\chapter{The Gallery} \section{Stepping into a Picture} The Pig stood on the tip of his toes to reach the doorknob. The heavy door opened slowly into the room. Inside Alice saw dozens of paintings. There might have been hundreds, she supposed, because the walls spiraled inward toward the center. The room seemed to be a maze of art. The Yellow Pig continued walking, but a painting caught Alice's attention. She stopped to look at it. It was a painting of what looked like a path that had been looped to form a figure-eight. Alice saw what looked like a small marching band of piglets posed in positions almost evenly spaced on the path. As Alice studied them longer, they began to move. Or at least that's how it appeared; she knew that her eyes must be playing tricks on her. The piglets were marching in single file around the loop. Suddenly the pig in front went around a bend and was walking upside down. He didn't seem at all disturbed by his bizarre change of position. Alice was afraid that he was going to fall off, but he was able to hold on to the path. ``He must be wearing special shoes,'' thought Alice. She wished she had such shoes that would enable her to walk upside down. The upside down piglet kept walking. The piglet behind him went around the bend and also turned upside down. So did the one after him. Soon more than half of the pigs were on the inside part of their path. Then, just when Alice thought they would all end up facing the wrong way, the piglet in the lead somehow became right-side up. The other piglets followed him. One of the piglets jumped off the path for a moment and dipped his left paws into a tray of bright blue paint. He dipped his right hooves into a tray of sunny yellow paint. Then he hopped back on. Now Alice could follow him more closely because as he walked he left colored foot prints behind him. Blue on the left and yellow on the right. But when he got about halfway around the loop, Alice noticed something interesting. The foot prints had switched places. The yellow foot prints were now on the left and the blue ones on the right. The piglet kept walking. He soon got back to the place where he had started leaving foot prints, and sure enough, the colors of the foot prints really were reversed! This confused Alice for a moment until she realized that it made perfect sense. The piglet was walking upside down. It was on the other side of the path from where it had been before. She had no idea how it had gotten there because she was sure she hadn't taken her eyes off the piglet. But since the piglet was upside down, its hoof prints had reversed. Another piglet walked on top of that piglet, and Alice saw that its left paw lined up with the upside down piglet's right paw. They were on opposite sides of the same path. ``How will the poor reversed piglet ever get back to the normal side of the path?'' Alice worried. But she didn't have to worry for long because just as the piglet had turned upside down without Alice seeing it, it turned itself right side up again. It traced over its original foot prints. Alice resolved to watch it more closely this time around to see if the piglet would somehow slip off again. This time Alice saw it walk around a twist in the path, and then it was upside down. She began to trace the wet paint marks. She walked around the painting, staring intensely at the hoof prints and trying to get a better angle from which to observe the piglet's path. At that moment, it occurred to her exactly how odd it was that the piglets in the painting were moving. She reached out to touch the painting so she could trace the piglet's path with her fingertip. Only instead of stopping at the canvas, her hand went right through the painting. Alice was beginning to feel rather queasy again. ``If my hand can go through the painting,'' she thought, ``maybe my whole body can, too.'' And though the idea disturbed her, she was so curious to learn how the piglet was able to turn upside down, that she took a step closer to the painting, closed her eyes, and leapt into the picture. Alice found herself standing on the middle of the path. She looked around, trying to familiarize herself with her surroundings. ``Hurry up,'' an impatient voice behind her said, ``you're holding up traffic.'' ``Oh, sorry,'' Alice apologized. The pig gave her a push and she began moving in a daze. She didn't even feel like she was lifting her feet, but something was propelling her forward nonetheless. ``Whoa,'' she cried out as the path turned over. She felt herself falling. ``Just tell yourself it's impossible to fall,'' the piglet helpfully recommended from behind. ``If you believe you can't fall, you won't. It's that simple.'' Alice didn't think that was very simple at all. In fact, she found it most confusing. But she didn't have any better ideas, and she had to do something to keep from falling, so she muttered to herself, ``I cannot fall I cannot fall,'' over and over again. It worked until she realized that she wasn't falling, at which point her surprise made her stop concentrating. She fell to the floor with a loud thud. She lay there feeling miserable, although not at all physically hurt. The piglet who had given her the advice passed her. ``Try it again,'' he said. ``You almost had it. Just don't start questioning how you are able to stay on the strip and you'll do fine.'' He helped Alice to her feet and led her back on the path. Alice walked bravely around the loop, hesitating only slightly when she got to the top part that she recognized as the bend where she had turned upside down before. ``I cannot fall I cannot fall \ldots .'' This time she didn't fall. She continued walking. ``Now you are at the point where you started,'' said the piglet. Alice felt faint again. ``But I can't be,'' she said indignantly. ``I was up there before and now I'm somehow the other way. It's not the same place at all.'' ``Ah, but it is,'' replied the piglet. ``The location is exactly the same. It is only you, or rather your perspective, that has changed.'' Alice was even more confused now than she had been watching the painting from outside. ``I'm upside down now.'' ``Yes, you could say that,'' remarked the piglet cryptically. ``But that's only because that's how you think of it. I prefer to think of myself as always being right side up. It's much less confusing that way.'' ``I can't be right side up, though,'' Alice said. ``I'm on the other side of the path.'' ``What other side?'' inquired the piglet. ``Do you remember changing sides? Did you jump off the path and then get back on again? No, of course not. You are a most confused girl. You see, there is only one side of this path.'' Alice was concentrating on what the piglet was saying so hard that she had managed to walk around upside down without falling off the path. She came to another twist and was right-side up again. She let out a sigh of relief. Everything made slightly more sense this way. ``Perhaps now,'' said the piglet, ``you feel more like you are back where you started. It has taken you not one, but two, revolutions around the strip, but you are now exactly as you started your journey. It took you two revolutions because the strip is one sided,'' repeated the pig. ``One sided?'' asked Alice. ``What do you mean by that? I agree that I have only walked on one side of the path, but surely there must be another side. Why, there's the side of the tracks that I haven't walked on. That's the other side.'' ``It would be,'' the piglet said, ``but there is no part of this path, as you call it, that you have not walked on. You have walked over each part of the path twice. That's because the path is not your usual surface. It is a very special surface known as the Moebius strip.'' ``Moebius strip,'' Alice repeated to herself, making a note to ask the Yellow Pig about them. Surely he would know what nonsense the little pig was babbling. ``A Moebius strip is a band with a twist in it. Think of a long thin strip of paper. Now connect the two ends together. Only before you do so, give it a half-twist. Now you are lining up the front side of the paper with the back side of the paper. The front merges into the back, and there is no distinction. The surface you have created is single sided.'' Alice was walking upside down again. The piglet's words seemed further and further away. Alice fell off the Moebius strip and back through the canvas. She landed in the art gallery. The painting was right in front of her, but the piglets had stopped moving. The Yellow Pig was behind her. ``There you are,'' he exclaimed. ``I was so worried that you had gotten lost. Where were you?'' ``I was \ldots .'' Alice pointed, ``I was in there. I don't know if I can explain it.'' ``Try,'' said the Pig, and so Alice explained as best as she could that she had fallen into the painting and had followed the piglets in a right side up and upside down promenade. The Pig nodded. When she was finished talking, the Pig pulled out his notebook again. This time, he ripped out a sheet of paper. ``I'm going to make a Moebius strip for you,'' he said. He tore off a thin strip of paper. He connected the two ends together to make a normal band. ``Instead of doing this,'' said the Pig, ``I'm going to put a twist in it first. Hand me a piece of tape, please.'' Alice fumbled about awkwardly. ``I don't have any tape,'' she informed the Pig. ``Of course you do,'' he replied. ``You should always carry tape when you travel.'' Much to her amazement, he produced a piece of tape from behind Alice's ear. ``How did that get there?'' she asked. ``It was there,'' said the Pig. ``Don't you know what you keep behind your ears?'' Alice was speechless, which wasn't a bad thing because the Yellow Pig wasn't waiting for her to say anything. He taped the two opposite ends of the paper together. ``There. That's a Moebius strip.'' He handed his creation to Alice for inspection. \centerline{\epsfbox{images/4-moebius.ps}} ``The Moebius strip differs from a regular band in that it has only one side. Watch,'' he said. The Pig took out a pencil, this time from behind his own ear, and proceeded to trace a path on the paper. He drew a line down the center of the surface of the band. He had to rotate the band several times in order to continue the line. It took longer than Alice expected. Finally the two ends of the line met. ``I've traced out the path,'' he said. ``Now I'll prove to you that the Moebius strip really is one-sided.'' He cautiously removed the tape with his hooves. He laid the strip of paper down flat. A thin line ran down its center. ``That's one side of our original surface. If I turn it over, we'll see the other side.'' He turned over the paper to reveal the other side which had an identical line in the middle. ``How did that get there?'' he asked. ``I only traced a line on one side of our Moebius strip. Well, it's simple. The Moebius strip only has one side. That's what the twist does to it.'' Alice had trouble believing all of this, but she could see no flaw in the Pig's logic. ``Can I try it myself?'' she asked. ``Of course,'' replied the Pig, handing her the rest of the paper, scissors, a roll of tape, and a pen. ``I'm going to look at that painting for a moment,'' he said pointing. ``It's one of Escher's `Circle Limits'. Escher created so much wonderful art.'' He headed in the direction of the painting, leaving Alice contentedly cutting another strip of paper. Alice lightly drew a line along one side of the paper before twisting it. She taped the two ends together, just as she had seen the Pig do. The side with the line was now connected to the side without a line. ``Aha,'' she murmured. Then, instead of drawing a line, she wrote an R on the right side of the strip and an L on the left. She continued labeling all the way around the strip. ``Those are the foot prints,'' she explained to herself. Then she held the Moebius strip up and turned it around slowly. On the reverse of every L was an R and on the reverse of every R was an L. The foot prints hadn't been reversing themselves in the painting after all. Alice had just been seeing the piglet cross over each part of the strip twice because of the twist. It was still confusing, but it certainly wasn't the most confusing thing she had seen that day. ``I think I get it now,'' she called out to the Yellow Pig. \centerline{\epsfbox{images/4-moebiuslr.ps}} He returned from the painting, still mesmerized. After a moment the glazed look disappeared from his eyes, and he spoke. ``The Moebius strip has a lot of special properties. That twist makes things even more complicated than you might imagine. What do you think will happen if I cut this strip down the middle?'' he asked. ``You'll get two strips,'' said Alice. ``Nope,'' said the Pig gently. ``If you cut a normal band, you would get two thinner bands. But this one has a twist, so my cut can't separate it. Try it.'' Alice cut the Moebius strip along the center. The paper fell not into two strips but into one longer band. It was half the thickness and twice the length as the original strip. ``Is that a one-sided band or a two-sided band?'' asked the Pig. Alice drew a line down the center. It looped around the band twice before the ends connected. The pen crossed over two pieces of tape twice. ``It's two-sided,'' she said. ``It has two twists in it, but otherwise it is just like a normal ring.'' ``Now, what do you think will happen if you cut it in half again?'' asked the Pig. ``The same thing as last time,'' conjectured Alice. ``I'll get an even longer and more twisted strip.'' This time the Pig didn't have to prompt her to try. She was already eagerly cutting the strip in half. The strip split into two linked rings. Neither of them, she verified, were Moebius strips. ``What happens if we start with a strip with three twists in it?'' asked the Pig. Alice wasn't sure what to think. ``Well,'' continued the Pig, ``it will be one piece because it has that odd number of twists. The opposite ends line up so we can't separate them with a cut.'' Since Alice still didn't quite know what would happen, she cut another strip of paper and taped it together. When she cut it, a tangled mess resulted. ``I can't even tell if that's one piece or two,'' she said, discouraged. ``You could trace it,'' suggested the Pig. Alice did. She drew a line down the center of the band. It looped around the whole paper, but only on one side. ``It's one band,'' said Alice, ``wrapped around itself with two sides. I wonder what will happen if I cut it again. Since it has two sides,'' she conjectured, ``it will still have two sides when I cut it again.'' It did. Cutting this strip in half resulted in two very twisted and linked double sided rings. ``What happens if we take our original Moebius strip with one twist and cut it in thirds?'' pushed the Yellow Pig, already making a model to cut. The result was one original length Moebius strip linked to one longer doubly twisted normal ring. ``Wow,'' said Alice. ``That's all incredible. We're able to make so many different kinds of loops out of one original loop. The Moebius strip is very special.'' ``It is,'' agreed the Pig. ``That's something mathematicians have known for awhile. It's part of a branch of mathematics known as topology. Topologists study all sorts of surfaces. Like the shape of a donut or bagel, which they call a torus.'' ``I like donuts and bagels. Once when I came home from school I wanted a poppy seed bagel, but my sister Lorina had eaten it,'' interjected Alice. ``Bagels have holes in them.'' ``Yes, they do, and that's what mathematicians like about them. The hole makes a donut fundamentally different from a sphere. Topology is about different surfaces. Two surfaces are different if one can't be easily transformed into the other. In topology a lot of surfaces are considered the same, or equivalent, because they are just contortions of each other. Think about a balloon; balloons are almost perfect spheres. You can deform a balloon by poking at it, and it's still roughly a sphere. And no matter what you do to that balloon, you can't make it look like a donut without cutting it. Topologists don't allow cuts in transformations. A donut cannot be the same as a sphere because it has a hole. Similarly, a donut with two holes is different from a donut with one hole.'' The Pig continued, ``You may think of a sphere as being a three-dimensional solid, but at least in this context, a sphere is just a surface like a plane. At any given point on a sphere, it's like being on a plane. Our world is spherical and yet it appears flat. A two-dimensional map can represent the globe, with some distortion of size. Topologists don't care about the difference in size. They are just concerned that there is a way to relate every part of the globe to every part of the map. And this can be done because a sphere is just a two-dimensional surface. Even though it is situated in three dimensions, it is only two-dimensional, just like the surface of a piece of paper.'' ``So spheres are just surfaces?'' asked Alice. ``What about donuts?'' ``To a topologist, yes. The term {\it torus} just refers to the surface of the donut. Topologists get their coffee cups and donuts confused. A coffee cup has one hole for the handle. Donuts have one hole in the center. Even though coffee cups and donuts look like very different things to us, topologists think of them as being the same. They say their surfaces are topologically equivalent. They are both tori, which is the plural of torus.'' Alice's mind was racing. ``If you can make a flat map to represent a sphere, can you make a map of a torus?'' she asked. ``You can,'' said the Pig. ``In fact, there's a popular game here based on such a map. It's called Pac-ham. In Pac-ham, you are a character that collects dots and avoids enemies. It takes place on a two-dimensional surface. But unlike a normal playing board where you can be trapped in corners, there is no definite top or bottom, nor left and right. Instead, the surface is a continuous loop. You can walk off the top and come back from the bottom. In the same way, the left and the right sides are connected. The surface is a loop. And that's what a torus is. Think of a piece of rubber. Now glue two opposite sides together to make a tube like a garden hose. Then attach the two circular openings together to make a closed ring. The outer surface is a torus.'' ``Spheres and tori are known as orientable surfaces because they preserve orientation. There are also non-orientable surfaces which are surfaces with a twist. On a non-orientable surface, left becomes right and clockwise becomes counter-clockwise. The Moebius strip like that. The piglets you saw in the painting appeared to reverse themselves. But a Moebius strip is just a path like a line. There are other surfaces like the Moebius strip. One is known as the Klein bottle. It's very hard to conceptualize a Klein bottle. Most people can't picture it at all.'' ``But Escher is one of those people who can. He's not a mathematician, but he has an incredible mathematical intuition. You can see mathematics at the center of his work. A lot of his art makes use of tessellations and those wallpaper symmetries. He has used all seventeen patterns in his works. The painting I stopped to look at before is a different kind of tessellation that makes use of hyperbolic geometry. In hyperbolic geometry parallel lines intersect each other. The sum of the angles in a triangle is less than $180^{\circ}$.'' ``Stop!'' cried Alice suddenly. ``You're confusing me horribly.'' ``Oh,'' the Yellow Pig stopped. ``I'm sorry,'' he apologized. ``We're here to see art, not talk about math. Let me show you a painting by another artist.'' And so Alice and the Pig left the Escher sketches and headed to the next room. \section{On Conics} The next room was considerably smaller than the last room had been. The room itself was an artistic endeavor. The walls of the oddly shaped room were covered with frescos. The ceiling was painted with a mural of pigs. The windows were stained glass. ``This room is such a funny shape,'' remarked Alice. The room was, in fact, shaped something like this: \centerline{\epsfbox{images/4-ellipse.ps}} ``That's so it has better acoustical properties. Our words bounce off of the walls and resonate. That's what causes an echo. And waves reflect off the walls at the same angle they hit it. Since the walls are curved like that, they reflect at many different angles.'' ``Oh,'' said Alice, listening to her ``oh'' echo over and over again, each time slightly fainter than the time before. The Pig continued, ``Music halls often have curved walls and domes and walls with rough surfaces. All of those intricacies are to play with the way sound reflects.'' He began to sing: \vbox{ \ssp \begin{center} {\it Oh beautiful for number fields, \\ For Dedekind domains, \\ For analytic function rings \\ Above the complex plane!} \end{center} \dsp } Alice clapped at the end of his performance. ``Now,'' said the Pig. ``Here's something really neat about this shape. You stand over there,'' he said, pointing to a small spot that was clearly marked on the floor. ``And I'll stand over here.'' This second spot he indicated was identical to the first but all the way across the room. ``Then we are going to whisper to each other and because of the shape of the room, we will be able to hear each other perfectly.'' They took their places facing away from each other, and the Pig whispered ``Ellipse.'' ``I can hear you,'' whispered back Alice loudly. ``No need to be so loud,'' whispered the Pig. ``I can hear you perfectly.'' ``But how is that?'' Alice asked in disbelief. ``You are all the way across the room.'' Alice was beginning to think that anything was possible, but she also knew that the Pig must have a good mathematical explanation. The Pig explained in a whisper, ``The sound is not traveling in quite the way you would expect it to. Sound travels in waves, and those waves travel in straight lines. But when you speak there is not just one wave of sound. You are emitting sound waves in all directions.'' ``But aren't most of my sound waves in front of me? I'm not talking out of the back of my head, am I?'' Alice was concerned for a moment that maybe she was, but that didn't seem right at all. ``There's a wall in front of me. How can you hear me when I'm talking into a wall?'' ``Ah,'' said the Pig, ``think about what I have said about reflection.'' ``You said that sound waves bounce off walls and that they reflect at the same angle as they hit,'' she repeated. ``Exactly correct,'' said the Pig. ``If you talk at a straight wall, the sound will bounce right back to you. If there is a wall at a $45^{\circ}$ angle, sound will travel at a $90^{\circ}$ angle, exactly perpendicular to you.'' \centerline{\epsfbox{images/4-bounce.ps}} ``It's like playing billiards,'' Alice said. ``You hit the ball at different angles because you are controlling the angle you want the ball to go.'' ``I never could play pool,'' said the Pig wistfully, ``my hooves are too short. I can't play ping-pong either.'' ``I'm sorry,'' Alice sympathized. ``But yes,'' continued the Pig, ``the way sound travels is an awful lot like how a billiard ball travels. Only a billiard ball is confined to just the surface of the table, or at least it should be. Sound travels in three dimensions. Maybe more.'' Alice thought for a moment. ``So when I speak, my words travel in sound waves in all directions from me. Then most of them hit this wall and bounce off. They must reflect somewhere behind me, and that's how you hear them?'' ``Almost,'' said the Pig. ``Think about a sound wave, or a line, directly in front of you. It hits the wall and comes straight back down the line marking a diameter of the room.'' The Yellow Pig walked over to Alice, stopping to pick up some billiard balls. ``What are those for?'' asked Alice. ``They are for you to find out where the other sound waves go,'' the Pig said. ``Oh,'' Alice said. The Pig handed the balls to Alice. ``Roll the first one just a little bit to your right.'' Alice did as she was instructed. The ball hit the wall and bounced further away from her. She turned around and watched it roll across the floor. It passed very close to the point where the Pig had been standing before. ``Look,'' squealed the Pig. ``That's the other marked spot.'' ``The Pig planned that one,'' Alice thought. ``I'll try throwing another ball in a different way.'' She grabbed another ball from the pile and aimed for a point well to her left. She watched the ball eagerly. It stopped near the same spot on the opposite side of the room. She tried again with a third ball. Like the previous two, it steered itself across the magical point. She tried a fourth ball and a fifth ball. And a sixth. Then the Pig collected the balls. He brought them to the point where he had been whispering to Alice. He gently rolled the balls at different angles toward the wall closest to him. All six balls bounced off the wall and aimed toward Alice. Fortunately, they lost momentum and stopped rolling before they assaulted her from every direction. Alice was surrounded by a semi-circle of balls. ``How did you do that?'' asked Alice. ``Easy,'' said the Pig. ``I picked those two points very carefully. If I try throwing the balls from anywhere else in the room, they won't meet at the same location. The shape of the room is a very specific one. It's called an ellipse.'' ``E-lips,'' Alice repeated. ``The two points that I chose are very special. They are known as the foci of the ellipse. Ellipses are everywhere. Planet orbits are elliptical in shape. Circles are one type of ellipse. But whereas circles have the same diameter everywhere, an ellipse varies in diameter. The largest diameter is called the major axis, and the shortest is the minor axis. From these two distances we can compute the area of the room. We have to use $\pi$. There are equations that describe ellipses as well,'' said the Pig. ``Ellipses are a type of quadratic, or second degree polynomial equation.'' ``But what makes them so special? And how do I make an ellipse without having to graph some complicated equation?'' asked Alice, whose head hurt whenever she thought about the quadratic formula. ``Well,'' said the Pig, ``I could tell you. Or you could figure it out for yourself, with a little bit of help. I happen to have a stencil of an ellipse with me. I'll trace you an ellipse on paper.'' He produced a stencil, as if from thin air, and drew an ellipse on another page in his notebook. He also pulled out a ruler from behind his ear. He marked two points inside the ellipse and two more on its perimeter. ``The two points on the inside are the foci,'' he said. \centerline{\epsfbox{images/4-ellipsef.ps}} ``That's like where we were standing,'' interjected Alice. ``Right,'' the Yellow Pig said. ``I want you to measure the distance from each of the outer points to both foci.'' This sounded slightly dull to Alice, but she knew the Pig must have some surprise up his sleeve, so she took the notebook from the Pig and began to measure with his punit ruler. She started with a point at the top of the paper. The distance to the further inner point was 12 punits. The distance to the closer inner point was 5 punits. She measured the distances to the next point. They were 10 and 7 punits respectively. Alice wrote down the results and stared at them for a moment. ``Hey,'' she exclaimed. ``Both pairs add up to 17! In the first one, 12 and 5 is 17, and in the second, 10 and 7 is 17. I'll bet if I pick another point on the ellipse and measure the distances, they will add up to 17 as well.'' The Pig nodded. Alice grabbed the pencil and added another point to the picture. Sure enough, when she measured the distances and added them up, the sum was 17. ``The 17 part isn't really important,'' said the Pig. ``I just happened to have an ellipse where that constant sum was 17. You can make ellipses for other numbers as well. By moving the foci further apart or closer together, you can create an ellipse that is longer and thinner or one that is more circular.'' ``I want to make an ellipse,'' said Alice. ``One that adds up to 20 and is thinner.'' ``No problem,'' the Pig said. ``All you need is a piece of paper, two push pins, a pencil, and some string.'' These he supplied. He instructed Alice to cut the string to 20 punits. She did. ``Now, pick two points to be your foci. You want them to be somewhat far apart to make a thin ellipse. Put your push pins at those points.'' Alice did as the Pig instructed. ``Tie the ends of the string to the thumbtacks.'' This took Alice a little while to do because the knots kept slipping. ``Here's where you trace out the ellipse. The string has a length of 20 punits. Your ellipse is the set of all points that are a certain distance from the two foci. So just hold the string taut with your pencil.'' Alice put the pencil on one side of the string and pulled so that the string was tight. ``There, that's one point on your ellipse. Move the pencil to another point so the string is still stretched as far as it can.'' Alice did this. ``That's another point on your ellipse,'' said the Pig. \centerline{\epsfbox{images/4-ellipsepins.ps}} ``I get it,'' said Alice. ``All of those points will be points of my ellipse. I can just put the pencil point down on the paper and trace out a path. If I make sure the string is tight, I'll be drawing an ellipse.'' ``Exactamundo,'' said the Pig. Alice traced an ellipse in the Pig's notebook. ``An ellipse is a conic section. Think about two ice cream cones.'' ``I like ice cream cones,'' said Alice. ``And I like ice cream and chocolate syrup.'' ``Me, too,'' agreed the Pig. ``Ice cream is yummy, and we are going to make all sorts of neat shapes out of our imaginary ice cream cones. Take two cones. Put the first one on a table with the circular opening on the bottom so it can sit there easily.'' ``Like a hat,'' interjected Alice. ``Yes. Now put the other cone on top of that, only this time facing the way you would hold an ice cream cone to eat out of it. So the two pointy ends of the cones are touching. The shape of the two cones combined is a double cone. It has a circle on the top and then it gets thinner and thinner until it is a point in the middle. Then it widens out again. Now instead of having an ice cream cone, think about something like a block of clay filling in the cones. You want to imagine something solid, but something that can be cut easily. \centerline{\epsfbox{images/4-cone.ps}} ``Have you ever cut cheese?'' he asked. ``I have,'' said Alice, wondering where the Pig was going with this conversation about dairy products. ``Well, normally you try to cut in straight slices, perpendicular to a cutting board or another surface. But suppose you wanted to cut thin slices off of the top instead. That's what we are going to do. We're going to cut our cone shape into slices or sections. If we cut our cheese cone with slices exactly parallel to the cutting board, we'll get a bunch of circles of different sizes. But it's not very likely that we will cut exactly parallel to the cutting board. I certainly can't. I'm not very good at cutting cheese even when I'm not trying to make special shapes.'' Alice tried to picture the Pig cutting cheese and could see where he might have some difficulties. ``So I end up cutting the cheese with slices that are just at a slight angle to the cutting board. Can you see what shape results?'' ``It's kind of like a circle,'' said Alice. ``If you are only cutting at a slight angle, you'll end up with what is almost a circle.'' ``Right,'' said the Pig, ``I'll get a sort of elongated circle. A shape that is rounded but not all points on its perimeter will be the same distance from its center. And that shape is our ellipse. That's why an ellipse is called a conic section. Because it is a section of a cone.'' ``That makes sense,'' Alice said. ``But what happens if you cut the block of cheese at a really sharp angle, like almost entirely straight up and down?'' ``Excellent question,'' said the Pig. ``What happens if we take a cross section of the double cone at a different angle, like one perpendicular, at a right angle, to the table?'' The Pig repeated the problem and paused to let Alice think about it. ``It will cut the right side up cone and the upside down cone. I'll have cut two pieces of cheese.'' ``Right. The cut will start going through the wider top of the cone. Then as the cone gets thinner, the cut will no longer be hitting the cone. You'll be cutting through air. Then the knife will intersect the cone again on the bottom half, creating two separate pieces. If you look at the edge that is left on the main block of cheese, you'll see another conic section known as a hyperbola. \centerline{\epsfbox{images/4-hyperbola.ps}} ``That's spiffy. Can we make any other shapes from our cone?'' Alice inquired. ``Yes, as a matter of fact, there is one more shape we can make. Hyperbolas and ellipses are created at a large range of angles. Circles are created only at one very precise angle. What other angle could be important in the double cone?'' Alice thought about this. She was puzzled. ``It's not the vertical cut,'' she thought to herself, ``because we already said that made a hyberbola. What other lines are there?'' She thought some more. ``The diagonal of the cone itself?'' she asked finally. ``Yup,'' said the Pig. ``If you cut a piece of cheese parallel to a side of the cone, you will only cut cheese from one part of the cone. Since you won't have two pieces, it's not a hyperbola. And it won't be an ellipse because the shape won't have the same roundness to the end. The shape created is known as a parabola. \centerline{\epsfbox{images/4-parabola.ps}} ``So those are our four conic sections?'' asked Alice. ``Circle, ellipse, hyberbola, and parabola?'' \centerline{\epsfbox{images/4-conics.ps}} ``Hyperbola,'' corrected the Pig. ``There are actually three more simple cases, but those are the four main conic sections. They are the different shapes you can make by cutting a cone at different angles. And now you see how the shape of this room is related to a wedge of cheese.'' ``Ellipses,'' said Alice. ``This room is an ellipse, just like the section on the end of a cut of cheese. Neat.'' ``Conic sections have neat properties. In our elliptical room, we can whisper to each other because the sound waves focus between two points. In a parabolic room, sound would be directed out in parallel waves. Hyperbolas are used in satellite dishes and in lithotripters to collect and direct waves at kidney stones. A lot of people, including my friend Isabel, have studied conic sections.'' He continued, ``We used to come down to this gallery together a lot before she got so busy with math and family. This was a place for us to take a break from math problems.'' He laughed. ``And here I am talking math. Let me stop for a little while and show you another painting.'' \section{And Another Picture} The Yellow Pig led Alice to the far end of the room. They stopped briefly to admire the roof, which was, as the Pig said, in the shape of a geodesic dome. In the back of the room was only an easel, covered by a thin sheet of a velvety material. ``This is a painting called `Sheep Fiction' by Sal V'doordolly,'' said the Pig, removing the cloth to unveil the painting. Alice studied the painting. One bottom corner of the painting appeared to be a series of sketches. There were several flat outlines of sheep followed by many more sheep that had been shaded so as to look three-dimensional. There were sheep that looked like they were leaping out of the picture, sheep that looked like they were falling into the picture, and even a two-dimensional sheep that was eating a flower. In the other bottom corner there were a bunch of oval-like shapes. These started out small on the left, got larger, and then receded in size again almost symmetrically. There was a sheep that looked like it had been flattened. ``Poor sheep,'' thought Alice to herself. There was a sheep that had its insides drawn. There was a sheep-like outline in dots. In the very center of the painting were two flat sheep, like cut-outs or sheep stencils. They were facing each other. On either side of them were two forward facing sheep. One sheep was black on the left side and white on the right side. The other was black on the right side and white on the left side. The top corners contained sketches of squares and cubes in such a regular way that Alice knew there was a pattern. At the top center was a large shape that looked much like a cross made out of alphabet blocks. The blocks seemed to almost pass through each other. Directly to the left of the cross was a small rectangle, like a door. To the right was a more stylized and block-like door. ``What's with the cross?'' Alice inquired of the Pig. ``Oh that,'' said the Pig. ``It's not really a cross. Well, it is, but that's just because it is unfolded.'' ``Unfolded?'' repeated a puzzled Alice. ``What do you mean by that?'' ``Well,'' started the Pig, ``it's like this. Think of a regular cross. He drew a series of squares in his notebook that took the shape of a cross. ``How many squares are there?'' \centerline{\epsfbox{images/4-cubenet.ps}} Alice counted the squares. There were four squares in a vertical line. Two other squares had been drawn on the left and the right of the second square down. ``Six,'' she informed the Pig. ``Good,'' said the Pig. ``How many sides, or rather faces, does a cube have?'' ``Six,'' said Alice after a pause, ``like dice.'' ``Right,'' said the Pig. ``I can fold up a cube from my net of six squares.'' Alice watched him cut out the cross and fold it into a cube. She helped him tape the edges together. ``Neat,'' said Alice. ``Can you make other shapes as well?'' ``I can,'' the Pig replied. ``What do you think this will make?'' he said, pointing at his notebook. He had drawn four triangles in which all of the sides had the same length. One triangle was in the center and the other three were attached to its three sides. \centerline{\epsfbox{images/4-tetranet.ps}} Alice stared at it. She could see the three outside triangles folding up to meet at a point. ``It's like a pyramid,'' she said finally. ``Yup,'' agreed the Pig. ``It's a pyramid with a triangular base. The other common kind of pyramid --- the Egyptian kind --- has five sides, one of which is a square base. In our pyramid, though, all of the sides are triangles and the same. Mathematicians call it a tetrahedron. The `tetra' part refers to the fact that it is made up of four faces. The cube is occasionally called a hexahedron because it has six faces. ``Two-dimensional shapes are much easier to understand. A hexagon has six sides, or edges. A hexagon also has six corners, or vertices. A three-dimensional solid has many faces in addition to edges and vertices. The cube has six square faces and the tetrahedron has four triangular faces. But how many vertices does a tetrahedron have?'' Alice folded the Pig's triangle to make a tetrahedron. She taped the edges together. Then she put it down on the floor so a point was facing up and a triangular face was on the ground. There were three points as the corners of that triangle and the one point on top. Alice turned the tetrahedron around slowly to make sure she wasn't missing any other vertices. She wasn't. ``A tetrahedron has four vertices,'' she reported. ``And how many edges?'' asked the Pig. The edges were a bit harder to count. Alice counted twice to make sure she had the right number. There were three edges on the triangle on the floor, and then there were three more coming up to the top point. ``Six,'' she told the Pig. The Pig thought that was right, but he had to count for himself to make sure. He kept losing count. He got another sheet of paper and drew his four triangles again. ``It's sometimes easier for me to see things in two dimensions,'' he explained. This time, instead of having his triangles connected to each other, he left a little space in between. ``Now,'' he explained, ``I have four triangles. Each triangle has three edges. That's three times four or twelve sides. When I fold up the triangles, two triangles touch along any edge. In other words, adjacent triangles share edges. So even though I have twelve edges or sides when the tetrahedron is flattened, when I fold it up, half of those edges disappear. That's half of twelve or six edges.'' This seemed to convince him that Alice had counted correctly. Alice thought he had a neat way of looking at something three-dimensional. ``A tetrahedron has four vertices, six edges, and four faces,'' he concluded. ``The faces are triangles. Three triangles meet at every vertex. `Vertex' is the singular form of `vertices','' he added. ``What about cubes?'' ``A cube has six square faces,'' said Alice. ``I guess you want to know how many vertices and edges a cube has too, don't you?'' ``I do,'' said the Pig. ``That is, if you don't mind counting.'' ``Not at all,'' said Alice, who was glad to be able to assist her guide, who seemed to have a much harder time counting the sides of shapes than she did. Counting the corners was easy. Again, she put the shape down on the floor. The four corners of one square lined up on the floor. Four corners of another square were at the top of the cube. Alice labeled them all so the Pig could see them more clearly. ``There are eight vertices.'' The Pig accepted this. ``How many squares come together at any point?'' Alice looked at the cube again. ``Three,'' she said, ``just like the last time.'' The Yellow Pig nodded. ``How many edges are there?'' ``That's easy too,'' said Alice. ``There are four edges bordering the top square. And four edges bordering the second square. And then there are four edges connecting the two squares. So there are twelve edges total.'' The Pig thought about this and tried to draw it out as well. ``There are three edges per vertex,'' he mumbled to himself, ``and eight vertices in all, and each edge is shared by two faces. Three times eight divided by two is twelve.'' ``So,'' said Alice, enjoying her role as teacher, ``a cube has eight vertices, twelve edges, and six faces. What other shapes are there?'' ``Not that many regular ones,'' said the Pig. ``A {\it regular} polyhedron is a three-dimensional solid in which all of the edges, faces, and angles between faces are the same. It's not like in two dimensions where there are regular triangles and squares and pentagons and hexagons and even 17-gons. In three dimensions, you can't just start out with the same shapes and angles and expect them fold together. In fact, there are only five such regular polyhedra; they are sometimes referred to as the Platonic solids because Plato knew of all of them.'' ``Those Greeks sure knew a lot about geometry,'' said Alice. ``They did,'' said the Pig. ``They were fascinated by shapes. And for good reason. Shapes are very interesting.'' ``So what are the other three solids besides the tetrahedron and the cube?'' Alice asked. ``They are a bit more complicated. The next shape is the octahedron. It has eight faces. These faces are triangular. Four of them meet at any point. Another shape is the dodecahedron. `Dodeca' means two and ten. Dodecahedra have twelve faces in the shape of pentagons. Three of them meet at any point. The final shape is the icosahedron. `Icosa' means twenty, so these solids have twenty faces. They are triangles, and five of them meet at any point.'' As he spoke, he sketched the polyhedra in order from least to most faces. \centerline{\epsfbox{images/4-polyhedra.ps}} ``There are no other regular polyhedra. You can make other solids, but you can't make any other solids where the faces are all identical regular polygons and the solid angles between them are also the same. That's what it means to be a regular polyhedron.'' ``How many vertices and edges do those shapes have?'' asked Alice. ``Well,'' said the Pig, ``You could count the vertices and edges, but I'll just tell you.'' He thought for a few moments. ``The octahedron has six vertices. It's like two square pyramids with their squares glued together. And it has twelve edges. Four from each of the pyramids and four where the pyramids' bases connect. The icosahedron has 20 faces and 12 vertices and 30 edges. The dodecahedron is in some ways the reverse of that. It has 20 vertices and 30 edges to go with its 12 faces.'' ``That's an awful lot of edges,'' said Alice. ``It is,'' said the Pig. ``Which is why it's surprising that a number as small and simple as two falls out of all of those numbers.'' ``What do you mean?'' Alice asked. She was intrigued by the idea of another magical number relating everything. ``Exactly what I said,'' the Pig said. ``The number of vertices, edges, and faces for these polyhedra is related. And it's the number two that relates them. Let me make a table.'' He wrote down in his notebook: \begin{center} \begin{tabular}{l|l|l|l|l|l} {\bf -hedron} & {\bf tetra-} & {\bf hexa-} & {\bf octa-} & {\bf dodeca-} & {\bf icosa-} \\ \hline {\bf face shape} & triangle & square & triangle & pentagon & triangle \\ {\bf faces} & 4 & 6 & 8 & 12 & 20 \\ {\bf vertices} & 4 & 8 & 6 & 20 & 12 \\ {\bf edges} & 6 & 12 & 12 & 30 & 30 \\ \end{tabular} \end{center} ``That's what we know about faces, vertices, and edges,'' said the Pig. ``Look at the tetrahedron again. Add up the number of faces and vertices. Then subtract the number of edges.'' Alice did as instructed. ``Four plus four is eight. Eight minus six is two.'' ``Now try the cube,'' the Pig said with a wink. ``Six plus eight is fourteen. Fourteen minus twelve is two. Is that where you are getting the two from?'' she asked. ``You mean I'll get two for the other shapes as well?'' The Pig just smiled. Alice could see that he wasn't going to give anything away, so she worked out the arithmetic for the octahedron. ``Eight plus six is fourteen, minus twelve is two again. Twelve plus twenty minus thirty is two. Twenty plus twelve minus thirty is two. They are all two,'' she exclaimed. ``That's neat.'' ``It doesn't only work for those five solids,'' said the Yellow Pig. ``It works for any other polyhedron as well. Just count up the faces and vertices and subtract away the edges and you'll get two. It's another beautiful result in mathematics that was worked out by Euler.'' ``Cool,'' said Alice. ``Mathematicians have been trying to understand dimensions for a while. One of the easiest ways to do so is to reduce an object to something of lower dimension. Having the cross-like net of a cube is one such way to do this. The net is two-dimensional. It's just a surface. Another way to think of a three-dimensional object in two dimensions is to project it onto the plane. Think of a clear glass cube over a white piece of paper and a light shining over it. The light passes through the cube but is blocked by the edges. So what results on the paper is a two-dimensional representation of the cube. This is one of the things that V'doordolly attempts to show in his painting. Projection and perspective are extremely important to artists. It's difficult to represent something three-dimensional on a flat surface. That's why we use the technique of perspective. Objects that appear closer to us are often drawn larger than objects in the background. Artists imagine a fixed point off in the distance to which everything is being projected. It's a complicated idea. Perspective drawing has undergone a lot of refinement, particularly during the Renaissance.'' Alice thought about paintings she had seen. ``Is that why things like streets in paintings get closer together in the distance? A road starts out with the two sides being very far apart and then the lines come toward each other.'' ``Exactly,'' said the Pig. ``Perspective has to do with how things appear to us from certain angles. How do we see three-dimensional things in art? How do we see three-dimensional things in the real world? Those are things I don't understand. Our perception and eyesight are pretty advanced.'' ``I know another way we can represent three dimensions,'' said Alice. ``We can rotate a three-dimensional shape slowly and then sketch it from several different angles. That way you can see the back. Or you can make a movie where something rotates. Movies are only two-dimensional, but they sure look real at times.'' ``They do,'' agreed the Pig, ``even when they aren't. Movies are able to portray motion. And the video camera tries to imitate what our eyes do on their own. Seeing a movie is a lot like seeing something real.'' He continued, ``I can think of yet another way to represent three dimensions using only two. Think of cutting an onion into slices of rings; think of a three-dimensional object as a series of slices so thin that they are smaller than slivers. A solid object is just all of those slices put together. If you cut an onion into rings, you are cutting in one direction. Your cuts are parallel planes that divide the spherical onion. A sphere becomes a series of circles. If you put those circles back together, you have a sphere.'' ``Onion rings aren't all the same size,'' said Alice. ``When you cut the top of an onion or the bottom, you get small rings. When you cut closer to the middle, you get larger rings.'' ``This is true,'' said the Pig. ``One slice of an onion isn't enough to tell you what an onion looks like. But a whole bunch of slices, if they are the right slices, will.'' ``What do our other shapes look like if we cut them?'' asked Alice. ``That depends on how we cut them,'' said the Pig. ``The easiest way to cut them is to put them on the floor, balancing on a face, and then cut pieces along parallels to the floor.'' ``I get it,'' said Alice. ``When you put down a cube and cut it, you get squares. They are all the same size. ``They are,'' the Pig agreed. ``That's not true if you cut a tetrahedron though.'' Alice placed the tetrahedron on the floor in front of her. ``My first slice is a tiny, tiny triangle from the top. It's almost like a point. My second slice is another triangle, bigger than that one. The next one is bigger than that. It's like the onion, starting off small and then getting bigger.'' ``Where's the biggest triangle?'' asked the Pig. ``At the bottom,'' said Alice. ``The base is the largest part. That's different from the onion which gets smaller again. All of these shapes are so different.'' ``They are,'' said the Pig. ``And it's much harder to see other ways to cut them. But what if instead of starting with the face of a cube, you started with a vertex, just as you did with the tetrahedron?'' Alice frowned. She tried to visualize this. She put the cube in front of her and held it so that one vertex was facing directly up. Another vertex was facing down. Held this way, the cube looked like a strange top. \centerline{\epsfbox{images/4-cubevv.ps}} ``How many faces meet at a vertex?'' prompted the Yellow Pig. ``Three,'' said Alice. ``Oh! So the first cut will intersect those three faces. It will be a very small triangle. Can that be right? There's a triangle inside the cube?'' ``It is right,'' said the Pig. ``If you cut a cube from vertex to vertex, you start off with a point, then small triangles that get larger. Things get funny in the middle.'' He traced a triangle around the cube. Then he traced a strange line around the middle of the cube. ``This line is parallel to the triangles.'' ``So that's another one of our slices?'' asked Alice. ``It looks funny.'' ``This line intersects every face of the cube,'' said the Pig. ``There are six faces, so it is a hexagon.'' ``So,'' said Alice, ``a cube when cut that way is just a bunch of triangles getting larger until they explode into hexagons. And then the slices go back to triangles and get smaller and smaller until they are a point.'' \centerline{\epsfbox{images/4-cubeslices.ps}} The Pig beamed. ``Wonderful,'' he said. ``What do you think happens if you hold the cube so that one edge is on the ground and one edge is facing up and then you cut it?'' Alice was getting good at this slice game. All she had to do was hold the object and visualize cutting it like cutting vegetables or cheeses. She liked cutting different shapes out of vegetables and cheeses. She was often accused of playing with her food too much before eating it. The Pig was waiting patiently for an answer. ``The first cut,'' said Alice, ``is just a line.'' She borrowed the Pig's pencil and drew a line around the cube a little bit lower down. She looked at it again. ``The next cut is a rectangle. There are a bunch of rectangles getting bigger. Now let's see what we get in the center. I think that's going to be the biggest rectangle and then the rectangles will get smaller.'' ``It makes sense that a cube contains lots of rectangles and squares, but would you believe that tetrahedra have rectangles in the middle? You can see them if you cut a tetrahedron from edge to edge.'' He lay a tetrahedron on the floor with only one edge touching the ground. There was another edge going in the opposite direction at the top of the tetrahedron. ``Cutting this one starts out with a line and ends with a line. In between are rectangles. Only they start out being wider than they are tall and end up being taller than they are wide. And in the middle is a square.'' Alice had some trouble seeing this, but she figured the Pig was probably right. ``There's one more thing,'' he continued. ``Think about how the dimensions are related. Take a point. A point is zero-dimensional. Now copy that point and slide it over somewhere to the right of that point. Connect the two points. Now you have a line. A line is one-dimensional. Now copy that line and translate it down. Connect the corresponding points on the lines. That makes a two-dimensional figure, a square. Copy that square and lift up the copy. Connect the squares together by their corners and you get a cube. If you make another cube and move it in a direction perpendicular to all of those directions and then connect the eight vertices, you will get a four-dimensional hypercube.'' \centerline{\epsfbox{images/4-hypercube.ps}} ``Why are you telling me all of this?'' asked Alice. ``There's a very good reason,'' the Pig said. ``It's to help you better understand the inhabitants of the left door.'' \section{Not Wonderland} ``The inhabitants of the left door?'' repeated Alice. ``You mean the rectangular door in the painting? We're going to meet people from this painting?'' ``Not exactly people,'' said the Pig with an air of mystery. ``But yes, you are going to go into the painting to talk to them.'' The thought of going into another painting made Alice somewhat queasy. The expression on her face must have made this quite clear, because the Pig reassured her, ``Don't worry. No Moebius strips. You will find everything in this land quite tame.'' Alice mostly trusted the Yellow Pig's judgment, so she relaxed considerably. ``What's behind the door?'' she inquired. ``A two-dimensional world,'' said the Pig. ``A two-dimensional world?'' Alice asked in disbelief. ``Yes,'' said the Pig. ``The inhabitants there are not people, but rather flat shape-creatures living on a flat plane. They are entirely contained within their plane and hence within two dimensions.'' ``What do they look like?'' asked Alice. ``Well,'' said the Pig, ``to us they look like simple polygons: triangles, squares, and even some pentagons and hexagons. But that's not how they see themselves,'' he continued. ``You see, because they are in the plane, so are their eyes which are located on one side of them. They can't see each other from the top.'' ``Weird,'' said Alice. ``Do they even know what they look like?'' ``Mostly,'' said the Pig. ``They talk to each other, and they can guess. Shape is pretty important to them so they've spent a lot of time studying it. What do you think they look like to each other?'' he asked. Alice wasn't sure. ``Think of a large glass table with cutouts of triangles and squares and pentagons,'' the Pig said. ``Now instead of looking at the table from above like you usually do, get at eye-level with the table. What do you see?'' Alice thought. ``Just a little bit of their sides. I can never see a whole shape, and the shapes look different at different angles. As the shapes turn, they must look completely different to each other.'' ``They do,'' said the Pig. ``With a triangle,'' continued Alice, ``I can sometimes see two sides and sometimes only one. It depends on if a point is facing me or not. I can usually see two sides of a square and two or three of a pentagon. Is that how they tell each other apart? By how many sides they see?'' ``Mostly,'' the Pig said. ``That must be awfully difficult for them. Why, they have to walk all the way around each other to determine their shapes.'' ``Oh, they do that,'' said the Pig. ``They have a much more elaborate greeting ritual than just handshaking. Both polygons walk around each other. It's impolite to ask someone what shape they are. Or rather, it's not so much that it's impolite, but it shows your ignorance. Any well-bred polygon has spent many years studying how to tell polygons apart. ``Wow,'' said Alice in awe, ``it all sounds so complicated.'' ``It is, and yet their world is so much simpler than the one we know. They are, after all, limited to only two dimensions. In the fourth dimension, or hyperspace, things are much more complicated. But,'' he said, `` I'll stop talking now and let you see a two-dimensional world for yourself.'' Alice cautiously reached up to the painting, opened the rectangular door, and stepped inside. ``Hello?'' called out Alice. Though the world she was in seemed empty, her words had no echo. ``Hello?'' she greeted the inhabitants again. She turned around and realized that the Pig was not behind her. She was all alone. She didn't worry about how she would get back out. She knew when the time was right, she would find herself on the other side of the painting's door. ``Is anyone here?'' she yelled softly, walking around this new world. She was walking on a small flat plane. It was a closed disk. She saw odd shapes which weren't moving and had things in them. These, she decided, were houses. ``How odd that their houses have no roofs,'' she thought. ``I can see right into their homes.'' She walked to one of the larger houses, and after a moment's hesitation, stepped inside. ``Hello?'' she said for the third time. ``What? Who is there? Where are you? What are you?'' asked a frightened voice. ``I'm a girl from the other side of the door. My name is Alice,'' she introduced with a curtsy. ``Please don't be afraid of me.'' Alice thought she should be much closer to the plane in which her new friend lived. ``I just need to lower myself,'' she thought. She somehow slid through the plane so only her head was above the plane and the rest of her was below it. The small shape screamed in horror. ``You're changing size and shape!'' Alice could see that he was a pentagon, not much larger than twice the size of her hand. Alice was confused by this outburst, but tried to soothe the pentagon. ``I'm sorry if I disturbed you by barging in on you, but you are the first person, err, shape I have seen here, and your house was wide open.'' ``My house was not at all open,'' said the pentagon indignantly. ``I assure you I always leave my doors closed.'' ``Yes, I suppose you did,'' agreed Alice. ``But you see, I came in through the roof.'' ``You came in through the roof?'' sputtered the pentagon. ``What sort of craziness is that? You came in through the roof? Why next you'll be telling me that there really is a Santa Claus.'' ``I did enter through your roof,'' said Alice. It occurred to her then why the pentagon was so confused. He lived in a two-dimensional world and could not see anything above him. The only directions that made sense to him were north, south, east, and west. Up and down were meaningless concepts. He had probably never thought or heard of a roof before. ``Where did you come from?'' the pentagon asked suspiciously. ``I'm afraid it may not make much sense to you,'' said Alice, ``but I came from above. Not from north, south, east, or west, but from another direction entirely. One that is perpendicular to those that I have described.'' ``Impossible,'' said the pentagon, ``you are spouting nonsense.'' ``You are right in a way,'' said Alice. ``Now that I think about it, I'm not sure I can explain it myself. But you see, it's very simple. You are in a two-dimensional world. And I came here from the third dimension.'' ``The third dimension?'' repeated the pentagon. ``Then what my father said is true. There really are magical shapes with special powers who visit us. You are lucky you chose this house to make your presence known,'' he warned. ``The town is of the opinion that such visitors are nothing but trouble, witches who are to be destroyed immediately to preserve the town. I've always wanted to meet a magical shape. You are safe here.'' This news worried Alice, but she decided to dismiss it. After all, if there was any trouble, she could just pull herself out of the plane. ``I'm not a magical shape,'' she told the pentagon. ``I'm just a person.'' ``But you do have special powers,'' he said. ``You can change shape.'' ``I can't change shape at all,'' said Alice. ``I can stand up or sit down or curl up into a ball, but I'm still really the same shape. Two arms, two legs, a head, and a body. Nothing different.'' ``You're doing it again!'' cried the pentagon. ``You are changing shape.'' ``Oh,'' said Alice. ``I see what you mean. But I'm not changing shape at all. You are just seeing different parts of me. You can't see all of me at once.'' ``Well, of course not,'' said the pentagon. ``No one can see all of another shape at once. You can't see the back side.'' ``You can't,'' said Alice. ``But I can see all of you, because you are flat.'' At this the pentagon seemed offended. She hastily continued. ``You see \ldots .'' Alice tried to come up with an analogy the pentagon would understand. ``It's like this. Think of a line segment or a bunch of such segments. All living on a longer line. Their eyes are at their front. So when they look at each other they just see a single point. All lines look the same. But when you look at them, you can look from the side. Then you see them as they really are --- as lines.'' ``Ah,'' said the pentagon. ``I understand now. Thank you for the explanation. I'm glad I live in two dimensions; those lines must lead an awfully dull life.'' Alice decided not to mention that two dimensions seemed pretty dull to her as well. She tried a different explanation based on something the Yellow Pig had told her not long before. ``I am three-dimensional,'' said Alice. ``You are a pentagon. When someone looks at you, they notice that you have a different width at different points. It's the same with me. I have a different thickness at different places. The third dimension consists of a bunch of planes, just like this one, stacked on top of each other.'' Alice could see that she was losing the pentagon with the phrase `on top of'. ``Not north or south or east or west,'' she said again, ``but on top of. That's another whole direction entirely.'' ``I'm afraid I don't think I understand,'' said the pentagon. ``But that's okay. Please tell me how you can change shape.'' ``I don't change shape,'' said Alice again, slightly frustrated. ``It was that you were seeing different parts of me. You can only see one side of the part of me that intersects your plane. And as I move up and down,'' she said demonstrating again, ``you see different parts of me. My feet, for example, are larger than my ankles. Then as I move down, the part of my legs that intersects your line of vision becomes thicker. Why, I probably look like two large ovals to you right now.'' The pentagon agreed. ``I am not any one of those shapes,'' concluded Alice, ``but the solid made up of all of those shapes in succession.'' Just then there was a knock on the door. The pentagon scurried across the room. ``You can see who it is, can't you?'' he asked Alice in a whisper. ``Yes,'' she answered back quietly. ``There are three circles.'' ``You better go then,'' said the pentagon. ``I am sure they know you are here and have come to find you. Meeting you was quite an experience. My father will be pleased to know I have met someone from your world. I'm sorry I couldn't follow your explanation very well. I shall think about it.'' They heard a knock on the door again. ``Please, leave before they come in,'' the pentagon requested. Alice lifted herself up so that only her feet were in the plane. Then she lifted herself up a bit more and was outside of their plane entirely. She waved goodbye to the pentagon, knowing that he couldn't see her at all. ``Curiouser and curiouser,'' she thought to herself. The door was just in front of her. She opened it and left the painting. \section{A Most Peculiar Sphere} ``That was most strange,'' said Alice, upon re-entering the art gallery. She stopped. The Yellow Pig was nowhere to be seen. ``Oh bother,'' thought Alice, ``I've lost him again. I suppose he has gone off to admire some other artwork. I'll just wait for him to return.'' She looked around again. Sitting on the floor beside the sheep painting was her teddy bear. ``There you are, teddy!,'' she exclaimed. ``I've missed you.'' She picked him up and gave him a hug. ``Where have you been? I've been looking all over for you. So has the Yellow Pig. I'd introduce you to him, but I seem to have lost him now. Oh dear. I'm going to put you in my pocket now for safekeeping until we get home.'' And assured that she would not lose her bear again, she turned her attention back to the sheep painting. It was very odd. She recognized a sheep projected onto a plane, sheep drawn in perspective, and even cross sections of sheep. ``Why, it's almost as if this painting is trying to explain sheep using only two dimensions. How odd,'' she said aloud. She knocked lightly on the second door. ``What's that?'' asked a voice. Alice looked up from the painting, startled. ``Wh-where did you come from?'' she asked. She realized that she was talking to a sphere, but that didn't seem too unusual, all things considered. ``I came through the second door in the painting,'' the sphere responded. ``I didn't see you,'' said Alice. She was a bit confused as she had been staring very intently at the painting and thought for sure she would have noticed the arrival of the visitor. ``Of course you didn't see me,'' said the sphere. ``You weren't looking the right way. Not that I expected you to.'' Alice found his attitude to be in rather bad taste. ``Which way should I have been looking?'' she asked timidly. ``In a completely different direction than you were looking,'' replied the sphere, as if that said it all. Alice tried a different tactic. ``My name is Alice. Who are you?'' ``I,'' said the sphere haughtily, ``am a hypersphere.'' ``A hypersphere? Like from the fourth dimension?'' asked Alice. ``Exactly. And I am visiting your wretchedly limited realm.'' ``Wow,'' said Alice, ignoring his remark. ``There really are four dimensions? I thought the fourth dimension was just something people talked about. A concept, you know. Something mathematicians like to think about. `Suppose there is a fourth dimension \ldots .'{''} ``There's no supposing about it,'' said the hypersphere. ``There is a fourth dimension, and I am from four-dimensional space.'' ``Prove it,'' said Alice. ``Prove it?'' said the hypersphere with a laugh. ``If I weren't from the fourth dimension, how did I get here?'' Alice failed to be impressed with that argument and told the hypersphere so. ``Here,'' said the hypersphere. ``Put me in one of your three-dimensional boxes and watch me escape.'' Alice did so, and the hypersphere got out of the cube with ease. ``I'm convinced now,'' said Alice. ``Not because you escaped, but because of how you escaped.'' Now the hypersphere was interested. ``What do you mean?'' he asked. ``When you escaped, it looked to me like you were getting smaller. You started out as a large sphere and then got smaller and smaller until you were just a speck. Then you disappeared from my sight entirely. That's when you must have been in a different space, parallel to my own.'' Alice wasn't sure she understood what she was saying at all, but it entertained her to realize that she knew more than this great hyperbeing. ``See,'' she thought to herself, ``it's just like when I went through the door to two dimensions. The pentagon could only see sections of me. No wonder he was so confused. But if two-dimensional beings don't understand the third dimension, it makes just as much sense that I wouldn't understand the fourth dimension.'' ``Hmph,'' said the hypersphere. ``I guess you are right. I hadn't thought about it that way. I've always just thought of all of the tricks I can pull on you three-dimensional beings.'' ``Like what?'' asked Alice. ``Well,'' said the hypersphere, ``like this. Take off your right shoe.'' Alice unbuckled and removed her shiny black shoe. The hypersphere took it from her and disappeared for a moment. When he returned he had a left shoe. Alice was very puzzled by this. ``How did you do that?'' she asked, after inspecting the shoe. ``It's easy,'' replied the hypersphere. ``There's no difference between a left shoe and a right shoe.'' But try as she might, Alice could not get the shoe back on her foot. Alice was really not in the mood for such tricks, but she needed the hypersphere to transform one of her two left shoes. ``Please,'' she begged, ``Do it again.'' The sphere was happy to do so, and this time he returned a right shoe. Alice put both shoes back on her feet before the hypersphere could suggest any more tricks. But the hypersphere had no more tricks. ``I'm bored,'' he said with a yawn. ``There's nothing to do here. I'm going home.'' ``Wait,'' said Alice. ``You got here through a door, right?'' The hypersphere gave some strange indication which Alice interpreted as a nod. ``Was it a door in a painting?'' ``It was,'' said the hypersphere. ``Well,'' asked Alice mischievously, ``was there another strange looking door in the painting?'' The sphere agreed that there had been. ``When you return,'' instructed Alice, ``knock on that door and wait for a response. You'll be visited by a being from the fifth dimension.'' ``The fifth dimension?'' said the sphere incredulously. ``Why that's impossible. Everyone knows there are only four dimensions.'' ``Really?'' asked Alice. ``I'm not surprised you find it hard to believe. It was hard for my two-dimensional friend to realize there were three dimensions. It was hard for me to realize there were four. But I'll bet there are more than four dimensions. I'll bet somewhere there's someone who finds your world flat and boring.'' The hypersphere was almost roaring by this point. ``You don't know what you are talking about,'' he shouted. ``I am the king of this universe!'' His voice bellowed though his size became smaller. And then he was gone. ``Well done.'' Alice turned around. It was the Yellow Pig again. ``You saw that?'' she asked. ``I didn't imagine that?'' ``I can't say if you imagined it or not, but if you imagined it, then so did I,'' replied the Pig. ``I only caught the end of your conversation I'm afraid. I think you put that hypersphere in his place very nicely.'' Alice beamed. ``Thank you,'' she said. ``And guess what?'' The Pig didn't guess, so Alice continued, ``I found my bear. He was just sitting by this painting. I have no idea how he got there. I'd like you to meet him.'' Alice pulled the bear out of her pocket. The Yellow Pig extended his hoof to the bear, and they shook paws in a fashion. ``Pleased to meet you,'' he said. ``How do you do?'' ``He won't answer you,'' said Alice. ``He just doesn't talk.'' ``I see,'' the Yellow Pig said with a smile. ``It's a lovely teddy bear.'' Alice put the bear back in her pocket. ``I think I understand about dimensions. Sort of. But those were really just threats I was yelling. I have no idea what will happen if he knocks on the door in his painting. Do you think there is a fifth dimension?'' ``I don't know,'' the Pig responded. ``But it wouldn't surprise me at all.'' ``I don't think anything surprises me anymore,'' said Alice. ``There is one thing I don't understand that maybe you can help with. How did he change my shoes?'' ``Oh,'' said the Pig. ``That is something I believe I can explain. Trace your foot prints on paper and cut them out.'' ``It's what my feet look like in a two-dimensional world,'' said Alice. ``Right,'' said the Pig. ``And how was your trip through the first door to the two-dimensional world?'' ``Very flat,'' said Alice. ``I met a nice pentagon. I think I confused him.'' ``That's okay,'' said the Pig. ``He will think about it a lot and over time it will make sense to him. He may grow up to be a great mathematician even. Anyway, when you were in the two-dimensional world, you could have picked up one of these foot prints and flipped it over. Then it would look exactly like the other foot print, and the pentagon would have no idea how you had done it.'' ``So,'' thought Alice, who was finding analogies to be an extremely useful tool, ``the hypersphere picked up my shoes out of the third dimension, rotated them around in the fourth dimension, and then put them back down?'' ``That's what I think, yes,'' said the Pig. ``So what is the fourth dimension?'' asked Alice. ``I still can't picture it.'' ``I'm afraid I don't know either,'' said the Pig. ``I sometimes think of it as being inside out. Maybe in the fourth dimension it's like being on a Moebius strip and there is no inside or outside.'' ``What do you mean?'' Alice asked. ``Well, instead of shoes, think about rubber gloves. How can you turn a right rubber glove into a glove that will fit on your left hand? You turn it inside out. Then it will fit perfectly. So maybe inside and out are the directions in the fourth dimension. Others may tell you the fourth dimension is time and its directions are past and future. Speaking of time,'' he said, looking at his watch, ``shall we go? It's getting awfully late.'' ``I guess it is,'' said Alice. ``I've enjoyed this gallery very much.'' ``Perhaps you'll come again someday,'' said the Pig, leading Alice around the other side of the room to the front door. The side wall was covered with a huge mirror. ``Hey!'' exclaimed Alice. ``Look at us in the mirror. I'm on your right side, but in the mirror I'm on your left side. And my shoes.'' Alice picked up her right foot and shook it. ``In the mirror, the shoe that moves is a left shoe. It's like the fourth dimension!'' ``It is!'' said the Pig, equally excited. ``I hadn't thought of that.'' ``Well,'' said Alice with a renewed sense of adventure, ``there's only one thing to do.'' She grabbed hold of the Yellow Pig's right hand, and the two of them went through the looking glass.