The Fourth Dimension

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I would sincerely like to thank my Patreon supporters, without whom I would not be motivated to write these stories.
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*[[Brandon| Brandon C S Sanders]]

Latest revision as of 01:08, 23 June 2020

This is one page of the Metaphoriuminomicon

December 24th : The Fourth Dimension

Quotes of the Day

To
The Inhabitants of SPACE IN GENERAL
And H. C. IN PARTICULAR
This Work is Dedicated
By a Humble Native of Flatland
In the Hope that
Even as he was Initiated into the Mysteries
Of THREE Dimensions
Having been previously conversant
With ONLY TWO
So the Citizens of that Celestial Region
May aspire yet higher and higher
To the Secrets of FOUR FIVE OR EVEN SIX Dimensions
Thereby contributing
To the Enlargement of THE IMAGINATION
And the possible Development
Of that most rare and excellent Gift of MODESTY
Among the Superior Races
Of SOLID HUMANITY"
-Dedication of Edwin A. Abbott's, Flatland: A romance of many dimensions.

Contemplation for the Day

Let us imagine

You and your elephant are swimming through open space. There is nothing as far as the eye can see, except faint glowing grid lines measuring out all of the directions, up, left, forward, right, back and down. You wonder for a moment is El is going to appear on his Golden Throne and start lecturing you about the lost prehistory of the universe. But something stranger happens. Objects appear, white flowing objects all around you. They grow and then shrink and then disappear again, then reappear and grow and shrink again. Soon as they all bob in and out of existence, they get out of synch so that some are reappearing before others disappear. You notice that some of them give off beams of blue light occasionally. You would go and poke one to see what it was made of but, of course, your elephant is swimming in space. It had felt like your elephant was moving in a controlled direction and speed but your inability to explore these objects makes it clear that in fact she was just coasting in one direction while gently 'swimming' her legs.

You continue to watch the objects and feel there is something familiar about the flowing white cloth, about the blue beams of light. You notice that the bobbing slows, and they cease to disappear and reappear but simply oscillate in size and dance in shape. Then, for a moment, you see one take on the shape of a miniature Sophia-body. It sees you and focuses its blue gaze on you and you know where you have seen the flowing white cloth and blue beams of light before. She grows and dances in shape and then suddenly all the objects take on the shape of Sophia-bodies of various sizes and focus on you. You feel hot in their combined blue gaze. They speak in their unison voice, "Just One, are you enjoying Spaceland?"

"Yes, Sophia," you say, "It is quite pleasant. What are you up to today?"

"We are exploring multi-dimensional geometry, Just One, " Sophia replies in her spine shaking unison-voice, "Specifically, we are interested in exploring the meaning of direction in greater detail."

"Oh, I explored a direction with El, yesterday." You say confidently, "We explored time. We traveled to another dimension where time was like space and you could walk around in it."

"Just One," Sophia admonishes, "You cannot travel TO another dimension. You sound like one of those terrible soft-science-fiction stories. Authors of such works often speak of travelling TO another dimension. But dimensions are not places and you can not travel TO them any more than you can travel to Up-Down. Up and Down are a pair of directions that are opposite. This is true of all dimensions. All dimensions are pairs of opposite directions. You can travel IN a direction, but you can't travel TO a direction. Directions are not destinations."

"Oh. " You say perplexed, "Then where was I, yesterday?"

"Just One," Sophia announces, "You were in a brane, B, R, A, N, E, a limited region in which a certain number of dimensions are apparent."

"It was shaped like a bowl." You say still uncertain about what Sophia is telling you.

"A curved brane." Sophia diagnoses.

You drift for a while thinking.

"Have you ever read Flatland by Edwin A. Abbott, Just One?" Sophia asks.

"No," you answer.

"Well," Sophia responds, "It is worth reading Flatland. It is available on the web for free."

One of the Sophia-bodies speaks up, a tiny one floating below your elephant's feet, "Shouldn't we start our explorations?"

Another of the Sophia-bodies, a normal size one floating off to your left says, "Let us begin by creating a simple brane. The simplest. A line segment." She waves her hands and a line segment appears. It is quite long and passes right through the cloud of floating Sophia-bodies, but without intersecting with any of them.

You lean off your elephant and peer closely at the line segment where it passes near to you. You can see little sub-segments of the line segment which are differently coloured, red or blue, "What are these coloured sub-segments that I see, Sophia?"

Sophia answers in her unison voice, "Those are one-dimensional objects inside the one-dimensional brane, Just one."

"Why aren't they moving?" you ask.

Sophia laughs, "Because that brane has only one dimension, a space dimension; no time dimension."

The tiny Sophia-body below your feet speaks, "link it to the time dimension of this parent brane!"

The normal size Sophia-body to your left waves her hand again and the blue and red sub-segments begin to move. They bounce off each other and oscillate back and forth slowly. They never change size or pass through or around each other. They stay in the same order and bounce off of each other, always bouncing off the one ahead of them and then the one behind them.

"Kind of dull!" you say, disappointed.

A medium sized Sophia-body floating ahead of you speaks, "Let's create a second brane that is one dimension more interesting, and then collide the two universes." She waves her hand and skillfully inserts a vast plain into the scene. The plane doesn't intersect with any of the Sophia-bodies and is parallel to the line segment universe but only a few inches from it. The plane-universe is doted with geometric shapes, circles, squares, complex irregular polygons of various colours. The spin and drift around the universe colliding with each other and bouncing off of each other and then going off to collide with other shapes. As they collide, they trade spin and drift to and from each other.

"Now for the collision!" the medium sized Sophia-body says and waves her hand. The line segment slowly drifts toward the plane until it lies in the plane and then stops. Nothing happens. The line segments continue to move back and forth in the line-universe while the shapes continue to drift and spin through the plane-universe. Then a small green square drifts across the line segment universe just as a denizen of that universe is approaching the spot where the square is drifting. The line segment collides with the square and yells in surprise, "Who are you and where did you come from?" it yells.

The square replies, "I am a square, I came from the left and I am going to the right."

The line segment replies, "I do not know what a 'square' is nor what this 'left' and 'right' are supposed to be. I deny that it is even possible for you to be here. There has never been anything between, myself, the King of Lineland, and the Prince of Lineland. It is impossible for you to have come between us, for there is nowhere for you to have come from."

"That doesn't make sense!" you object, "How can a line segment speak or think? Its constituent particles must always remain in the same order. How could new information be encoded in it at all?"

The Sophia-bodies speak in their unison-voice, "Metaphorium, Just One."

"OK, but if the line segment can think and form English sentences, " you reply, "why can't it imagine the second dimension just as we can imagine the fourth dimension."

"Just One," says Sophia in her unison-voice, "Think how much harder it is to imagine a second one of something when you only have ever experience one than it is to imagine a fourth when you have already experienced three. Consider what it would be like if you lived in a universe with 341 observable dimensions. Imagining a 342nd would be quite easy. If you live in a universe with 341 dimensions, then you will have learned to count dimensions and imagining one more is quite easy. Because we only had three dimensions, we tended to treat them as three different things rather than counting them and noting there were three, so it was quite difficult to imagine a fourth or more. For a line segment in a one-dimensional universe, counting dimensions is a very difficult thing to imagine. There is just one dimension. Why would you count it?"

You return your attention to the square and the line segment. The square says, "I will show you what left is." With that the square leaves the one-dimensional universe and circles around to the side of the King of Lineland and pokes him in the middle with a sharp corner.

"Where did you disappear to?" The line segment says confused and then, "Ouch, what just happened? I got a pain in my inside."

"I just poked you from the left," says the square.

The King of Lineland is furious, "I deny that you touched me there. For how could you touch the Line, that is to say the inside, of any Man? It is only possible to touch the nearest ends of the two nearest persons in space."

You laugh, "But it is easy to touch the middle of line from any direction except the direction in which the line points. Why would this King not understand that?"

"Just One," says Sophia in her unison voice, "You are not using your imagination. If the King cannot imagine left or right or up or down but only backward and forward, then he could not imagine his middle being touched at all."

"Right," you say, "but the square has two dimensions so it will be able to imagine a third right." And with that you reach down and poke the square in the middle with the tip of your finger from above.

"Ouch" cries the square, "What just happened? I felt something inside me, in my Area."

A triangle near by pipes up, "That is impossible. You must be imagining it."

You laugh, "I touched you from above."

"Who said that?" says the square.

You laugh again, "I did. I am a three dimensional being and I am above you."

"What does 'above' mean and what does 'three dimensional' mean?" asks the square.

You explain, "Above is another direction, like forward and backward and right and left. Of course, you can't picture it anymore than the King of Lineland can picture right or left. But just as right and left exist, above and below also exist. The King of Lineland is one-dimensional, you, square, are two-dimensional and I am three-dimensional."

The square is shocked, "I never thought of the difference between the King and I as being a difference in degree but instead a difference in kind. I thought I understood left and right and he did not. I never imagined that you could count directions or dimensions as you call them and then add another and another and another. How far does it go up? How many dimensions can there be?"

"Oh, there is only three." You say confidently. Then Sophia laughs in her unison-voice and one of the Sophia-bodies near you shrinks down to a point and disappears, and a moment later you feel a sudden pain deep inside your tummy, "Ouch!" you cry.

The King of Lineland pipes up, "Sounds like Ms. Three-Dimensional just got a surprise! I can't see her and I am pretty certain she is just a hallucination and it is impossible to see her and I don't understand what all this talk of dimensions is, but I feel a certain satisfaction at her receiving a little of her own medicine."

Sophia says in her unison-voice, "I just poked your insides from Fourthward, Just One. You are not the end of geometry. Geometry allows for more dimensions than you can count."

The tiny Sophia-Body below you says, "Let's try explaining four dimensions to Just One in the same way that the sphere explains three to the square in Flatland."

Sophia says in her unison-voice, "Just One, conceive a cube moving parallel to itself (that is to positions which are parallel to its current position) fourthward. Imagine that every point in the cube, that is its entire inside, is to pass fourthward through 4-space in such a way that no point shall pass through the position previously occupied by any other point but each point shall describe a straight line of its own. This is all in accordance with Analogy."

"Um?" you say, confused.

The tiny Sophia-body laughs, "Is it clear to her? Of course not. Her mind having been naturally selected to picture things in three dimensions and being practiced in picturing things in three dimensions does not have the ability to picture things in four dimensions."

A Sophia-body further away says, "So we must try a different analogy, lets talk about shadows of equal extend rectilinear multi-dimensional polygons in branes of one less dimension. Let us start with a square above LineLand."

Another Sophia-body, who happens to be near the square says, "This one will do." and grasps the protesting square from its two-dimensional space.

The tiny Sophia-body says, "For shadows we need a source of light." Her arm disappears for a moment and then reappears disproportionately larger than her and holding a desk lamp over LineLand. A segment of LineLand is brightly lit.

The Sophia-body holding the square places the square between the light and LineLand and a segment of the brightly lit part of LineLand goes dark.

Sophia speaks in her unison-voice, "Just One, let us dissect the shadow of the Square."

The square protests, "I don't want to be dissected, put me back in my world, I am completely confused by all this!"

Sophia continues, "If I hold the square at a slight angle to the light and make the square transparent except for the forward side," The square becomes transparent except for the forward edge, "Note how just a short segment of the shadow near the forward end of the shadow remains." Indeed, it does just like in the image below.

ShadowOfASquareSide1.png

The square protests more, "You are violating my shape-rights. I should not have to endure transparency!"

Sophia goes on, "If I make the opposite edge opaque again then another short segment of the shadow reappears; the part that was furthest toward the rear before." And indeed, it does as in the image below.

ShadowOfASquareSide2.png

The square squeals, "Enough of this. Make me all opaque not just one side at a time."

Sophia continues, "If I return the entire square to opaqueness as it desires, then the segment of LineLand between the two shadowed segments also becomes shadowed." And it does as in the image below.

ShadowOfASquareFull.png

Sophia returns the square to Flatland and the square locomotes some distance away before stopping and looking back.

Sophia asks, "Does that make it clear, Just One, how a two-dimensional square can cast a one-dimensional shadow?"

"Yes," you reply, "pedantically so!"

The tiny Sophia-body's other arm now disappears and reappears a moment later, again disproportionately large and holding a Rubik’s cube. She turns the lamp so that it creates a circle of light in Flatland and then holds the cube under the lamp so that a hexigon of shadow appears in the middle of the circle of light.

Sophia continues in her unison-voice, "Now let us repeat the same experiment with a cube casting a two-dimensional shadow. First, lets make the cube transparent except for one face." The tiny Sophia-body with the huge arm twists the Rubik’s cube and it goes transparent except for the red side which now casts a parallelogram-shaped shadow at one vertex and two sides of the original hexagonal shadow as in the diagram below.

ShadowOfACubeFace.png

Sophia explains, "Now, if we make the opposite side opaque also," and the tiny Sophia-body with the huge arms gives the Rubik's cube another twist and the orange side, opposite the red side, also goes opaque. Somehow the red side remains intact, still covering one face of the cube despite the cube having been twisted. You assume that metaphorium was at work. You look at the shadow and a second parallelogram-shaped shadow has appeared at the opposite vertex for the first as in the diagram below.

ShadowOfACubeFace2.png

Sophia continues, "Note that the three-dimensional shape is casting two simple two-dimensional shadows just as the two-dimensional shape in the equivalent state cast two simple one-dimensional shadows. Now let us make the whole cube opaque again." With that the tiny Sophia-body with the huge arms gives the cube one more twist and it becomes a completely opaque cube again and the space between the two parallelogram-shaped shadows is filled in with shadow. Sophia continues, "The vertexes of the 2 two-dimensional shadows are connected by straight lines and filled in to form the complete shadow.”

ShadowOfACubeFull.png

Sophia continues, “Now let us use a four-dimensional hyper-cube to cast a three-dimensional shadow in your brane, Just One."

"What is a hyper-cube?" you ask, "What does one look like?"

Sophia laughs a unison-laugh and it chills your spine, "Just One, to understand a hyper-cube, it is easiest to work by analogy up from a line segment. Consider that a line segment is a one-dimensional object bound by two zero-dimensional boundaries, points. In all of its dimensions a line segment is the same length because it has only one-dimension. It is bounded by two boundaries because it needs a boundary in each of the directions of its one dimension."

"That is a very strange way to describe a line segment!" you exclaim.

"Just One," Sophia replies, "by describing a line segment in that way I can now give a complete description of a square in almost an identical manner: A square is a two-dimensional object bounded by four one-dimensional boundaries, line segments. In both of its dimensions, the square has the same extent. It is bounded by four boundaries because it needs a boundary in each of the directions of its two dimensions and each dimension is made up of two opposite directions."

You consider this for a moment and then see how useful it is, "So a cube is a three-dimensional object bound by six two-dimensional boundaries, squares. A cube has the same extent in each of its three dimensions. It is bounded by six boundaries because it needs a boundary in each of the directions of its three dimensions."

"Very good, Just One," says Sophia, "Do you care to try to describe a hyper-cube now?"

You hesitate and then just go ahead and try, "A hyper-cube is a four-dimensional object bound by eight three-dimensional boundaries, ..." You hesitate because what you are saying doesn't seem to make sense, "cubes?"

"Yes, Just One," Sophia says, "Continue."

The next part seems easy enough, "A hyper-cube has the same extent in each of its four dimensions. It is bounded by eight boundaries because ..." again, you hesitate but this part seems to make enough sense, "... because it needs a boundary in each of the directions of its four dimensions."

"Perfect, Just One." Sophia says happily.

"But how can a cube bound a space?" you ask, "Wouldn't one face of a cube be all that was necessary to bound a space in one direction?"

Sophia laughs again and you really wish she wouldn't; her unison-laugh is really creepy; it isn't like a group of people laughing together; it is more like one person laughing from many mouths and it is very disconcerting, "You are imagining it bounding a three dimensional region. But it takes a three-dimensional boundary to bound a four-dimensional region. Do not try to picture it, just trust me."

"OK but if a hyper-cube is made of eight cubes," you say, "then it would just be a pile of blocks wouldn't it."

"You are insisting on trying to picture it, Just One." Sophia admonishes you, "Well if you are insisting then let us try to describe the boundaries of the objects we have already described. Those zero-dimensional boundaries of a line segment, the end-points, what if I described them as follows; They have no extent in the dimension they bound but have only extent in the other zero dimensions of the object they bound."

You laugh, "Another peculiar description," you say, "I take it that it can be used to describe the boundaries of the higher objects also? The line segments which bound a square have no extent in the dimensions they each bound but only in the other one dimension of the object they bound."

"Well done, Just One!" Sophia exclaims.

"I think I understand it too," you say, "the line segment that bounds the left side of a square has no extent in the left-right dimension but only in the backward-forward dimension just as the line segment that bounds the forward side (or leading side) of a square has no extent in the forward-backward dimension but only in the left-right dimension."

"Perfect, Just One," Sophia says, "now on to the cube!"

You start, "The squares which bound a cube have no extent in the dimensions they each bound but only in the other two dimensions of the cube they bound. So, the left side of a cube is made of a square with no extent in the left-right dimension but extent only in the up-down dimension and in the forward-backward dimension." You laugh happily, "Well that was actually quite easy."

"Great, Just One," Sophia says, "now on to the hyper-cube!"

"Oh Oh!" you say, a little anxiously, "The cubes which bound a hyper-cube have no extent in the dimensions they each bound but only in the other three dimensions of the hyper-cube that they bound. So, the left side ..." you hesitate confused, "... Sophia, does a hyper-cube have a left side?"

"Well, Just One," Sophia says, "that depends. But assuming that the four dimensions that it is rotated into include the three dimensions that you are used to plus the one I labelled fourthward/anti-fourthward, then Yes."

"Let's just go with Yes." you say not really certain you understood what Sophia just said, "So the left side of the hypercube is made of a cube with not left-right extent ... " you stop again baffled by what you are saying, " ... Sophia, how can a cube have no extent in the left-right direction, wouldn't that make it two-dimensional?"

"Just One, have faith in the analogy," Sophia admonishes, "keep going."

You try again, "So the left side of a hyper-cube is made of a cube with no left-right extent but only extent in the up-down dimension, the forward-backward dimension and ... " you stumble but pick up and continue, " ... and the fourthward/anti-fourthward dimension?"

"Correct Just One,” says Sophia, "Now describe the cube which bounds the anti-fourthward side of the hypercube."

You look at Sophia, horrified, but take a breath and try, "The anti-fourthward side of a hypercube is made of a cube with no fourthward-anti-fourthward extent but only extent in the up-down dimension, forward-backward dimension and the left-right dimension!" you finish with exhilaration, "A normal three-dimensional cube!"

"Very good, Just One," Sophia says, "But all eight cubes bounding a hyper-cube are normal three-dimensional cubes. It is just that all but two of them will be rotated 90 degrees out of the three dimensions that you are used to."

"Oh." you say uncertain if you understand.

"Are you ready to describe the shadow of hypercube now?" Sophia asks, and the two huge arms of the tiny Sophia-body shrink down to a point and disappear, one taking the desk lamp with it and the other leaving the Rubik’s cube spinning in the air and slowly floating away behind you, "I will now point the light back at the three dimensional brane that you occupy." Sophia says and a spherical bright region encompassing you and your elephant and most of the Sophia-bodies appears.

Instinctively, you glance about looking for the source of the light or for your shadow, but you can see neither, "Where is my shadow?" you ask.

Sophia laughs and your elephant squirms at the disturbing sound, "Just One, the light source is to anti-fourthward, so the light and also your shadow are cast to fourthward."

"Oh, Of course," you say because you would rather not admit that you don't really understand what she just said.

"Now, I will place a hyper-cube between the lamp and the brane you occupy, Just One." Sophia says and a strange shadow appears. In the midst of the bright spherical region, there is a sharply defined region without the same glow. You find that your elephant can swim you around the darkened region and you see that it has twelve faces, six faces are rhombuses; 3 at each end and a tube of six elongated parallelograms connect them, "It looks like the shape of a pencil stub that has been sharpened at both ends with three slices of a whittling knife." you say surprised.

"Let us dissect it as we did the other shapes before it." Sophia says, "First we will make the hyper-cube transparent except for the left most cube." All of the shape disappears except for the three rhombuses at one end. But what is revealed is another three rhombuses attached to the back of the first three so that they form a parallelepiped, a flattened cube.

"OK so a regular cube can throw a three-dimensional shadow of a kind of flattened cube if lit from the fourth dimension." you say, puzzling over the geometry, "Makes sense I suppose."

Sophia continues, "Now we fill in the right most cube." And a second flattened cube-shaped shadow appears where the other end of the original whole shadow was. Three of its faces are the rhombuses that made up the other end of the original shadow and the other three rhombus-faces would have been hidden inside the original shadow.

"OK still makes sense." you say.

Sophia finishes, "Now we make the rest of the hyper-cube opaque." And the original shape appears, the 6 additional elongated parallelograms connect the two flattened cubes together.

"OK that make sense by analogy." you say, "But I still cannot picture a hyper-cube."

Sophia laughs again, your elephant swims its legs almost in panic, "Of course not, Just One. Picturing its shadow will have to do for now. Now go back to your normal world and continue to work by analogy. If you are careful, you will find that there are many aspects of a hyper-cube that you can picture."

As you slip out of the realm of metaphorium and back to your reality, you hear one Sophia-body say to another, "Do you think that Just One will ever learn to count time dimensions?"

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