And He Built A Crooked Post

My friends, I cannot premise this post any better than by saying that I'm sorry for not updating in what feels like forever. Unfortunately college is a daunting thing, and I find myself either trying to catch up on work or catch up on relaxing. As relaxing as writing these posts is, I have been finding many other hobbies to occupy myself with that have more immediate results (e.g. playing piano/guitar). I hope you know that I love you all and infinitely appreciate the support you have given me though, so please don't take my absence as an affront! Now, onto the post that I promised you all awhile ago: a follow up to the short story And He Built A Crooked House.

The main thing that confuses a lot of people about this story is the implications behind the applications of the fourth spatial dimension (something I briefly started to talk about in Episode 1 of Time Log). The fourth spatial dimension is a confusing, weird, and yet a boundless opportunity to study the intricacies of the universe. Let me sum up briefly what the fourth spatial dimension is and how you can learn to understand it.

THE FOURTH DIMENSION (ooooooohhh....scaryyyyy....)

Okay, so before I get into the fourth dimension, why don't I do a simplistic breakdown of the first, second, and third dimension so that you have a good mental comparison?

The First Dimension:

The first dimension is pretty simple to understand. Take any straight line (it has to be theoretically perfectly straight and perfectly thin. Any curves or thickness automatically qualify it in a higher dimension). This straight line with no thickness actually represents an instance of (and I use that term specifically, I'll explain in a bit) the first dimension. Now imagine a single point, like a dot drawn with a pencil, but imagine that it has no width, length, or depth. This point when travelling in the first dimension can only travel along that line, both forward and back. This starts to get complicated though, because the first dimension doesn't really exist by itself in our third-dimensional world. Theoretically, for a line to be able to "exist", it must have a width as well as length, otherwise it is non-observable to us. So the usage of a line to represent the first dimension is actually more analogous than anything for the application of the first dimension. So a first-dimensional object can't exist for us!

The Second Dimension:

The second dimension is even easier to understand because we encounter these types of objects all the time. To add a second dimension to our understanding, all we have to do is add a "one-dimensional" line at 90 degrees to a second "one-dimensional" line. This should make an L-shape, which by itself is what we call a two-dimensional object. If you want more examples, take a sheet of paper and a pencil. Draw any shape you can think of. Any combination of lines in any combination of angles. The shape you just drew is a two-dimensional object. Such an object can have width and length, but not depth. However, you may have picked up on this concept, but again, your drawing of two-dimensional shapes actually has a depth to it similar to the dot drawn in the first-dimensional example! So also similar to the first-dimensional object, it also can't exist for us! So your drawing of shapes is only considered a representation of two-dimensional objects.

The Third Dimension:

I won't talk much about the third dimension because we all live in the third dimension so I wouldn't imagine it would be too hard to conceive of a lot of different examples! What I will say is how the third dimension relates to the other two. So I mentioned that the second dimension differs from the first by a second straight line drawn at a 90 degree angle to the first one. The third dimension differs from the second by drawing a third straight line at a 90 degree angle to the other two. These lines can also be thought as axes (see below) where each axis represents a new dimension!

1st Dimension

2nd Dimension

3rd Dimension

So now we get into the harder thing to visualize, which is the fourth spatial dimension. The quickest way to explain it with minimal complexity is to consider the fourth spatial dimension as a fourth straight line at a 90 degree angle to each line in the third dimension. Unfortunately, it really isn't possible to physically draw an accurate representation of all four dimensions, so let's think of this in a more mentally-visual manner. Let's say you have a one-dimensional line. A two-dimensional object has access to a dimension that the line does not. So an object that has the option to move in the second dimension could theoretically touch any point on the one-dimensional line without having to move through any other points (e.g. endpoints) to touch that spot. Let's bring it up one dimension. Let's say you have a two-dimensional object, for example a circle, and a third-dimensional object, for example a pencil. Similar to the previous example, that pencil can access a dimension that the circle cannot. Therefore, that pencil can touch any point in that circle without moving through any other point in the circle (e.g. the border) simply by utilizing the third dimension. Bringing it up one last dimension, we can take a third-dimensional object, such as a solid metal cube. If we had some theoretical fourth-dimensional object, it could actually touch any point in that cube without touching any other point by accessing this "fourth dimension".

So how can I explain this mystical house that the story described? The simplest thing is to understand that the fourth dimension isn't visible to us three-dimensional objects! Similar to how a second dimensional object cannot see the fourth dimension, but it can theoretically observe third dimensional "slices" of a fourth dimensional object. But since those slices would have to not have a fourth dimensional "thickness", the fourth dimension cannot be visible in the third dimension. Hence why this fourth dimensional house disappeared! The obvious question, though, is why were they able to see the house from the inside? Unfortunately, as described above, this was where creative licensing was used. Regardless of state of position or frame of reference, you cannot change whether you see a fourth dimensional object or not. Theoretically, we can imagine though that being inside the house was an isolated "slice" of the third dimension that they were a part of (instead of just observing), and that could somehow allow them to see that slice, and that slice only.

I apologize if after this your brain hurts. Visualizing the fourth dimension is incredibly difficult for a number of reasons, most importantly that we were raised to visualize a maximum of three dimensions (and there really would be no evolutionary reason as to why we would need to visualize higher). So all mental gymnastics aside, this story is a subtly revolutionary piece that really helps sow the seeds of "higher dimensional thinking" into pop/modern culture.

Again, I'm so sorry for the massive hiatus, and I hope to be getting back into the swing of writing entries soon! Thank you always for your support. If you think that someone you know might even be remotely interested in any of these pieces, please let them know! As I have always said, the more activity/response I get, the more I'm inclined (and feel obligated) to write. Thank you all!

--J

Time Log Episode 1!

Hey everyone! So I know that I should be writing the follow-up post to "And He Built A Crooked House", but instead I got excited about writing this new miniseries! So come watch and enjoy! Thanks as always, and let your friends know that I exist...

--J

A Story of the Future (Part 1)

This story below is called "--And He Built A Crooked House". If you have read it before, then I'd still recommend reading it again, and if you haven't read it yet, keep reading! This was one of the first physics/math-related stories that I ever read (closely followed by The Martian Chronicles [Ray Bradbury. Depressing at points but brilliant] and The Foundation Trilogy [Isaac Asimov. Not depressing and a fantastic sci-fi trilogy]). This story mainly talks about the weird and not-so-serious implications of the fourth spatial dimension. After I finish posting the whole story (which may take a few posts) I will add a special post afterwards to try to explain what a tesseract (a.k.a. a hypercube) is! The story is in three parts and go continuously from part to part (just for your convenience!). Enjoy the story.

—And He Built a Crooked House
by Robert A. Heinlein


Americans are considered crazy anywhere in the world.

They will usually concede a basis for the accusation but point to California as the focus of the infection. Californians stoutly maintain that their bad reputation is derived solely from the acts of the inhabitants of Los Angeles County. Angelenos will, when pressed, admit the charge but explain hastily, "It's Hollywood. It's not our fault—we didn't ask for it; Hollywood just grew."

The people in Hollywood don't care; they glory in it. If you are interested, they will drive you up Laurel Canyon "—where we keep the violent cases." The Canyonites—the brown-legged women, the trunks-clad men constantly busy building and rebuilding their slap-happy unfinished houses—regard with faint contempt the dull creatures who live down in the flats, and treasure in their hearts the secret knowledge that they, and only they, know how to live.

Lookout Mountain Avenue is the name of a side canyon which twists up from Laurel Canyon. The other Canyonites don't like to have it mentioned; after all, one must draw the line somewhere!

High up on Lookout Mountain at number 8775, across the street from the Hermit—the original Hermit of Hollywood—lived Quintus Teal, graduate architect.

Even the architecture of southern California is different. Hot dogs are sold from a structure built like and designated "The Pup." Ice cream cones come from a giant stucco ice cream cone, and neon proclaims "Get the Chili Bowl Habit!" from the roofs of buildings which are indisputably chili bowls. Gasoline, oil, and free road maps are dispensed beneath the wings of tri-motored transport planes, while the certified rest rooms, inspected hourly for your comfort, are located in the cabin of the plane itself. These things may surprise, or amuse, the tourist, but the local residents, who walk bareheaded in the famous California noonday sun, take them as a matter of course.

Quintus Teal regarded the efforts of his colleagues in architecture as faint-hearted, fumbling, and timid.

· · · · ·

"What is a house?" Teal demanded of his friend, Homer Bailey.

"Well—" Bailey admitted cautiously, "speaking in broad terms, I've always regarded a house as a gadget to keep off the rain."

"Nuts! You're as bad as the rest of them."

"I didn't say the definition was complete—"

"Complete! It isn't even in the right direction. From that point of view we might just as well be squatting in caves. But I don't blame you," Teal went on magnanimously, "you're no worse than the lugs you find practicing architecture. Even the Moderns—all they've done is to abandon the Wedding Cake School in favor of the Service Station School, chucked away the gingerbread and slapped on some chromium, but at heart they are as conservative and traditional as a county courthouse. Neutra! Schindler! What have those bums got? What's Frank Lloyd Wright got that I haven't got?"

"Commissions," his friend answered succinctly.

"Huh? Wha' d'ju say?" Teal stumbled slightly in his flow of words, did a slight double take, and recovered himself. "Commissions. Correct. And why? Because I don't think of a house as an upholstered cave; I think of it as a machine for living, a vital process, a live dynamic thing, changing with the mood of the dweller—not a dead, static, oversized coffin. Why should we be held down by the frozen concepts of our ancestors? Any fool with a little smattering of descriptive geometry can design a house in the ordinary way. Is the static geometry of Euclid the only mathematics? Are we to completely disregard the Picard-Vessiot theory? How about modular system?—to say nothing of the rich suggestions of stereochemistry. Isn't there a place in architecture for transformation, for homomorphology, for actional structures?"

"Blessed if I know," answered Bailey. "You might must as well be talking about the fourth dimension for all it means to me."

"And why not? Why should we limit ourselves to the—Say!" He interrupted himself and stared into distances. "Homer, I think you've really got something. After all, why not? Think of the infinite richness of articulation and relationship in four dimensions. What a house, what a house—" He stood quite still, his pale bulging eyes blinking thoughtfully.

Bailey reached up and shook his arm. "Snap out of it. What the hell are you talking about, four dimensions? Time is the fourth dimension; you can't drive nails into that."

Teal shrugged him off. "Sure. Sure. Time is afourth dimension, but I'm thinking about a fourth spatial dimension, like length, breadth, and thickness. For economy of materials and convenience of arrangement you couldn't beat it. To say nothing of the saving of ground space—you could put an eight-room house on the land now occupied by a one-room house. Like a tesseract—"

"What's a tesseract?"

"Didn't you go to school? A tesseract is a hypercube, a square figure with four dimensions to it, like a cube has three, and a square has two. Here, I'll show you." Teal dashed out into the kitchen of his apartment and returned with a box of toothpicks which he spilled on the table between them, brushing glasses and a nearly empty Holland gin bottle carelessly aside. "I'll need some plasticine. I had some around here last week." He burrowed into a drawer of the littered desk which crowded one corner of his dining room and emerged with a lump of oily sculptor's clay. "Here's some."

"What are you going to do?"

"I'll show you." Teal rapidly pinched off small masses of the clay and rolled them into pea-sized balls. He stuck toothpicks into four of these and hooked them together into a square. "There! That's a square."

"Obviously."

"Another one like it, four more toothpicks, and we make a cube." The toothpicks were now arranged in the framework of a square box, a cube, with the pellets of clay holding the corners together. "Now we make another cube just like the first one, and the two of them will be two sides of the tesseract."

Bailey started to help him roll the little balls of clay for the second cube, but became diverted by the sensuous feel of the docile clay and started working and shaping it with his fingers.

"Look," he said, holding up his effort, a tiny figurine, "Gypsy Rose Lee."

"Looks more like Gargantua; she ought to sue you. Now pay attention. You open up one corner of the first cube, interlock the second cube at the corner, and then close the corner. Then take eight more toothpicks and join the bottom of the first cube to the bottom of the second, on a slant, and the top of the first to the top of the second, the same way." This he did rapidly, while he talked.

"What's that supposed to be?" Bailey demanded suspiciously.

"That's a tesseract, eight cubes forming the sides of a hypercube in four dimensions."

"It looks more like a cat's cradle to me. You've only got two cubes there anyhow. Where are the other six?"

"Use your imagination, man. Consider the top of the first cube in relation to the top of the second; that's cube number three. Then the two bottom squares, then the front faces of each cube, the back faces, the right hand, the left hand—eight cubes." He pointed them out.

"Yeah, I see 'em. But they still aren't cubes; they're whatchamucallems—prisms. They are not square, they slant."

"That's just the way you look at it, in perspective. If you drew a picture of a cube on a piece of paper, the side squares would be slaunchwise, wouldn't they? That's perspective. When you look at a four-dimensional figure in three dimensions, naturally it looks crooked. But those are all cubes just the same."

"Maybe they are to you, brother, but they still look crooked to me."

Teal ignored the objections and went on. "Now consider this as the framework of an eight-room house; there's one room on the ground floor—that's for service, utilities, and garage. There are six rooms opening off it on the next floor, living room, dining room, bath, bedrooms, and so forth. And up at the top, completely enclosed and with windows on four sides, is your study. There! How do you like it?"

"Seems to me you have the bathtub hanging out of the living room ceiling. Those rooms are interlaced like an octopus."

"Only in perspective, only in perspective. Here, I'll do it another way so you can see it." This time Teal made a cube of toothpicks, then made a second of halves of toothpicks, and set it exactly in the center of the first by attaching the corners of the small cube to the large cube by short lengths of toothpick. "Now—the big cube is your ground floor, the little cube inside is your study on the top floor. The six cubes joining them are the living rooms. See?"

Bailey studied the figure, then shook his head. "I still don't see but two cubes, a big one and a little one. Those other six things, they look like pyramids this time instead of prisms, but they still aren't cubes."

"Certainly, certainly, you are seeing them in different perspective. Can't you see that?"

"Well, maybe. But that room on the inside, there. It's completely surrounded by the thingamujigs. I thought you said it had windows on four sides."

"It has—it just looks like it was surrounded. That's the grand feature about a tesseract house, complete outside exposure for every room, yet every wall serves two rooms and an eight-room house requires only a one-room foundation. It's revolutionary."

"That's putting it mildly. You're crazy, bud; you can't build a house like that. That inside room is on the inside, and there she stays."

Teal looked at his friend in controlled exasperation. "It's guys like you that keep architecture in its infancy. How many square sides has a cube?"

"Six."

"How many of them are inside?"

"Why, none of 'em. They're all on the outside."

"All right. Now listen—a tesseract has eight cubical sides, all on the outside. Now watch me. I'm going to open up this tesseract like you can open up a cubical pasteboard box, until it's flat. That way you'll be able to see all eight of the cubes." Working very rapidly he constructed four cubes, piling one on top of the other in an unsteady tower. He then built out four more cubes from the four exposed faces of the second cube in the pile. The structure swayed a little under the loose coupling of the clay pellets, but it stood, eight cubes in an inverted cross, a double cross, as the four additional cubes stuck out in four directions. "Do you see it now? It rests on the ground floor room, the next six cubes are the living rooms, and there is your study, up at the top."

Bailey regarded it with more approval than he had the other figures. "At least I can understand it. You say that is a tesseract, too?"

"That is a tesseract unfolded in three dimensions. To put it back together you tuck the top cube onto the bottom cube, fold those side cubes in till they meet the top cube and there you are. You do all this folding through a fourth dimension of course; you don't distort any of the cubes, or fold them into each other."

Bailey studied the wobbly framework further. "Look here," he said at last, "why don't you forget about folding this thing up through a fourth dimension—you can't anyway—and build a house like this?"

"What do you mean, I can't? It's a simple mathematical problem—"

"Take is easy, son. It may be simple in mathematics, but you could never get your plans approved for construction. There isn't any fourth dimension; forget it. But this kind of a house—it might have some advantages."

Checked, Teal studied the model. "Hm-m-m—Maybe you got something. We could have the same number of rooms, and we'd save the same amount of ground space. Yes, and we would set that middle cross-shaped floor northeast, southwest, and so forth, so that every room would get sunlight all day long. That central axis lends itself nicely to central heating. We'll put the dining room on the northeast and the kitchen on the southeast, with big view windows in every room. Okay, Homer, I'll do it! Where do you want it built?"

"Wait a minute! Wait a minute! I didn't say you were going to build it for me—"

"Of course I am. Who else? Your wife wants a new house; this it it."

"But Mrs. Bailey wants a Georgian house—"

"Just an idea she has. Women don't know what they want—"

"Mrs. Bailey does."

"Just some idea an out-of-date architect has put in her head. She drives a new car, doesn't she? She wears the very latest styles—why should she live in an eighteenth century house? This house will be even later than this year's model; it's years in the future. She'll be the talk of the town."

"Well—I'll have to talk to her."

"Nothing of the sort. We'll surprise her with it. Have another drink."

"Anyhow, we can't do anything about it now. Mrs. Bailey and I are driving up to Bakersfield tomorrow. The company's bringing in a couple of wells tomorrow."

"Nonsense. That's just the opportunity we want. It will be a surprise for her when you get back. You can just write me a check right now, and your worries are over."

"I oughtn't to do anything like this without consulting her. She won't like it."

"Say, who wears the pants in your family anyhow?"

The check was signed about halfway down the second bottle.

[End Part 1]