flatland in the basement of a Pakistani restaurant. ``The Maxwell Equations'' by Anatoly Dnieprov is the story of a mathematician and a Nazi war criminal who is forcing mathematicians to be computers by manipulating their brain frequencies. George Zebrowski's ``Gödel's Dream'' is about a computer program that must run forever. ``Cubeworld'' by Henry H. Gross discusses the implications of turning the Earth into a cube. As my thesis is a longer work, these short stories did not have any significant direct influence on Alice in Mathland, but I found many of them delightful to read and was encouraged by the number of authors who have written mathematical pieces. The site http://math.cofc.edu/faculty/kasman/MATHFICT/default.html led me to a number of books; many more examples of mathematical literature are listed there.
Using some of these as examples, I experimented a lot with the style of my thesis. I began envisioning something with clear divisions between the creative stories and the mathematical instruction. I anticipated having something like the dialectic vignettes (Platonic dialogues) of Gödel, Escher, Bach that would be both entertaining and serve as jumping off points for delving into further mathematical exploration. But over time these more creative sections grew in proportion to the solely instructive sections, and they encompassed the math that I had anticipated having in those meta-sections. In order to demonstrate that mathematical creative writing can be done, I wanted to merge the two distinct styles. Fortunately, my writing was doing this on its own, and in a way that improved the story. As a result of having the math and the story intertwined, I think my thesis flows better and is able to emphasize more strongly my idea that mathematics and creative writing are not mutually exclusive. I believe I have succeeded in creating something which is both mathematically instructive and entertaining.
In addition to drawing heavily on examples of mathematical literature, my thesis is derived from non-mathematical literary models, including the dialogue and the journey. Much as in The Divine Comedy, I have a Virgil and a Dante. My Virgil is the Yellow Pig, a fantastical and entertaining math teacher. My Dante is Lewis Carroll's Alice, a curious and inquisitive young girl with whom I hope the reader can identify. My story is a physical journey through a mathematical wonderland as well as a mental journey in which Alice is exposed to mathematics and learns to enjoy it. I have chosen Alice as my character because I see Lewis Carroll, an author and a mathematician (logician), as one of my inspirations.
My story consists of five chapters, each etching the surface of a branch of mathematics with topics that I hope are interesting. The first chapter introduces geometry, including the Pythagorean theorem; the second explores numbers - p, e, the golden ratio, and primes; the third discusses combinatorics, the Pigeonhole principle, graphs, and groups; the fourth considers more geometry and topology, the study of surfaces; the fifth describes probability, game theory, and symbolic logic. Each chapter contains several sections of dialog between Alice and the Yellow Pig. At the end of my thesis are a handful of appendices in which I consider these mathematical topics in more depth. This is my way of experimenting with different forms and providing more historical and mathematical information. I encourage readers not to neglect these appendices.
Although it contains a lot of mathematics, Alice in Mathland is not a textbook. It is meant to be both informative and entertaining. My story is meant not for mathematicians, but to educate the general public. As such, even though the topics considered are advanced, very little mathematical background is assumed. My goal is to make mathematics accessible. In his introduction to Flatland, Isaac Asimov comforts the reader: ``Fear not, however. It contains no difficult mathematics and it won't sprain your understanding. [It is] a pleasant fantasy. You will have no sensation of `learning' whatsoever, but you will learn just the same.''
I hope you, like Alice, find your journey into my mathematical world a pleasant and rewarding one.
It was an unusually warm day, and Alice had taken the opportunity to walk to a field not far from her house. She had just prepared a tea party for a select few of her stuffed animals. They were finishing their tea, though she never actually saw them drinking. Alice was sitting on a comfortable patch of grass, chaperoning to make sure their napkins didn't blow away. She detected some motion out of the corner of her eye and got up to investigate. Alice hoped it wasn't her neighbors' dog. The last time she had had a tea party, he had nearly made off with her stuffed llama. Fortunately there was no dog this time. Instead she saw a group of small animals scampering by in a pink and yellow speckled flurry. Snatching up her stuffed animals, she chased after them. They led her around in circles a few times and finally stopped just near the picnic blanket where they disappeared down a surprisingly large hole.
Cautiously, Alice approached the hole. She knelt down beside it, leaned over, and peered in. It appeared to be a long tunnel, but she couldn't see how long it was or where it went because of the darkness. She stood up and inspected her dress to make sure it hadn't gotten dirty. Mother would throw a fit if it had. Another little yellow creature leaped from behind her and jumped down the hole. Startled, Alice jumped too, losing her grip on her favorite teddy bear. ``Oh no,'' she cried out, as she watched it fall down the hole. She thought for a moment. Then, she carefully wrapped up her other animals in the blanket and put them in the basket she had used to carry them to the field. ``I'll be back soon,'' she said, kissing them each once. ``Do not worry.'' And in another moment, down went Alice after her bear and the pink and yellow creatures, never once considering where the hole would lead her and how in the world she was to get out again.
The hole went straight on for some way. It was like being on a roller coaster or a slide. Alice held her skirt to keep it from blowing. She supposed she was going downward, though she really couldn't tell. ``I'm falling, so it must be downward,'' she rationalized. Down, down, down, with nothing but the whoosh of the wind. As she slid, she wondered if she would ever stop falling. She wondered how far she had traveled and where she was. She tried to calculate how far she must have fallen and how fast she was falling, but she found it difficult to remember her sister's physics lessons while falling. She found that fact to be rather inconvenient. ``What's the point of such learning,'' she thought ``if I can't use it when I might need it?''
She wondered if she would end up in the center of the earth. Or perhaps she would come out the other side. Or, even stranger still, she could return to where she started, only to find everything changed. Or maybe she would find everything the same but that she had changed. She thought she wouldn't like that very much at all, but then she wouldn't be herself anymore so maybe she would. And here Alice began to get rather sleepy, until suddenly she landed - thump - on a pile of yellow books, and the fall was over. There was nothing except for the books. Alice had fallen into a dark cave with cold stone walls. Elaborate torches lined the walls.
She was not a bit hurt, but she was slightly disoriented, and she thought she saw a yellow pig. Or, it is more accurate to say, she thought what she saw couldn't have possibly been a yellow pig even though she was certain that was what she was seeing. Things became still curiouser as she suddenly found herself singing under her breath a song she had never heard before.
Alice looked around for her teddy. On the ground beside her lay a thick blue velvet ribbon. This she recognized as the ribbon she had tied around the bear's neck. The bear was nowhere in sight. ``But what was that pig-like animal?'' she asked herself. ``Perhaps it can help me find my bear.''
Alice ran off in the direction of the peculiar animal, but it seemed to get smaller and then it vanished before her eyes. She stopped running just before a wall which did not appear to be a wall at all. It was a mirror that reflected Alice and what was behind her, causing her to see an infinite tunnel of Alices. She thought it very strange that a pig had been there and was no longer, but as it wasn't much odder than seeing a yellow pig in the first place, she tried to dismiss it. It was, after all, a Thursday.
She looked to her left and right. On either side of her was a stone wall. There was no source of light, but somehow Alice was able to see. The corridor looked as though it were frequently traveled, as it didn't seem either dusty or lonely. She briefly considered asking it if it were lonely, but didn't for fear that the girl in front of her who looked just like her would think she was a ridiculous child, talking to hallways as though they could answer. ``That's silly,'' she said aloud. ``It's just my reflection, and it won't think anything of me talking to the hall.'' Sure enough, as she spoke, so did the other girl. She would have said more, but it occurred to her that someone might arrive, and then wouldn't she look even more ridiculous, talking to her own image!
Alice turned around and walked back down the corridor. All the way at the other end was a red door. Above it was a sign that read ``Enter'' and below it another sign that said ``Exit''. Poor Alice, knowing not whether to enter or exit, sat in front of the door considering her predicament for quite some time. At last she decided to open the door without either entering or exiting. ``After all, I'm only entering if I think I am going somewhere. And I'm only exiting if I think I'm leaving somewhere. But I don't know where I would go, and I don't know where I am, and I'm certainly not thinking very clearly at all today.'' And so she opened the door slowly and cautiously.
Alice had expected to find herself in another room, but instead she found herself outside. She must have been outside because there was sunlight. Unless she had been outside and in this strange land the sun was only inside. But that didn't seem right. Alice stepped out, or rather, through the door into one of the most magnificent meadows she had ever seen. Flowers dotted the grass as far as she could see. A bubbling creek wound its way through the flower beds. To one side was a grove of trees; above Alice was an expansive bright blue sky, a backdrop on which wispy white clouds had been painted. The aroma of the flowers was stronger than anything Alice thought she had smelled before, and though it was entirely pleasant, it made her dizzy.
Always drawn to water, Alice walked to the small pond which the creek had formed. The water was clear and it sparkled like liquid diamonds. She cupped her hands and dipped them in the pond and then drew the water to her mouth. The water was colder than she expected, but it felt good as the sun was quite warm. The water tasted clean, and immediately Alice felt refreshed. She lay down on a patch of flowers and dipped her long hair into the pond. Staring upward, she watched the clouds change shape. ``That one looks like a bunny,'' she thought, ``and that one a dragon.'' The odd shapes and their boundaries entertained her until she detected a slight motion out of the corner of her eye.
It was a yellow pig; she was now quite certain that it was a yellow pig, as it appeared to be both yellow and a pig. She called out to it, ``Hello, Pig.'' Startled, the Pig turned around and ran toward the trees. For the second time that day, Alice took off after him. She ran and ran through the green blur of trees. The trees were very thin, but they seemed to be laid out in a square grid as if to trap her, and as she ran through the spaces between the trunks, she had to be careful so as not to run into them. ``Running into trees would not be good,'' she thought to herself.
She was gaining on him. Suddenly he stopped and turned around. ``Do you know why I will be able to outrun you?'' he asked. Without waiting for an answer, he chortled, ``Because I am running irrationally!'' And with that, he resumed running.
``Wait,'' Alice called out after it. ``What do you mean?'' Again, he stopped and turned around. He stared at Alice for a very long moment. In this moment Alice was finally able to observe the mysterious pig. He looked like a normal pig, though perhaps he was a bit larger than most pigs. He had a yellow pencil tucked behind his right ear and a calculator tucked behind his left ear. He was a deep yellow in color, somewhere between a golden orange and the color of lemons, with some darker spots on his belly.
``Would you really like to know?'' he asked.
``Oh, yes, very muchly,'' replied Alice, who was intrigued by the talking pig and didn't want it to run off again. Surely this pig could help her find her bear and tell her how to return home. She had fallen such a long way.
And so the Pig bounded over to Alice and motioned for her to sit down. He proceeded to stand upright on his two hind legs, remove the pencil from behind his ear, and speak, in a manner that wasn't much different from preaching.
The Pig began: ``The trees in this forest are laid out in a most regular pattern, as I'm sure you have already noticed. Consider not the trees, but the center of each tree trunk. If you look at all of these points, they make up a rectangular lattice.''
``A rectangular lettuce?'' interrupted Alice.
``Not a lettuce, a lattice,'' responded the Yellow Pig. ``A rectangular lettuce would be unproductive, not to mention silly. A lattice is just a grid, like the corners of squares. Or, like the intersections of streets.'' And so saying, he picked up a stick and drew a series of evenly spaced parallel lines in the dirt. Then he drew more evenly spaced lines that intersected those at right angles. ``All of these points of intersection are lattice points.'' Indeed, an aerial view of the forest would have looked very much like a square grid of evenly spaced points.
``In a unit square lattice, points are separated by one unit from their horizontal and vertical neighbors. It doesn't matter what this one unit is, but it's the same distance.''
Again Alice interrupted, ``But those two points,'' she said pointing, ``are further apart than those two.''
``Let's call them Lorina and Edith,'' Alice suggested.
``Well, I was thinking of simpler names than that,'' the Pig explained. ``Let's call this point at the bottom (0,0). And that one just to the right of it (1,0), then (2,0), and so on. And the ones going up in the left column (0,0), (0,1), (0,2), ... . The first number in the pair refers to how far to the right the point is, and the second number refers to how far up it is. The two points you mentioned are (0,0) and (3,4). They are in both different rows and columns. You see how the naming works? The name of a point identifies its location.''
``But those are boring names. Lorina and Edith would be much better.''
``Perhaps, but my naming is more logical, because with numerical names, we can calculate the distance between any two points. The distance between (0,0) and (0,1) is 1. I have described the points using what are known as ordered pairs. The first number in both pairs is the same: 0. It is only the second number that has changed. To determine the distance between them, we need only to subtract 0 from 1. Similarly, the distance between (1,0) and (3,0) is 2. This time the second number in the pairs is the same so we subtract 1 from 3. Now do you understand? It would make no sense at all to subtract Lorina from Edith.''
``Yes, but you haven't attempted to subtract Lorina from Edith either,'' pointed out an indignant Alice, who didn't see why the Pig wouldn't take her naming suggestions. ``You picked two points that are in the same row or two points that are in the same column. Lorina and Edith aren't in the same row or column, so they wouldn't have either the first or second number in their pairs in common. How could you subtract them?''
``Well, it would be quite a bit more complicated to do,'' admitted the Pig, ``but I assure you I can. I want to explain a little bit about how I was running before I answer your question. There are all kinds of diagonals in this lattice. You pointed out one of them to me already. The diagonal through (0,0) and (1,1) goes through (2,2) and (3,3) and (n,n), for any n. There's another diagonal parallel to it that goes through (0,1), (1,2), (2,3), and so on. And there are diagonals perpendicular - that is, at a ninety degree angle - to these, such as the one that goes through (0,5), (1,4), and (2,3). Or the one through (3,4) and (4,3). I only need to specify two points to determine the line. That's a very important fact. I can refer to each line by an equation or just in terms of how steeply it slopes.''
``Oh, you are making my head hurt,'' said Alice. ``I feel as though I am in a math class with all of this talk about equations.'' Alice found math class to be confusing.
``Equations are just a way of expressing something, just as my point-pairs are a way of identifying points with two values, an x and a y. Pictures are another way of expressing concepts,'' he said, pausing to sketch some diagrams in his notebook. Alice watched as he drew a grid of points and labeled several lines. She was impressed at how well he could draw, being a pig and thus not having the advantage of a thumb.
``The y decreases by 1! So if I chose an entirely different two points, like your (0,0) and (1,3), then as the x increases by 1, the y increases by 3.'' Alice's head still hurt, but not as much anymore.
``Absolutely correct,'' praised the Pig, and Alice beamed. ``The slope in that case would be 3. Now maybe you'll see what slope has to do with running. I was running in a straight line with a certain slope. Only I couldn't run in a straight line with slope 1. Look at our graph. The line from my origin - that's the point (0,0) on the graph - with a slope of 1 intersects a point. That point represents a tree, and I certainly didn't want to run into a tree. I couldn't run in a straight line with a slope of 2 or 3 or 5 because I would eventually hit a tree then, too. Nor could I run in a line with a slope of 1/2. That is, whenever the x changes by 2, the y changes by 1. Slope is just the change in y divided by the change in x. Running at a slope of 1/2 is not very different from running at a slope of 2, and so I would hit a tree in the same amount of time. Running at a slope of 3/2 or 4/3 isn't any better. Because if you look at lines starting from that bottom left corner with those slopes, they all intersect points.''
``I think so,'' said Alice, hoping that maybe the Pig would make more sense if he continued.
``Good, because here is the tricky part. I wanted to run in such a way that I would never hit a tree, but running at any slope x/y would cause me to hit a tree. I didn't want to just run so I wouldn't hit a tree for a long long time, but so that I would never ever hit a tree. So I just picked a number that isn't x/y. That way, I will never intersect one of those (x,y) tree-points,'' the Pig said, waving his pencil gloriously.
``A number that isn't x/y?'' repeated Alice.
``There are a lot of numbers, surely you can't expect to be able to represent them all so easily.'' He paused and scratched his forehead. ``Oh dear, I have to explain to you about numbers. Well, think about a number line. And pay close attention because I have a lot of terms to define here. On a number line are integers, numbers like -17, -16, -15, 0, 1, 2, 3, and 1729, to name a few. The positive integers are all the ones greater than 0; that's 1, 2, 3, and so on. They are called the natural numbers or counting numbers. And between these natural numbers are fractions like 1/2, [1/ 17], [17/ 42]. These are called rational numbers because they express ratios between two integer numbers. Even if we draw all of these rational numbers on a line, we would still be missing most of the points on the line. Because in between the rational numbers, there are so many more non-rational numbers. The number line contains the rational and irrational numbers; that's all of the real numbers.''
``Please do explain. But first, tell me a few things,'' requested Alice, who was always full of questions. ``What are irrational numbers? What did you mean by there being more irrational numbers than rational ones? Have you seen a teddy bear? Are you a yellow pig? Oh,'' she said, remembering her manners, ``My name is Alice.''
``I am a yellow pig, and I'm pleased to meet you. Welcome to Mathland,'' said the Pig.
``Mathland?'' repeated Alice. ``Is that where I am?''
``It is,'' said the Pig, ``so we should do math. You asked about irrational numbers. Irrational numbers are simply those that aren't rational. It's hard to understand this without any examples, so if you wait a little while I'll tell you about the first irrational number to be discovered. It's also difficult to explain why there are more irrational numbers than rationals. It has to do with being able to list numbers. How many natural numbers are there? Remember, that's the numbers 1, 2, 3, ... .''
``A lot,'' said Alice. ``More than I could count even if I count for my entire life.''
``Right,'' the Yellow Pig said. ``But for mathematicians, that's not always precise enough. There are an infinite number of natural numbers. But this doesn't bother them, because they can still describe the natural numbers. That's because they are ordered. If you give me a natural number, any natural number, I can give you the next natural number by adding one to it. We can list all of the natural numbers: 1, 2, 3, ... , n, n+1, n+2, ... .''
``I see,'' said Alice.
``Now, here's where it starts to get tricky,'' warned the Pig. ``How many integers are there? Integers are all of the natural numbers, all of the natural numbers with negative signs in front of them, and zero.''
``Infinitely many,'' said Alice.
``Right,'' agreed the Pig. ``But compare that `infinitely many' with the number of natural numbers. Are there more integers than natural numbers? Less?''
``There must be more,'' concluded Alice, ``because all of the natural numbers are integers. There are twice as many integers as natural numbers because there are the negative numbers as well. And there's a zero, so that's even more than twice as many.''
``It seems so, doesn't it?'' the Pig replied. ``But think about that number zero. How much difference does one number make? You already have an infinite number of numbers. Why consider it at all? Isn't it enough to say you have twice as many integers as natural numbers?
``It seems so. One number is meaningless when compared with so many,'' Alice agreed.
``Then by the same logic,'' taunted the Pig, ``what difference does twice as many numbers make? Twice something infinite is just something infinite.''
``I guess so,'' said Alice. ``I hadn't thought of that. All infinity is infinity. There's nothing bigger.''
``Not quite,'' said the Pig. ``There are different kinds of infinities, but we don't need to get into that. The number of elements in the set of natural numbers is the same as the number of elements in the set of the integers. Or rather, their order or cardinality is the same. That's just a fancy way of saying that there is a one to one correspondence between the natural numbers and the integers. So, we can list all of the integers. Any idea how?''
``We can list all of the positive integers first, then zero, and then the negative integers,'' Alice suggested.
``We could,'' said the Pig, ``but I'm afraid we would never get past the positive numbers to list the negative ones. We are much better off writing them like this: 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, ... . You see, it is just like our listing of natural numbers, only we have stuck a zero at the beginning and inserted negative numbers after each positive one. This makes it clear how the integers are related to the natural numbers.''
``You mean to tell me that both of these lists have the same number of elements, even though one list is contained in the other?'' asked Alice incredulously.
``More or less. They have the same cardinality. I know that sounds fishy,'' said the Pig.
Alice wasn't sure she believed him, but it did seem that one infinity shouldn't be bigger than another. Certainly not significantly bigger, so she let it go. She didn't quite understand what the Pig meant by cardinality, but it sounded interesting. She wondered if he would ever get back to the distance between Lorina and Edith.
``Now we have another set of numbers,'' the Pig said, ``known as the rational numbers. Those are the numbers in the form x/y, where x and y are both integers. Thinking of how to list all the rational numbers is much trickier, but there is a way. We make a table with the positive rational numbers. It could begin like this.'' He wrote:
|
|
|
|
|
|||||
|
|
|
|
|
|||||
|
|
|
|
|
|||||
|
|
|
|
|
|||||
|
|
|
|
|
``In the top row we have a bunch of fractions with 1 as their numerator. That means 1/1, 1/2, 1/3, 1/4, 1/5, et cetera. In the second row we have fractions where the numerator is 2. Like 2/1, 2/2, 2/3, 2/4, 2/5. What do the columns look like?'' he asked.
``Each column has the same whatdoyoucallit? The bottom number in the fraction.''
``Denominator,'' supplied the Pig.
``Thank you,'' said Alice politely. ``The first column has 1/1, 2/1, 3/1, 4/1, 5/1. The second column has 1/2, 2/2, 3/2, 4/2, 5/2.''
``Right,'' said the Pig. ``And these numbers go on and on in two directions. The rows and the columns. There are infinitely many rational numbers in the table.''
``Wait,'' interrupted Alice. ``Some of the numbers on your table occur twice, like 1/2 and 2/4. How can you be sure that there are infinitely many rationals when you have the same numbers so many times?''
``Excellent question,'' said the Pig. ``If we consider only fractions reduced to their lowest terms, like 1/2 instead of 2/4, we avoid that problem. At any rate, every single positive rational number is contained in our table. We may have duplicates, but we aren't missing any numbers. And because our rationals are derived from the natural numbers, there are the same number of rational numbers as natural numbers. We can order them, too, though that is a bit more complicated.''
``But aren't they already in order?'' asked Alice, confused.
``I suppose they are,'' said the Pig, ``but I meant something more specific. If I were to write a list in this order, it would start 1/1,2/1, 3/1, 4/1, 5/1, º, 1/2,º. But that's no good, because it would have lots of huge gaps. When I say I want to list the rational numbers, I mean that there shouldn't be leaps in the list. To accomplish this we need to look at some diagonals in our table.'' Alice watched him draw arrows:
``Wow,'' Alice exclaimed, looking over the Pig's shoulder, ``that's neat. We've ordered an awful lot of numbers.''
``We have,'' the Pig agreed. ``But you've gotten me off on a tangent. I meant to be telling you about distance, not cardinality.''
``Well, then do that, too,'' instructed Alice. ``But,'' she asked again, ``have you seen a stuffed teddy bear?''
``I'm afraid I haven't,'' he replied. ``Why?''
``Because I have lost my bear,'' said Alice sadly. ``I dropped it down a hole, and I climbed in after it to try to find it. But I found only its ribbon.''
The Pig looked at her consolingly. ``Don't worry. You'll find your bear. I'll help you. But first, I'll explain about triangles so we can subtract Lorina from Edith.''
``I'll explain to you how to find the distance between the two points you chose in the lattice. Think about your two points as being opposite corners of a square,'' he said drawing a square.
``The question is to find the length of the diagonal of the square with side length one. Let's look first at the area of the square. It is 1 square unit because the area of a square is just the length of the side squared. Now I'm going to add another square to our diagram, this one with the diagonal as a side. This new square has a center at (1,1). And each of its corners is one unit away from the center. Do you know what its corners are?'' asked the Pig.
``I think so,'' said Alice. ``The point above (1,1) is (1,2). The point below (1,1) is (1,0). The point to the left of (1,1) is (0,1), and the point to the right of it is (2,1). So those are the four corners.''
``The Greeks were just concerned with geometry; what we think of as algebra was, for them, just another way of representing geometry. We can rewrite many problems of algebra in terms of geometry. We can also rewrite problems in geometry as problems in algebra. From our picture, we get the equation side2 = 2. Do you know what a square root is?'' he asked.
Alice nodded. ``A square root is a funny sign that looks like a cross between the long division sign and a check mark. It means the opposite of squaring a number. So since 32 is 9, the square root of 9 is 3.''
``Right,'' said the Pig. ``Actually, the square root of 9 is also -3, but we don't want to worry about negative numbers. They don't make any sense geometrically. To solve for the length of the side, we need to take the square root of 2. That is, the length of each side of the larger square is ÷2.''
``That's not equal to 1, or 2, or 3, or any such number,'' remarked Alice, puzzled.
``It isn't,'' said the Pig. ``It isn't even equal to a fraction. It's an irrational number.''
``I don't think it's nice to call a number irrational,'' said Alice.
``Perhaps not, but that's how the Greeks thought of them, as illogical numbers, as numbers that didn't fit into their way of thinking of ratios. And the square root of two was the first number to puzzle them in this way.
``Now I'm ready to tell you about the distance between any two points. This is where we use the Pythagorean theorem. The Pythagorean theorem says that a2+b2 = c2, where a and b are the lengths of the sides of a right triangle and c is the length of the hypotenuse.''
``Hippopotamus?'' asked Alice.
``No, hypotenuse.''
``A hippo in a noose? What would you do with a hippo in a noose?''
``No, not a `hippo in a noose' either. A high-pot-en-oose. The hypotenuse of a right triangle is the longest side, the one across from the right angle.''
``Oh. That's a really silly name for it. Is that always the longest side?'' asked Alice.
``Always,'' replied the Yellow Pig. ``And not only is it the longest, but we have an explicit formula for finding its length, given the lengths of the other two sides. That's our Pythagorean theorem,'' he said, writing:
|
``So this hypotenuse thing is always the same length?''
``In relation to the other sides, yes. I'll show you. Let's see ... today is a Thursday, so I'll give you the Thursday proof.''
``There's more than one proof?'' Alice asked.
``Why, pig-heavens, I bet there are over thirty-seven proofs, and they all explain the same thing in a different way. I'll show you a few of the proofs. This first one is a proof by picture. Our algebraic expression a2+b2 = c2 is represented geometrically by this picture.'' Alice studied the picture carefully.
``The area of the outer square with sides of length (a+b),'' supplied Alice.
``Right. So when we say that in a right triangle a2+b2 = c2, what we mean is that the sum of the area of two squares with side lengths a and b is equal to the area of a larger square with side length c. I can draw a square on side a and a square on side b and their combined area will equal the area of the square on side c. Does that make sense?''
``Yes, I think so,'' said Alice.
``Good. Now, for some algebra,'' continued the Pig. ``The area of the larger square is (a+b)2. And that has to equal the area of the small inner square, which is c2, plus the area of the four triangles surrounding it. Each of these right triangles has sides of lengths a and b. So the area of each triangle is 1/2ab, and since there are four of them, the combined area of the triangles is 2ab. Now we need to find the value of (a+b)2.''
``Isn't that a2+b2?'' Alice inquired.
``No,'' said the Pig, ``try an example.''
Alice thought aloud, ``(1+2)2 = 32 = 9, and 12+22 = 1+4 = 5. I guess it isn't,'' she concluded. ``So what is (a+b)2?''
``You have to be careful when multiplying polynomials - expressions like a+b. It's like when you learned to multiply numbers. Think about squaring 17, which is really the same as squaring (10+7). First, we multiply 7 by 7. Then, we multiply 7 by 1, which is really 10. This gives us 17 times 7. Next, we multiply 1, or 10, by 7, and then by 1. This gives us 17 times 10. We add these two results together to get 17 times 17.''
He continued, ``The same principle applies to squaring (a+b), that is, calculating (a+b)(a+b). We use a method known as FOIL. That stands for first, outer, inner, last. We multiply the first two terms, the outer two terms, the inner two terms, and the last two terms. Then we add all of those together. We can represent this with a diagram.'' He wrote:
``I guess it does,'' agreed the Pig. ``And it shows us how to obtain the product (a+b)(a+b) using the FOIL method. The first two terms are a and a, so we get a2. The outer terms are a and b; we multiply those to get ab. The inner terms are b and a, yielding ba, and the last terms are b and b, or b2. We add all of the terms together to get a2+ab+ba+b2 or a2+2ab+b2.
``So, the area of the whole square is a2+b2+2ab,'' said the Pig. ``And it is also c2+2ab. This gives us the equation a2+b2+2ab = c2+2ab. Both sides have a 2ab so we can cancel them out. We are left with precisely what we wanted to prove: a2+b2 = c2.''
``That's the first logical thing I've seen all day,'' Alice remarked.
``Yes, it's very logical. Most of what you will see here is logical, although it might not appear to be so at first. This is a world ruled by mystery and logic. Let me show you a second picture.''
``Yup. In this diagram it's the outer square that has side length c. And the inner square has side length (b-a). Again, there are four triangles with legs of lengths a and b.''
``So, the area of those four triangles is still 2ab,'' Alice thought out loud.
``Right. And the area of the larger square is c2, and the area of the smaller square is (b-a)2.'' The Pig looked at Alice. She remained silent. He continued, ``b-a is really just b+(-a), so we can use the FOIL method again.'' He drew:
``Correct,'' said the Pig, ``and that's just a2-2ab+b2.''
``Right,'' agreed Alice. ``I think I can finish the proof now. The area of the small square and the four triangles has to equal the area of the large square. So a2+b2-2ab+2ab = c2. And the -2ab+2ab part goes away, leaving us with a2+b2 = c2.''
``Absopositivelutely correct,'' praised the Pig. ``Those are my two favorite proofs of the Pythagorean theorem, but there are many others that are quite different. One, credited to a mathematician named Legendre, is based on the idea of similar triangles. Similar triangles are triangles with different side lengths but the same angle measures. I'll show you another geometric proof that you should be able to understand if you analyze this picture carefully,'' he said, sketching a right triangle and forming squares on all three sides. He then split the largest square into a small square, which was the same size as the small square on one of the sides of the triangle, and four equally shaped quadrilaterals. He also split the medium sized square into the same four equally shaped quadrilaterals. ``Now you can see that the combined area of the small square and the medium sized square on the sides of the triangle is equal to the area of the larger square.''
``Back to Lorina and Edith. Lorina was the point (0,0) and Edith was the point (3,4) in my naming scheme. We can make a right triangle from Lorina and Edith, like so,'' he said, drawing:
``I see,'' said Alice. ``The square of the distance is the square of the hypotenuse which is equal to 32+42. That's 9+16 or 25. Then the distance between Lorina and Edith is ÷[25] which is 5,'' she concluded. ``Is that right? Is the distance 5 units?''
``It is,'' said the Pig. ``I'm very pleased that you were able to calculate the distance between Lorina and Edith. Now you know how to calculate the distance between any two points.'' Alice was pretty pleased herself. She thought that the Pythagorean formula was quite useful.
``Let's use our formula to calculate some more distances,'' the Pig continued. ``If we know the horizontal and vertical distances a and b, we can calculate the diagonal distance c. Most of the time c is an irrational number, like ÷2.''
``But for 3 and 4, we got 5,'' remarked Alice.
``We did,'' said the Pig. ``There are an infinite number of such integer solutions to a2+b2 = c2. Even though there are infinitely many integer solutions, it's not very clear how to find them. The Greeks knew of a few triples with integer values for side lengths. The smallest of these is our (3,4,5). Two more triples are (6,8,10) and (9,12,15). They work because they are larger versions of (3,4,5). We say they are multiples of (3,4,5).''
The Pig continued, ``It turns out that all such triples can be written in the form (p2-q2,2pq,p2+q2).''
``What do you mean?'' asked Alice.
``Just pick two whole numbers, p and q, with p greater than q.''
``Like 2 and 1?'' asked Alice.
``Good example,'' said the Pig. ``Then p2-q2 is 22-12 = 3. And 2pq is 2 ·2 ·1.''
``That's 4,'' said Alice.
``Right,'' said the Pig. ``And do you know what p2+q2 is?''
``Let's see,'' started Alice, ``it must be 22+12 which is 4+1 or 5.''
``Exactly,'' said the Pig. ``See? That's our triple: 3, 4, and 5.''
``Neat,'' said Alice. ``Can I make another triple?''
``You sure can,'' said the Pig.
``I'll try 2 and 3,'' Alice said. ``That makes the first number in the triple is 32-22 or 5. The second number in the triple is 2 ·3 ·2 which is 12. And the third number is 32+22 = 9+4 or 13. Is that a Pythagorean triple?''
``We can check: 52 is 25, 122 is 144, and 132 is 169. What's 25+144?''
``That's 169,'' answered Alice. ``And 52+122 = 132, so it works. But,'' she paused, ``why does it work?''
``I'm glad you asked. In mathematics it's important not to accept everything, but to try to understand why things are true. It's fairly difficult to find that magical formula,'' said the Pig, ``but fortunately, we already know it, so it's fairly easy to see that it will produce Pythagorean triples. We can verify that it works in the same way we checked that (5,12,13) was a triple. We just use substitution. The Pythagorean theorem says that a2+b2 = c2. We let a = p2-q2, b = 2pq, and c = p2+q2. That's a lot of numbers and variables, so hold on to your hat,'' he warned.
``I'm not wearing a hat,'' said Alice, slightly confused.
``It's just a figure of speech,'' the Pig said with a smile. ``It means that you should pay close attention. Here's what we want to show: (p2-q2)2+(2pq)2 = (p2+q2)2.''
``Well, (p2-q2)2 is p4 - 2p2q2 + q4. And (2pq)2 = 4p2q2. So when you add those two together you get p4 + 2p2q2 + q4. That's a2+b2. Our c2 is (p2+q2)2 or p4 + 2p2q2 + q4, which is the same thing we got for a2+b2. Sure enough, a2+b2 = c2. Did you follow that?''
Alice had to admit that while it seemed logical, she would have to look over the details before she could really be convinced. She borrowed the Pig's notebook and slowly worked out the algebra for herself. ``It sure is neat,'' she said.
``It is,'' agreed the Pig. ``And there's even a Pythagorean triple with the number 17. That's my favorite number.''
``What's so special about seventeen?'' Alice asked.
``A lot,'' said the Yellow Pig. The Pig continued talking, but Alice was having trouble following him, for he had suddenly become almost frighteningly excited. Instead, she dozed off and took a short nap, filled with dreams of hippos and a's and b's.
When she woke up, Alice found herself back in the woods, lying on a bed of leaves and covered by a blanket of five-pointed stars. The Pig was sitting nearby mumbling to himself and scribbling notes on a pad. Noticing she was awake, he stopped scriblling and said, ``I'm sorry I got so carried away before. I was a bit irrational, I'm afraid. I could be more irrational, though. You know how I told you there were all of those irrational numbers? Well, what I didn't tell you is that some irrational numbers are more irrational than others? It's kind of like all pigs being equal.'' He chuckled.
Alice wondered if he would be terribly upset if she interrupted him. His lectures so far had been interesting, but she hadn't had anything to eat since early that morning and was now very hungry. And he was terribly confusing. What did he mean about pigs being equal? About being more irrational? She thought he was already very irrational, though she dared not say so.
The Pig looked back toward his notepad and continued, ``The first troubling irrational number that the Greeks discovered was ÷2, but there are many other irrational numbers that are even more interesting. Two of my favorite irrational numbers are known as p and e. They are both very important numbers, especially in geometry and calculus.''
Alice sighed lightly and shifted her position, trying to ignore her growing hunger. Startled by the noise of her movement, the Pig looked up. ``I'm sorry, I'll stop now. You've had an awful lot of math for one day. And you must be hungry,'' he said.
Surprised and slightly embarrassed by the Pig's perceptiveness, Alice felt she had to apologize. ``I really am enjoying the math. It's just that I haven't had anything to eat and that makes it hard to concentrate.''
``Well, then,'' replied the Pig, ``let's get some food. I'll take you back to my cabin. It's not far from here, just back by that grove of trees.'' Alice saw a small clearing in the direction that he pointed.
The Pig stood up and collected his belongings. Standing next to him, Alice guessed that he was about three feet tall. He had a funny way of walking, a fast somewhat bouncy skip. He had to take several little steps to keep ahead of Alice. The two walked in near silence, giving Alice time to examine the Pig more closely. His ears were now pointy and standing on the top of his head. Before they had been sort of floppy and drooped on either side of his head. His eyes, she noticed, were different colors. His left eye was bright blue while his right eye was a dark green. He had a curly little tail which Alice was very tempted to tug. She didn't though, because she thought that would be rather rude.
The trees were getting more congested. The Pig led Alice on a small cobblestone path. The terrain became much hillier. ``It's just over this hill,'' he said. The path was slightly overgrown with bushes, and a canopy of taller trees shaded it from the sun. Alice saw the clearing ahead. There was a semi-circle of rocks in front of two very large trees. As they walked around the trees, Alice saw a large rock with a chimney sticking out of it. On the side of the rock was a small cabin.
The cabin had a very small door, in front of which was a welcome mat and above which read the inscription ``Y. Pig''. Alice followed the Pig inside, ducking so she could fit through the door. ``This is my home,'' the Pig said almost timidly. ``I don't have many visitors.''
The cabin was not the least bit spacious. It was the sort of place Alice imagined a real estate agent describing as ``cozy'' because ``cramped and cluttered'' didn't sound nice enough. To be fair, Alice thought, it probably wouldn't seem nearly as small if she were as short as the Yellow Pig. The kitchen was big enough and had a barstool as there was no dining room. It looked like the Pig slept in the living room on a pile of hay. Surrounding the hay were piles of papers, jigsaw puzzles, and a Rubik's cube. The most impressive aspect of the cabin was the full wall of books.
``What would you like to eat?'' asked the Pig, interrupting Alice's thoughts.
``What do you have?'' Alice asked, afraid that the Pig might only have foods that would interest a pig, though she didn't know what exactly a pig, especially a yellow pig, would eat.
``I don't have very much food. I have some fruit pies: strawberry, blueberry, and key lime. I also have numbers, my favorite snack.''
``Numbers? You eat numbers?'' asked Alice.
``Of course I eat numbers,'' the Pig replied. ``How do you think I learned so much math?'' Alice thought that he was serious for a moment, but his blue eye twinkled merrily and the corners of his mouth were twitching.
``So what are these edible numbers?''
``They're crackers in the shape of numbers. They're especially yummy when dipped in numeral soup, but I don't think I have any of that.'' He took out a plate of the number cookies for Alice. They were small, and there were dozens of them. An awful lot of them were 17's, but Alice saw other whole numbers and even some decimals and fractions.
``Oh, they're like animal crackers!'' exclaimed Alice. ``I like animal crackers. My sisters and I often get them on the way home from school.'' Here Alice grew pensive for a moment, wondering when she would have animal crackers again. She could do without school and maybe even her sisters, she supposed, but she would like to go home. How would she ever get out of this strange land? She had fallen quite a long distance. ``Animal crackers come in all different shapes: elephants and cows and pigs. I like to eat them slowly, saving their heads for last.''
``Pig heads?'' the Pig gasped. ``You eat pig heads?''
``Oh no,'' Alice clarified. ``They aren't real pig heads. I would never eat pig heads. Well, I suppose I like bacon, but that's not from the head, is it?'' She could tell that she was only making things worse. The Pig had turned a very pale shade of white. ``I'm sorry,'' Alice apologized again. ``I would never eat yellow pigs.''
``One of my brothers is a blue pig,'' said the Yellow Pig rather irritably.
``I would never eat blue pigs either,'' said Alice. ``Or orange pigs or purple pigs. Or even silver pigs. I won't eat pigs anymore. Please don't be angry with me,'' Alice pleaded, now almost close to tears.
``I'm not angry with you,'' said the Pig after a pause. ``Try one of the numbers.'' Alice gingerly picked up a number 3, afraid that eating a number 17 might be sacrilegious. She didn't want to offend the Pig again.
The number was sugary and somehow crunchy and chewy at the same time. Alice helped herself to another. The Pig had one as well. He chose a number 17. Alice supposed she was allowed to eat them. After Alice and the Pig had eaten a sizeable portion of the number cookies, the Pig brought out a small blueberry pie. ``My pies are perfectly circular, or rather cylindrical,'' he said, ``and each pie has a diameter of 2 punits.''
``Punits?'' Alice asked. ``What's a punit?''
``Why, it's one pig unit, of course,'' said the Pig in a way that made it sound as if he found the entire matter perfectly obvious and was surprised that Alice would ask such a simple question. ``A punit, in this case, is between two and three inches long. So my pies are about 5 inches in diameter.''
``What do you mean by `in this case'?'' Alice further inquired.
``Exactly that,'' said the Pig. ``What makes the punit such a wonderful unit of measurement is that it changes. Punits for pies may be different from punits for the height of ice cream cones or the shortest distance across a mud puddle. It's the most natural thing in the world to want to refer to completely different lengths as being one punit.''
``If you say so,'' conceded Alice. It sounded horribly confusing to her, but she didn't want to argue with the Pig when he was being so illogical.
``Oh, you'll be glad we're dealing with punits soon,'' the Pig said. ``It's much easier to do arithmetic on punits than messier arbitrary units. My pies have a diameter of 2 punits, and the radius is half the diameter. So each of my pies has a radius of 1 punit. Try that with your inches.''
``I guess you're right,'' Alice said, thinking it best to agree.
``Of course I am,'' said the Pig. ``Now, what's the circumference of this pie?'' he asked. ``That is, what is the distance around the outer crust? I mean, how does the distance around the crust compare to the distance of the diameter?''
Alice stared at the pie. ``It's certainly more than twice the diameter. Though I wouldn't think it's more than four times the diameter.''
``It isn't,'' the Pig confirmed. He took out a piece of string. ``I can make a square with this string around the pie, so that the pie is just touching the square on the center of each side. The length of each of these sides is 2 punits, the same as the diameter. And there are four of them for a perimeter - that's what we call the circumference of things which aren't round - of 8 punits. And that's larger than the perimeter of the circle.
``I know!'' interrupted Alice excitedly. ``We can wrap the string tightly around the pie and mark the length of the circumference. Then we can straighten out the string and measure it against your ruler.''
``Wonderful!'' the Yellow Pig exclaimed.
And that's just what Alice proceeded to do. She held up the string to the ruler. ``It's a little over 6 punits,'' she announced triumphantly.
``That's not precise enough,'' said the Pig. ``Fortunately, I have a very special magnifying ruler and calculator. It will show quarter-markings and third-markings and so on. Why, it will divide your punit into hundreds and thousands and even quintillions of equal parts if you want. It does lots of other things too.'' He typed the number 3 on the small keypad on one end of the ruler, and third-markings appeared on the punit.
``The string doesn't reach the first mark, so it's less than 6 1/3 punits,'' the Pig explained. He set the magical ruler to 4.
``It goes beyond the first marking. So it's more than 6 1/4 punits,'' said Alice. ``More than 6 1/4 and less than 6 1/3. How about tenths?'' The Pig showed her how to reset the ruler, and they saw that it was less than 6 [3/ 10].
``[3/ 10] is less than 1/3,'' said the Pig. ``Which means we have lowered our upper bound for the length of the string. This ruler has a special button to compare two numbers. It turns the fractions into decimals and then displays them in order from smallest to largest.''
``Neat,'' said Alice. ``You said I could break up a punit into as many pieces as I wish?'' The Pig nodded. ``What about 100 pieces?'' she asked, and he punched in 100.
``Oh my, that's hard to count,'' Alice exclaimed.
``You don't need to count it,'' the Pig said. ``I told you it was a very special ruler.'' He showed her a button on the ruler to display the number of the marking just to the left of the string and the number just to the right of it.'' Alice looked on amazedly. ``Oh, that doesn't work with every piece of string,'' explained the Pig. ``This is a magnetized string for use with this ruler.'' The ruler's LCD screen displayed 28 and 29.
``That means the length of the string, and therefore the circumference of the pie is between 6 [28/ 100] and 6 [29/ 100]. Or between 6.28 and 6.29,'' said Alice, pleased to show off her knowledge of fractions and decimals.
``Right-o,'' said the Pig. ``You can set the ruler to thousandths to find the next decimal place.'' He did so and announced, ``The circumference of the pie is just over 6.283 punits.''
He and Alice worked out a few more decimal places. But each time Alice told the Pig a new number, he shook his head, which looked sort of funny because it made his ears, which were floppy again, swing back and forth, and said ``That's not precise enough.''
Alice was starting to get frustrated. ``If your ruler is so advanced,'' she asked, ``why does it take so long to measure the string?''
``That's an excellent question,'' said the Pig. ``It actually can determine the length itself, but not as precisely as I would like.''
``Well, I'm not going to measure it anymore,'' said Alice defiantly. ``It's one of those tricky irrational numbers and I'll never be able to tell you the exact length.''
The Pig smiled. ``You're right of course. The roundness of the pie makes the length of the string an irrational number. You can stop measuring it now. But you are also wrong. I can tell you the exact length of the string.''
``How?'' Alice asked.
``The same way we dealt with the diagonal of a square. We can't write out that length as a decimal, but we know it is ÷2.''
``You mean, this 6.283 ... number is the square root of some whole number?'' Alice asked.
``Unfortunately not,'' the Pig explained. ``If you square it, or cube it, or raise it to any power, it's still not going to equal any normal fraction. This number is different from ÷2 in that it is not the solution to a regular polynomial equation. We say it is a transcendental number.''
``Transcendental?'' repeated Alice. ``That makes it sound all mythical or something.''
``Well, it is in some ways. It's certainly mysterious. Actually, if you take our number and divide it by 2, you find a very special number: 3.14159 ... . This number is mysterious because it shows up all over the place in mathematics, especially with circles. It shows up so often that it has its own name. And that's how I can tell you the exact length of the circumference. Just as the punit is a sort of made up unit for convenience, so is this one. Mathematicians call this number pi after pies like my own of course. They write p, which is a Greek letter. So the circumference of the pie is 2p,'' the Pig concluded simply.
``That's it?'' Alice asked. ``You expect me to be satisfied that we know the distance of the circumference of the pie just because we've given it some funny Greek name? That's a bunch of hogwash. No offense.''
``None taken. It would be more accurate to call it a bunch of math-wash. You see, mathematicians are often much more concerned with concepts than with numbers. It's enough that they are able to communicate a complicated idea with a little symbol. Just like our punit. We used punits to simplify a problem and communicate.''
``I guess that is sort of convenient,'' Alice agreed.
``Now that we've solved that problem, let us eat pie!'' squealed the Pig in delight. And he set out to cut the pie into two equal pieces.
The pie was rich and moist, and it was so very small that they soon had finished off every last crumb. Alice was about to ask the Pig to get out another of his yummy pies, when he asked, ``What is the pie's area?''
By this point Alice was not at all surprised that the Yellow Pig had another riddle for her. He was full of questions. She thought for a moment. ``The pie fit inside that string square we made earlier. And the square has an area of 22 or 4 punits, so the area of the pie is less than 4 punits.''
``That it is,'' agreed the Pig. ``And I can tell you how much less than 4 punits. The area of the pie is precisely p square punits!''
``p punits?'' Alice asked. ``How can that be?''
``That's p square punits. The square part is because we are talking about area. And I'll show you,'' the Pig said, taking out another pie. ``I'm going to cut this pie into very small pie wedges.'' And he cut and he cut until the pie was in dozens of itty-bitty pieces. ``Now, I'm going to put all of the pieces back together into what is almost a rectangle. Because we know how to calculate the area of a rectangle.''
``The area of a rectangle is length times width,'' Alice interjected.
``Yuperdoodle. Now here's how I make a rectangle out of our circle.'' Alice watched. The pieces were very nearly triangles, with two of their sides the same length. He took two slices and put them touching so that the sharp point of one was next to the crust of the other. He did this again for each pair of slices, and then he put all of the pieces together in that same way. ``Ta-da!'' he exclaimed. Sure enough he had made a rather long strip of the curved triangles that looked a little like a rectangle.
The Pig didn't seem to understand her comment. ``It's not a perfect rectangle,'' he explained, ``but that's only because I didn't cut small enough pieces. If I had cut each of these pieces in half, the curves would be less noticeable. And if I had cut each of those pieces in half, you would hardly see the curving at all. And so on and so on. In fact, if I had cut an infinite number of pieces, more than you could ever count, they would be infinitely thin, so small that you couldn't even make out the crust. And then I would have a perfect rectangle.''
``I think I understand,'' said Alice, though she was not entirely sure that she did. There was a limit to the amount of the Pig's logic that she could take at one time.
``But anyway,'' said the Pig, ``let's just pretend or suppose, as mathematicians like to say, that we have a rectangle. What's the length of a rectangle?'' He paused. ``Well, what was the length or circumference of a circle?''
``2p times the radius punits,'' supplied Alice.
``Correct. Mathematicians call the radius r and say 2pr punits. The circumference, or crust of the pie, borders the two long sides of our rectangle, half on each side. So the length of the rectangle is half the circumference of the pie, or pr punits. Since the radius of our pie is 1 punit, that's exactly p punits. Now what's the width of the rectangle?''
Alice studied the rectangle closely. Finally she saw the answer. ``It's the radius of our pie. Because the distance from the crust to the center of the pie is the radius. And all those sharp points on the slices are what was the center of the pie.''
The Pig beamed. ``Exactly right, Alice! So the area of the rectangle, which is the same as the area of the pie when it was in the shape of a circle, is the length p·r times the width r punits. That's p·r ·r punits or p·r2 punits. So the area of a pie is always pr2. That means the area of our pie with radius 1 punit is p punits. That's how much pie we've eaten. Well, not really, but let's eat another pie before I explain.''
Alice was confused, but hungry, so she didn't question the Pig and his tricks. The two ate the second pie quietly. When they finished, the Pig continued, ``We calculated the area of the pie, which is like knowing what size plate we would need to put the pie on or how much frosting we would need to put a thin layer across the top. We could even calculate how much frosting we would need for the sides because we know the circumference. But the area we have is not quite what we want to know, because area is only two-dimensional. We want something three-dimensional. We want to know the volume of our cylindrical pie. A cylinder is a circle with height. Mathematicians use the letter h to represent height.''
``They aren't very original,'' interjected Alice.
``I suppose not,'' agreed the Pig. ``If we multiply area by the height, we get the general formula for the volume of a cylinder: pr2h. The height of my pie is 1/2 punit, so the volume of the pie is p·12 ·1/2 = [(p)/ 2] cubic punits. That's the volume of one of my pies. Do you want more pie?'' he asked. Alice said she did, and the Pig took out the third pie. After they had finished that pie, they decided to go outside and lie in the sun while they digested their sugary meals.
Outside, the Pig helped Alice climb up onto the big rock, where they then lay resting for a few moments. The Pig took out his notebook and began to write again.
Alice, afraid of interrupting him, finally peered over at his notebook. He had written:
|
``Egads!'' she thought. ``What a horribly long fraction. It looks as though it will never end,'' she said to the Pig. ``It's like those irrational numbers, going on and on forever.''
``Yes, it is in some ways,'' said the Pig. ``But not all fractions that go on and on are irrational. It's the same with decimals. For instance, 0.33333 ... is a repeating decimal that is rational. It is equal to 1/3. Even though it is endless, it is regular. Decimals with more complicated patterns are rational too, like 0.248248248248 .... My fraction does go on forever, but I won't write it forever because there's a pattern. Do you see the pattern?'' he asked Alice.
She thought about it for a moment and the Pig offered her his notebook and pencil. ``Well, between every fraction bar is something plus one over something. And there are an awful lot of 1's. The first number is a 2 and then there are larger even numbers and the 1's. It goes 1, 2, 1, 1, 4, 1, 1, 6, ... . Two 1's and then the next even number. So I would guess that the next three terms are 1, 1, and 8.''
``Absolutely correct,'' the Pig said. ``I can stop writing out the fraction now that you understand it. Do you want to know what this fraction looks like as a decimal?''
``Yes,'' said Alice. ``It must be awfully strange. How can we calculate it when the fraction never ends?''
``We can approximate it, just as we did with p.'' explained the Pig. ``We'll compute the values of parts of the fraction and see what they look like.''
``But how can we calculate even the part of the fraction that you wrote?'' asked Alice, more than slightly daunted by the large fraction looming before her and the Pig.
``There's no reason to be intimidated by that fraction,'' said the Pig, ``but we can start out by calculating a smaller fraction, such as 2+[1/( 1+1)]. That's the beginning of our continuous fraction. We start from the bottom of that, and work our way up and to the left. So, 2+[1/( 1+1)] = 2+1/2. We can simplify that to the single fraction 5/2, which is equal to 2.5. Do you understand?''
``I do,'' said Alice, ``but that was a pretty short fraction.''
``Are you ready for something longer?'' asked the Pig. Alice nodded. ``How about this one?'' He wrote:
|
``Now that looks a lot harder,'' said Alice.
``It isn't really harder,'' said the Pig, ``but I guess it does look more complicated. Just remember that we need to work our way up from the bottom of the fraction, simplifying it in several steps. What's at the very bottom?''
``The fraction ends with 2+1,'' answered Alice.
``Right. So our fraction is the same as 2+[1/( 1+1/3)]. Next we consider the 1+1/3 part. That's 4/3.''
``I see,'' said Alice. ``The fraction is 2+[1/( 4/3)]. Now what?''
``Do you know what [1/( 4/3)] is?'' Alice looked puzzled. The Pig continued, ``That's 1 ½4/3. Dividing by a fraction is the same as multiplying by its inverse. The means we flip the 4/3 to get 1 ·3/4. And that's just 3/4, so our fraction is 2 + 3/4, or [11/ 4]. Written as a decimal that is 2.75. That wasn't so bad, was it?''
Alice agreed that it wasn't. ``Good,'' said the Pig, ``because I have a longer fraction for you to simplify.'' He wrote:
|
``Oh my,'' said Alice.
``Just start at the bottom,'' advised the Pig, handing Alice his pencil.
Slowly, Alice added together fractions:
|
She paused. ``Try flipping the fraction,'' the Pig suggested.
|
``Whew,'' said Alice, letting out her breath. ``What is that as a decimal?''
The Pig reached for his calculator. ``It's about 2.714. You did an excellent job with that fraction, by the way. Since it would take an awfully long time to simplify the whole fraction I wrote before and even longer to simplify that fraction with the new terms you suggested, I'll work those out on my calculator.'' He rapidly punched buttons for a minute or two and then announced his results: ``[1257/ 463] or about 2.7149 and [23225/ 8544] or 2.718281835.''
``Those numbers are awfully similar,'' observed Alice. ``They look like they are approximating another special endless number. Is there a name for this number?'' she inquired.
``As a matter of fact, there is,'' the Pig replied. ``The number 2.718281828 ... is called e. It was named after Leonhard Euler, a famous mathematician.''
``Oiler?'' repeated Alice.
``Yup,'' said the Pig. ``We'll come across more of his math later. But back to e. It's extremely important in calculus for limits and for computing continuously compounded interest.'' The Pig could see that he was losing Alice again. ``We can derive e as a limit in another way,'' he said, writing:
|
``That looks like some horrible mathematical expression,'' said Alice. ``How am I ever going to understand that?''
``It's not as bad as it looks. Just ignore the `lim' part and think of it as the value of (1+1/n)n for a really large integer n. Let's try some computations with different values of n,'' said the Pig. And so they did:
| ||||||||||||||||||||||||||||||||||||||||||||||
``Why, it is that very same number,'' Alice exclaimed. ``How does that number keep showing up? Just like p did!''
``Both numbers are very important in different branches of mathematics: p is in some sense a basis of geometry and e is a basis of calculus, which is the study of limits. Limits are pretty neat.
``Here's an old riddle known as Zeno's paradox. Let's say I'm running from here to that tree,'' said the Pig, pointing at a tree in the distance. ``I can run very quickly and accurately. So in the first second, I run half the distance to that tree. Then, in the second instant, I run half the remaining distance, or one-fourth of the original distance. At the third moment, I run half of the now remaining distance which is only one-eighth the original distance. I continue doing this advancing [1/ 16], [1/ 32], and [1/ 64] of the total distance in the next three steps. Each time I go half the distance that I had gone the time before. Mathematicians say that on the nth turn, I will have (1/2)n of the total distance left. The paradox is that I will never reach the tree. I can keep taking steps forever, but they are so small that I will never get to the tree.''
Alice thought about the paradox. In order to get to the tree, first the Pig would need to get halfway to the tree. After he got halfway to the tree, he would have to cover half of the distance remaining between him and the tree. And after that, the Pig would have to traverse half the still remaining distance. It seemed that he would never reach the tree, but Alice knew that in reality the Pig could get to any tree that he wanted to. ``How odd,'' she exclaimed.
The Pig continued, ``The total distance that I have covered is the sum of all the individual distances. Mathematicians like talking about sums. They like talking about sums so much that they have a special notation for dealing with endless sums. Instead of writing 1/2 + 1/4 + 1/8 + [1/ 16] + º+(1/n)2 + º, mathematicians use the Greek letter Sigma, written S.''
``Sigma?'' Alice repeated. ``Like that guy Sigma Freud?''
``No,'' said the Pig patiently. ``That's different. This S is just a letter to the Greeks, as is p. And mathematicians love to use Greek letters. They like writing confusing things like this:
|
``The limit that I wrote before reads `the limit as n approaches infinity ...'. Similarly, we read this as `the sum where n goes from 1 to infinity... .'
``Now, if we add up all of those numbers, we'll find that we get really close but don't quite reach 1. That's the paradox. Mathematicians go even further and talk about an infinite number of steps and limits and the sequence created by partial sums as converging to 1. They say things like series and least upper bound and Cauchy. Sometimes they even say sequentially compact, totally bounded, and clopen. Something is clopen if it is closed and open at the same time. Isn't that silly?'' asked the Pig.
Alice agreed that it was very silly. It didn't make much sense for something to be both open and closed. She was still trying to digest what the Pig had told her, that 1/2+1/4+1/8+[1/ 16]+º was 1. She thought maybe the sum would be less than 1 because the terms were so small, but then she thought it would be greater than 1 because there were infinitely many terms. Neither was true; the Pig said that the infinite sum was exactly 1. It sort of made sense. She could see by adding the first few terms together that the sum was close to 1. Adding more terms didn't make too much difference because each term was smaller than the one before it.
The Pig continued, ``Here's another sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, º. It's formed by the numbers 1/n as n goes from 1 to infinity. Now, let's look at the sum of all those terms. How would you write that sum using sigma?''
Alice looked at what the Pig had written before and wrote:
|
``Correctomundo. Now, what do you think this sum is equal to?'' he asked.
``Something not too large, I would guess. The terms are all getting smaller and smaller.'' The Pig didn't say anything. Alice thought about it more. ``Wait, it would have to be larger than 1 because our old sum is contained in this sum.'' Now the Pig nodded.
``Watch this,'' he said, and he proceeded to write out the sum, grouping some of the terms with parentheses.
|
``I've put only 1/2 in the first set of parentheses,'' the Pig explained. ``The next set of parentheses contains the numbers up to 1/4, our next power of 2. What I'm going to do is add up the groups of numbers within the parentheses. Then I will have infinitely many partial sums to add together. Look at the (1/3+1/4) part. Instead of adding those two fractions together, I am going to approximate them with something that I know is less than their sum. Listen carefully: 1/3 is greater than 1/4 and 1/4+1/4 = 1/2, so we know that 1/3+1/4 > 1/2. The next partial sum we have is 1/5+1/6+1/7+1/8. There are four numbers and they are all at least as big as 1/8. That means there sum is greater than 4 ·1/8 or 1/2. We can do the same thing again for 1/9+ º+[1/ 16]. There are eight numbers, each at least [1/ 16] in value for a total that is bigger than 8 ·[1/ 16] or 1/2 again. Each mini-sum grouped by parentheses represents a number that is greater than or equal to 1/2. So the sum of the part that I have written out is larger than 2.''
He continued, ``What is really neat is that there are infinitely many such partial sums. I can always group together subsequences that add up to values of at least 1/2. And since there are infinitely many such subsequences, the total sum will not stop at one number like our last series did. Since these partial sums are not getting smaller, the total sum will always get larger. Mathematicians say that this sequence, known as the harmonic series, diverges. Unlike the sum from Zeno's paradox which converged at 1, this one doesn't converge to any value. When you add more terms to the sum, it will always get larger. So you see, these two sums are fundamentally different.''
Alice was quite impressed with the Pig's little proof. ``So the first sum never actually equals 1, does it?'' she asked.
``That's right,'' confirmed the Pig, ``but it converges to 1. It's like Euler's limit (1+1/n)n thing. It keeps getting closer and closer to e. That's one way we can deal with irrational numbers. They are just limits. We can't write out an exact decimal representation, but we know what the number is approaching. All this talk about limits is making me thirsty,'' he said abruptly. And he picked up his notebook and pencil, and the two went back inside.
Inside, the Pig offered drinks. Alice had grape juice, and the Pig had orange. The Pig picked up his glasses and a deep blue cape which he wrapped over his shoulders. ``Shall we go into the heart of the garden?'' he asked. ``It's a most beautiful place.''
Alice, delighted by the scenery so far, was eager to see the garden. So the two of them set off back down the path toward the garden. On the way, the Pig told Alice of another irrational number. It was, the Pig told her, not a transcendental number, but an algebraic one because it was the solution to a polynomial, not polymer, equation.
The Pig began, ``My most favorite irrational number is often represented by another Greek letter, the letter f (phi). It is also known by a bunch of different names including the golden mean and the golden ratio. It's another of those infinitely many numbers that cannot be expressed as the ratio of two whole numbers, but like ÷2, p, and e, it's another very useful number. The value of f is [(÷5+1)/ 2]. That's the solution to the polynomial equation x2-x-1 = 0. It is approximately equal to 1.61803398875 ... .
``The number f has lots of exciting algebraic properties. For example, I'll bet you would be surprised if you calculated the value of f2 or [1/( f)].'' Alice made a note to try those two calculations sometime. ``The number f also shows up in geometry. Take a look at this star,'' the Pig said, stopping for a moment to draw a five pointed star in his notebook. ``This pentagram was the sign of Pythagorean brotherhood.''
He continued, ``For some reason, this ratio is just a wonderfully pleasing proportion to see, especially in art and architecture. The Greeks used the value of f in designing the Parthenon. The divine ratio shows up in the art of the Renaissance. What I find impressive about f, is how frequently it occurs in art and nature. But,'' said the Pig, ``I won't tell you about that yet. Instead, I will let my garden show you.'' The trees were becoming less dense again, so Alice figured they were near the garden.
``Your garden knows about this number?'' Alice asked. ``How can that be?''
``That's the mystery,'' the Pig said. ``Nature, artists, and mathematics. All are founded on beauty. And f is the most beautiful number there is. Except for maybe 17, of course.''
``Of course,'' agreed Alice, since 17 seemed so important to the Pig.
``We're almost at the golden garden,'' said the Pig. ``But first, I want to tell you about another sequence of numbers, known as the Fibonacci sequence. The numbers in this sequence start with 1 and 1. Successive numbers can be found by adding the previous two numbers. So the next number is 1+1=2. The number after that is 1+2=3.'' He continued generating a list in his notebook:
| |||||||||||||||||||||||||||||||||||||||||||||||||||
He motioned for Alice to sit down on the grass. She did so and he stood next to her. ``So the Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,'' explained the Pig. ``What's the next number?''
``The next number would be the sum of 55 and 89,'' said Alice, ``which is 144.''
``Right. The Fibonacci sequence is a neat sequence. Like our special irrational numbers, it shows up all over the place. As an example, let me explain to you the White Rabbit problem.''
``The White Rabbit problem?'' asked Alice. ``I had a dream about a white rabbit with a problem once. He was always late.''
``Well, this problem doesn't have to do with being late, but it does have to do with time and an awful lot of rabbits. Suppose at the beginning of the year there is 1 white rabbit couple, one boy and one girl. At the end of the January, they give birth to a boy bunny and a girl bunny. There are now 2 rabbit couples. At the end of February, the younger couple isn't old enough to have bunnies yet, but the original pair has another set of twins, another couple. Now there are 3 sets of rabbits. At the end of March, the first couple has two more babies. Additionally, the next couple is now two months old which is old enough for bunny reproduction. So that couple has two bunny babies as well, for a total of 5 couples of March hares. Each rabbit couple gives birth to a boy bunny and a girl bunny every month. At the end of April, there are three sets of rabbits to have bunnies, and they bring three new bunny pairs into the world. Now there are 8 pairs of rabbits. By the end of May, all except the three new pairs of rabbits can have babies and they give birth to one pair each of course. That's five new rabbits so there are 13 rabbit pairs altogether. Things get very hairy very quickly. At the end of June there are 21 pairs of rabbits. How many pairs of rabbits are there at the end of the year if the rabbits keep reproducing in the same way?''
``The Fibonacci sequence grows fairly quickly. What is neat about it is the rate at which it grows. Let's look at fractions formed by successive Fibonacci numbers,'' the Pig said, writing:
|
He gave Alice his calculator. ``Here, compute the decimal values for these fractions.'' She did and wrote them into his notebook so that it read:
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
``I get it,'' said Alice. ``Those fractions are getting closer and closer to each other. They look like they are ... what's that word? ... converging to a number. And they look as though they are converging to your special number f.''
``That they are,'' the Pig said. ``The golden ratio and the Fibonacci numbers are closely tied together. And now that you know that, I think you are ready to fully appreciate my garden. I worked in this garden for many summers as a younger pig,'' he told Alice.
About thirty feet in front of them was a large, well-sculptured hedge which seemed to surround the garden. The Pig led Alice around the side toward a wrought-iron arched gate. The gate had a complicated combination lock on it. ``This is to keep uninvited people out of the garden,'' he explained. ``The combination is 1-1-2-3-5-8. You are welcome any time.''
The gate to the garden opened, and the Pig ushered Alice inside. None of the incredible things Alice had seen and experienced that day compared to the golden garden. ``You should feel very honored that you have been allowed into this garden,'' the Pig told her, and she did feel very special to know that the Pig was sharing something this wonderful with her. The garden was like another world. Alice didn't see anyone else in the garden, though it would not have surprised her if there were several other pigs romping about or perhaps an extended family of bunnies frolicking in some shrubbery. The garden was positively teeming with life. It was sunny, but there was no bright sun overhead. Instead, the light seemed to be coming from within the garden in almost the same sort of way that light reflects off freshly fallen snow. The garden was its own world, and Alice found it hard to remember that there was anything outside the garden. She could feel the energy in the air.
Alice looked up and was astonished by the glorious sight of the sky, which was almost dripping in light. Instead of being the usual blue or gray, it was swirled with tints of colors that were just slightly brighter than pastels. Pinks and purples spun around one another. Blues and greens filled in their gaps, and bits of yellows and oranges shone through, all somehow twinkling as if, instead of having clouds, someone had sprinkled iridescent glitter all over the sky. Parts of the sky seemed to be winking at her, somehow inducting her into this awesome world with a private display of beauty.
``It's - it's wonderful,'' Alice whispered breathlessly. The Pig reached over and held her hand in his hoof.
``Watch as the colors change,'' he said. ``They change slowly enough so that you cannot tell where one color ends and another begins. Yet they fade into one another in swirling spirals that are almost dizzying.'' Sure enough, the colors began to move inward so that what was once blue had become purple and the greens had been replaced by the blue. When Alice thought they could wind themselves no tighter, the colors began to move back, expanding until Alice was certain that the outer pinks had escaped the garden entirely. The sky was like a blazing fire, only much more soothing. The air was cool.
Alice somehow managed to turn her attention away from the mesmerizing sky. In the center of the garden was a circular fountain. Spiraling out around it were dozens of different types of flowers, all growing in perfect health. ``Everything in this garden is beautiful,'' said the Pig. ``This garden has no place for ugly mathematics.
``I'll take you around the outer path, on what I like to call the Fibonacci tour,'' he said. ``We'll start with threes and fives. Lilies, irises, and trilliums are all flowers with three petals,'' he said, pointing them out as they walked around the garden's perimeter. ``Columbines, buttercups, hibiscuses, and larkspurs have five petals on their flowers.'' They stopped in front of a large red and white rosebush. ``Wild roses also have petals in multiples of five. Three and five makes eight, another Fibonacci number. Delphiniums and bloodroot have eight-petaled flowers. Over here we have corn marigolds which have thirteen petals.''
``The Fibonacci numbers run this garden, don't they?'' Alice said questioningly.
``You could say that,'' responded the Pig. ``Or maybe the garden runs the Fibonacci numbers. Personally, I think it more likely that they share a common sense of aesthetics.'' The Pig continued his tour. ``Asters have 21 petal parts. They aren't really petals, you see. And daisies behave as if they know of even larger Fibonacci numbers. Their parts frequently occur in 34's and 55's.''
Alice was completely in awe of the garden. She was impressed by the mathematics that the Pig was sharing with her, but even more so she was overwhelmed by the beautiful flowers. It was an ideal garden for a tea party. Her teddy bear! Why, she had almost forgotten. She wondered where he had gone off to.
``Fibonacci numbers don't stop at the flowers, though,'' continued her guide. ``They apply to all parts of the plant, including stems and leaves.'' He picked off a branch from a small pear tree. ``Look at the bottom-most leaf. The next highest one is not directly above it, but a slight twist away. Then there is another about the same distance up and the same distance around the stem. Let's keep going until we get to a leaf that is in the same position as the first leaf.'' He counted the leaves aloud to Alice. ``One ... two ... three ... four ... five ... six ... seven ... eight.''
``I can't explain it entirely,'' said the Pig. ``It's just one of those mysterious things about nature. The term for the leaf arrangement that we have been studying is the phyllotactic ratio. My guess is that the plant has evolved to make use of the most effective way for its leaves to get sunlight without blocking each other. The plant doesn't actually know about Fibonacci numbers; it is just that having a Fibonacci number of leaves is optimal. We can learn a lot from nature if we study it. We can learn a lot from numbers if we study them too.''
The Yellow Pig led Alice down a path that shone gold from fallen pine needles. The air smelled strongly of pine sap, and Alice caught the occasional whiff of perfume from the surrounding flowers. The Pig picked up a pine cone. ``Pine cones also have Fibonacci numbers nested in their spirals. ``There are two sets of spirals in the pine cones. There are the ones that go out clockwise and the ones that go out counter-clockwise. Both of these have spirals with different, successive Fibonacci numbers. Different pine cones may have different Fibonacci numbers depending on the tightness of the spiral.'' The Pig carefully labeled the pine cone so Alice could see for herself. ``Again, they do that because at the top of the pine cone, their kernels are so tightly packed together. When it unwraps around itself, that's just how it ends up.'' Alice looked at the pine cone.
``Golden mean, golden ratio, golden angle. I see why you call this the golden garden,'' said Alice. ``Everything is golden. It's amazing.''
``Sometimes I sit in this garden for hours working on mathematics or just staring at the flowers,'' the Pig confided. ``I like to sit over there under the golden tree. It gives me inspiration. When I was younger, my friends and I would camp out under the tree, staying awake talking until dawn. I'm just an amateur mathematician, but some of my friends are quite accomplished now.'' He paused. ``Would you like to meet a few of them?'' he asked Alice. ``Two of them live nearby.'' Alice, curious to meet other residents of this magical world, said she would, and hesitantly the Pig and Alice exited from the garden to which Alice knew she would one day return.
Alice and the Pig walked across a small meadow. ``Thank you for showing me the garden,'' Alice said to the Yellow Pig. ``It's one of the most beautiful things I have ever seen.''
``You're welcome,'' said the Pig. ``I'm glad you liked it. My mathematician friends live right over here. They are named Isabel and Gus the Rascal.'' They approached a door to a cabin that looked much like the Pig's from the outside, only it was considerably larger. The Pig knocked twice.
A lamb answered the door. She was wearing two pieces of jewelry around her neck: a cross and a triangle. ``How nice to see you,'' she addressed the Yellow Pig.
``It has been a long time since I have seen you and Gus. And I was just over in the garden, and I thought I would stop by. I have a friend that I would like to introduce to you. Isabel, meet Alice. Alice, this is Isabel.''
``Hello,'' said Alice shyly. Isabel shook her hand warmly. Alice found shaking hands with a lamb funny, but didn't want to laugh. She looked at her jewelry instead.
``I'm afraid Gus is out,'' Isabel told the Pig, ``but he will be back shortly.'' She turned to Alice. ``Would you like to know why I am wearing a triangle?'' Alice nodded. ``Let's go into the living room where we can sit down. Would either of you care for drinks?'' The Pig asked for two glasses of water. Isabel disappeared momentarily into the kitchen and returned with them.
``Isabel, can I look through your books?'' asked the Pig.
``Certainly,'' said Isabel. ``You know where my study