Section 16. How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland
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As soon as the receeding Peace-cry of my Wife had died away, I tried to approach the Stranger myself, to get a closer look at him.
I wanted to invite him to be seated, but his appearance was so astonishing that I found myself unable to move or speak.
Despite not having a single angle that I could See, every moment I looked at him he seemed to change right in front of my eyes, changing from different brightnessess and sizes in a way I’d never imagined possible.
The thought suddenly crossed my mind that the Figure in front of me could be a burglar or a murderer, some monstrous Irregular Isosceles who had faked the voice of a Circle to get past my servants into the house, and was now about to stab me with his invisibly sharp point.
In a dry living room, with no Fog to help me discern angles, and at such a short range, Sight Recognition wasn’t something I could trust.
Desperate with fear, I suddenly regained my ability to move and rushed forward, blurting out an unceremonious, “You must permit me, Sir--!” I didn’t even bother finishing the phrase before I felt him with my side.
My Wife was right. There were no angles at all, and not even the smallest roughness or texture. Never in my entire life had I met a more Perfect Circle.
He stood still while I walked completely around him starting at his eye and circling back to it again. Circular he was, all the way around. Perfectly Circular, there could be no doubt about it.
This was followed by a conversation, which I will do my best to write down as clearly as I can remember, leaving out only a few of my many apologies – because I was so humiliated and ashamed of the idea that I, a common Square, had dared to touch and Feel a Perfect Circle.
This conversation was started by the Stranger, who finally got impatient with me.
The conversation went like this:
The Stranger: Haven’t you felt enough? Isn’t it time we move past the introductions already?
Me: Please forgive me most esteemed Sir, I’m normally so much more prepared for these kinds of visits, but I wasn’t expecting such an important guest to appear so late at night without warning, and my nervousness got the better of me. I beg you not to tell anyone else how rude I’ve been, especially not my Wife. But before we say anything else, can I ask where you are visiting from?
The Stranger: From Space, from Space, Sir! Where else?
Me: Pardon me, my Lord, but aren’t we in Space right now?
The Stranger: Pooh! What do you know about Space? Define Space.
Me: My Lord, Space is Height and Width, stretching on forever.
The Stranger: See, you don’t even know what Space is! You think it’s only Two Dimensions, but I’ve come to tell you about a third. Space is Height, Width, and Length!
Me: Then you will be pleased to know that we have a fourth name for the Two Dimensions too; sometimes we say ‘Thickness’ instead of ‘Width’, the way you use the word ‘Length’ to mean ‘Height’.
The Stranger: I don’t mean three different names, I mean Three different Dimensions.
Me: Would your Lordship please show, or explain to me, what direction this Third Dimension is in?
The Stranger: It’s where I came from. It’s above you, and below you.
Me: So you mean it’s to the North and South.
The Stranger: I mean no such thing. I’m talking about directions you can’t look in, because you don’t have an eye on your side.
Me: Pardon me, my Lord, but if you look at me you’ll see that I have a perfectly good eye at the point formed by two of my sides.
The Stranger: Yes, but to see into Space, you would need to have an eye on your side, not your Perimeter. I mean, you would probably call it your insides, but in Spaceland we would call that your side.
Me: An eye on my insides! An eye in my stomach? You’re joking with me!
The Stranger: I’m not in a joking mood. I’m telling you I came from Space, but since you don’t understand what Space actually means, we’ll call it the Land of Three Dimensions. Just a little while ago I was looking down at your flat Plane, which you mistakenly call Space, and from up there I saw everything that you think of as “solid”, which to you just means enclosed on the sides. I saw into all your flat houses, your churches, your safes and drawers, and yes, even into your insides and stomachs! All of it lying there out in the open for me to see!
Me: It’s very easy to say things like that, My Lord.
The Stranger: Meaning you won’t believe me without proof. Well here’s your proof: when I came down here, I saw your four Pentagon Sons, each sleeping in his own room.
The Stranger: I saw your two Hexagon Grandsons, the youngest of which was with you for a while before he went to bed, leaving only you and your Wife in the living room.
The Stranger: I saw your three Isosceles servants, eating their supper in the kitchen, and the little errand boy in the laundry room. And then I came here. How do you think I got in?
Me: Through a hole in the roof, I assume.
The Stranger: Wrong. You know very well that your roof was just recently repaired, and the walls are so solid that not even a Woman could find a crack to fit through. I’m telling you again, I’m from Space – Isn’t what I’ve just told you proof of that?
Me: Your Lordship must know how easy it would be to learn what you just told me, just from asking my neighbors. It wouldn’t be difficult, especially for someone of your status.
The Stranger, muttering to himself: What else can I try? Wait – I’ve got an idea that might work!
The Stranger, to me: When you see a Straight Line – your wife, for example – how many Dimensions do you think she takes up?
Me: Ah, your Lordship is confusing me for one of the common rabble, who doesn’t understand Mathematics at all, and thinks that a Woman is literally a Straight Line who only exists in a Single Dimension! No, no, my Lord, I assure you, we Squares are better educated than that, we know just as well as you do that a Woman, though she is commonly called a Straight Line, is, scientifically, just a very thin Parallelogram, taking up the dimensions of both length and width, or thickness, if you want to call it that, just like the rest of us.
The Stranger: But just the fact that a Line is visible tells you that it has a Third Dimension.
Me: My Lord, I just said that we know a Woman has both Length and Width. We see her Length, and we can infer her Width. Even though it is very small, it can be measured.
The Stranger: You don’t understand what I’m trying to say. I mean that when you see a Woman, you can see her Length, and you infer her Width, but you also see what we call her Height, though with your country, that’s not very high at all. If a Line, seen from the side, had nothing except Length, without “Height”, it would cease to exist, and be completely invisible. Surely you understand this?
Me: I have to confess that you’re right and I don’t understand you at all, your Lordship. When we in Flatland see a Line, we see Length and Brightness. If the Brightness disappears, the Line is destroyed, and, as you put it, no longer occupies Space. But am I understanding you right, that you’ve decided that “Brightness” is a Dimension, and what what I call “Bright”, you call “High”?
The Stranger: No, that’s not right. By “Height”, I mean a Dimension like your Length, but with you, “Height” is hard to measure, because of how incredibly small it is.
Me: My Lord, this can easily be tested. You say I have a Third Dimension, which you call “Height”. Now, Dimension implies both direction and measurement. So just measure my “Height”, or just show me which way my “Height” extends, and you can consider me a true believer. But if even you can’t do this, then you’ll just have to excuse me for being ignorant myself.
The Stranger, to himself: I can’t do either of those things. How can I convince him? Surely, a simple statements of the facts...followed by a visual demonstration ought to work!
The Stranger, to me: Now, listen Sir. You are living on a Plane. What you call Flatland is the vast surface of what I would call a Fluid. You and your countrymen move on or maybe in the top of it, without floating above or sinking below.
The Stranger: I am not a Plane Figure like you, I am a Solid. You call me a Circle, but I’m not actually a Circle. You could say I’m an infinite number of Circles all placed one on top of the other, ranging in size from a single Point at the smallest, to a Circle of thirteen inches in diameter at the largest.
The Stranger: When I float inside your Plane like I am doing right now, the only part of me that exists in your Plane is a section that you correctly think of as a Circle.
The Stranger: The only way you Flatlanders will ever be able to see a Sphere in Flatland is as a Circle. A Sphere, by the way, is the proper name for my shape where I come from.
The Stranger: Don’t you remember the dream you had last night? I saw it – I can see everything in Flatland from up in Space, and I saw the dream written on your brain. Don’t you remember how you entered the realm of Lineland?
The Stranger: And the only way the King could make sense of you was as a Line, not a Square, because the Linear Realm didn’t have enough dimensions to represent all of you at the same time? You could only exist in Lineland as small slices of yourself, one at a time.
The Stranger: It’s exactly the same way with myself and your Flatland! Your country of only Two Dimensions isn’t enough to let all of me exist here at once, so you can only see single sections of me at any time. And this is why you see me as a Circle, even though I am a Three Dimensional Sphere.
The Stranger: The brightness of your eye has dimmed, which means you still don’t believe me. But get ready to see the proof of what I say. It’s true that you can’t see more than one of my Circular sections at a time, since you can’t lift your eye out of the Plane of Flatland. But – you can at least watch how my sections become smaller as I lift myself out of Flatland. Watch – I will rise above you, and you will see that my Circle will become smaller and smaller, until it is nothing but a Point, and then vanishes completely.
[ID: A black and white diagram showing a Sphere intersecting a line, with an eye on the right corner of this line labeled "my eye". At first the Sphere rests with the line intersecting the middle of his body, creating a large circle. This is labeled, "The Sphere with his section at full size". Then we are shown "The Sphere rising", with the line now in a lower section, creating a smaller circle. Finally the sphere is almost above the line completely, forming a tiny circle, labeled, "The Sphere on the limit of vanishing". End ID.]
There was no “rising”, at least not that I could see. But he did get increasingly smaller until he disappeared.
I blinked once or twice to make sure I wasn’t dreaming again, but it wasn’t a dream. From the depths of nowhere, seeming to come from close to my heart, I heard a hollow voice ask, “You see how I disappeared? Do you believe me yet? Well keep watching and I’ll slowly return to Flatland, and you’ll see my section become larger and larger.”
Every reader in Spaceland will easily understand that my mysterious Guest was speaking the simple truth. But to me, even though I was well-trained in Flatland Mathematics, this was a very difficult concept to grasp.
The diagram I have provided above will make it clear to any Spaceland child that the Sphere, moving upwards in the three positions indicated, obviously would have appeared to me, or any other Flatlander, as only a Circle that was becoming smaller and smaller until he was nothing but a Point, before vanishing.
But even though I had all the evidence in front of me, the reasons for it were just as confusing and unbelievable as ever. The only thing I could comprehend was that the Circle had somehow made himself smaller until he vanished, and now he was making himself reappear, and grow larger again.
When he regained his largest size, he let out a deep sigh, because it was obvious from my silence that I still didn’t believe or understand him.
And to tell you the truth, I was now starting to believe that he wasn’t a Circle at all, but some kind of entertainer – maybe a juggler. It was either that, or all the old wives’ tales were true, and Enchanters and Magicians were real.
After a long pause, the Stranger muttered to himself, “One option is left, if I’m going to avoid drastic action. I’ll have to try explaining using an Analogy.”
Then he was silent for an even longer time, until he started speaking again, continuing our dialogue:
Sphere: Tell me, Mr. Mathematician; if a Point moved Northward, and left a trail of light, what name would you give this trail?
Me: A Straight Line.
Sphere: And a Straight Line has how many Terminal Points?
Me: Two.
Sphere: Now imagine that Line moving parallel to itself, to the East or West, so that each Terminal Point leaves behind it a light trail in a Straight Line. What do you call this Shape? Lets assume that the Line has moved a distance equal to its own original Length. What do you call this Figure, I ask you?
Me: A. Square.
Sphere: And how many sides does a Square have? How many angles, or Points?
Me: Four sides, and four angles.
Sphere: Now, stretch your imagination a little, and imagine this Flatland Square moving parallel to itself by moving upward.
Me: What? Northward?
Sphere: No, not Northward; upward; out of Flatland altogether. If it moved Northward, the Southern Points would have to move to the spots where the Northern Points had started out, but that’s not what I mean.
The Stranger: I mean that every Point in you – since you’re a Square, I’ll use your shape to illustrate my point – every Point in you, in what you call your inside, needs to move upwards, through Space, in a way so that no Point touches a spot where another Point started out. Each Point will create a new Straight Line of its own. All of this follows the rule of Analogy. Surely you understand now?
Restraining my impatience – since I was now very tempted to rush blindly at my Visitor and shove him into Space, or out of Flatland, or anywhere really, as long as I could get rid of him –
I replied, “And this Figure that I’m supposed to be creating by moving it in the direction you are pleased to call ‘Upward’ – What is it like? I assume you’ll be able to describe it to me in the language of Flatland.
Sphere: Oh, certainly. It’s all very simple, and strictly follows the Analogy – but, by the way, you shouldn’t call the result a Figure, it’s not a Figure, it’s a Solid. But I’ll describe it to you, or rather, I’ll describe it using an Analogy.
The Stranger: We begin with just a single Point, which, obviously, is made up of One single Terminal point.
The Stranger: One Point then becomes a Line with Two Terminal Points.
The Stranger: One Line becomes a Square with Four Terminal Points.
The Stranger: Now you can answer your own question: One, Two, Four, the Geometrical Progression is obvious. What number comes next?
Me: Eight.
Sphere: Exactly. The Square produces a Solid that you do not have a name for, but we call a Cube, with Eight terminal Points. Now do you believe me?
Me: So does this Creature have sides, as well as angles, which you call “Terminal Points”?
Sphere: Of course, and all according to Analogy. But, by the way, I don’t mean what you call sides, which we sometimes call edges. It would have what we call sides, which you call Solids.
[ID: A black and white diagram showing a square, then a cube. One of the lines on the square has two arrows pointing to it, labeling it, "A Flatland 'side' (What Spacelanders call an 'edge')". Then one of the flat surfaces on the cube is labeled, "A Spaceland 'side' (What Flatlanders call a 'solid')". End ID.]
Me: And how many solids, or “sides” will this Being that you call a Cube have, when I am done creating him by somehow moving my inside in an “upward” direction?
Sphere: How can you ask that and still call yourself a mathematician? The side of the thing is always doubled with the increase of the Dimension. A Point has no Dimensions, so it has zero sides. Next we have a Line, whose points we will consider sides for this theory, which it has Two of. Next is a Square, with Four sides. We go from 0, to 2, to 4. 0,2,4. What kind of pattern do you call that?
Me: A Mathematical one.
Sphere: And what is the next number in that pattern?
Me: Six.
Sphere: Exactly! See, you’ve answered your own question: the Cube will have Six sides, or what you call your insides. You understand it all now, eh?
((A note from the second editor. I have chosen to present next section of dialogue by the Author in both his original phrasing, as well my “translation” into more casual language, because I think the original is very funny, and don’t want to deprive my readers of the joy of experiencing it. End of second editor’s note.))
Translation: “Monster!” I shrieked, “I don’t care if you’re a juggler, an enchanter, a dream, or a demon, I won’t tolerate your jokes anymore! Either you’ll die, or I will!”
Original: "Monster," I shrieked, "be thou juggler, enchanter, dream, or devil, no more will I endure thy mockeries. Either thou or I must perish."
And with these words, I charged him.
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