you know modern science, need to stop thinking of, "the fourth dimension", (Four-dimensional space, the concept of a fourth spatial dimension. Spacetime, (a infinite cyclical corkscrew, d law 2017 copyright), the unification of time and space as a four-dimensional continuum.), The fourth dimension is all space that one can get to by travelling in a direction perpendicular to three-dimensional space. Whenever an uninitiated person hears this, they start pointing their finger around in the air, trying to figure out how it's possible for such a direction to exist. Such a short explanation gives them no intuitive feeling of the fourth dimension.
the fourth dimension
In order to give you a better understanding the fourth dimension, I will begin with a method that follows a sequence of n-hypercubes that starts with the zeroth dimension and progresses up to the fourth dimension. An n-hypercube is the generalisation of the cube within n dimensions, with a 3-hypercube just being the traditional cube. By seeing each n-hypercube build up from the previous one, you should have a better understanding of the final step, from the third dimension to the fourth dimension.
Step 1: Zeroth Dimension
Imagine a point in space. It is a 0-hypercube. A point is zero dimensional because it has no width, length, or height, and is infinitely small. Every point is exactly the same and has the same measurements, because it has no dimension. Below is a picture of a point, representing the zeroth dimension.
Step 2: First Dimension
Take the zero-dimensional point and extrude it in any direction, creating a line segment, which is a 1-hypercube. All line segments are one-dimensional because they differ in size by only one measurement, length. They all have the same width and height, which is infinitely small. If you expanded the line infinitely, it would cover one-dimensional space.
Step 3: Second Dimension
Now take the line segment and extrude it in any direction that is perpendicular to the first direction, creating a square, which is a 2-hypercube. All squares are two dimensional because they differ with each other in size by two measurements, width and length. They all have the same height, which is infinitely small. All of the edges are the same length, and all of the angles are right angles. If you expanded the square infinitely, it would cover two-dimensional space.
Step 4 - Third Dimension
Take the non-infinite square and extrude it in a third direction, perpendicular to both of the first two directions, creating a cube, which is a 3-hypercube. All cubes are three dimensional because they differ with each other in size by all of the three measurements that we know of - width, length, and height. Just like the square, all of the edges within a single cube are the same length, and all of the angles are right angles. If you expanded the cube infinitely in all directions, it would cover three-dimensional space.
Step 5 - Fourth Dimension
Now, the final step. Take the non-infinite cube and extrude it in yet another direction perpendicular to the first three. But how can we do this? It is impossible to do within the restrictions of the third dimension (which will I refer to as realmspace in this webpage). However, within the fourth dimension (which I call tetraspace), it is possible. The shape that results from this extrusion of a cube into tetraspace is called a tesseract, which is a 4-hypercube. All tesseracts differ from other tesseracts in size by four measurements (equal to each other within a single tesseract) - width, length, height, and a fourth measurement, which I call trength. Looking back to the previous n-dimensional cubes, they all have the same trength, which is infinitely small. Just like the cube and square, all of the edges within a single tesseract are the same length, and all of the angles are right angles. If you expanded the tesseract infinitely, it would cover four-dimensional space.
There are several ways to view the tesseract, and I will show three of them here. The first one is called an inner projection, and it is formed from a projecting the tesseract into realmspace with a perspective projection. The parts of the original tesseract that are farther away appear smaller in the inner projection. The original cube cell that existed before the extrusion into a tesseract is in gray, the paths of the vertices are in teal, and the stopping point of the extruded cube cell is in blue. The real tesseract isn't shaped like the inner projection shown below - the inner projection is a very distorted "image" of the original tesseract. All of the edges you see in the image are actually the same length as each other, and all angles between edges are right angles.
The second way to view a tesseract isn't actually a normal tesseract, but a parallel projection of a skewed tesseract. To make this shape, first you make a tesseract, then shift the top cube cell a short distance in a diagonal direction, parallel to realmspace. Since this shift is parallel to realmspace, it can actually be in any direction that you can point to. After the shift, you trace the shadow of the skewed tesseract's edges. The result is a shape that has two cubes with their vertices connected together. In the orignal shape, all of the edges within the cube cells are the same length and have right angles with each other. However, they don't have right angles with the teal connection edges, and the teal connection edges are slightly longer than the cube cells' edges.
The third way to view a tesseract is a parallel projection into realmspace. It is the same as a skewed tesseract, but without the top cube cell shifted. Since the edges of the tesseract were extruded in a direction perpendicular to realmspace, when the shape is projected back into realmspace, the edges of the blue cube cell are projected straight back onto the gray cube cell's edges. The resulting projection is a simple cube. This didn't happen with the inner projection, because that projection was a perspective projection.
This last step of trying to view a tesseract shows the difficulties in portraying objects from tetraspace within the limitations of realmspace - there is an extra perpendicular direction that we can't depict within our own space without distorting the original object. Because of these problems, it takes many examples in order to begin understanding the nature of the fourth dimension.
You have now seen a glimpse of the fourth dimension. This is only the beginning
the second dimension to the third dimension
Tetraspace (four dimensional space) is a fascinating place. Events that would be bizarre and mind-bending to us puny little realmic beings would be common-place and taken for granted in tetraspace. Jumping straight into thinking about tetraspace is enough to boggle any human mind. An indirect route must be taken if there is any hope of understanding the possibilities of tetraspace. The method I use here is the one pioneered by the books Flatland and Sphereland, and is the one used in just about every other text about the fourth dimension. First, the I tell of the adventures of a planar being (a two-dimensional being) attempting to understand realmspace. Going from planespace (two-dimensional space) to realmspace will shed light on how to conceptualize going from realmspace to tetraspace. While going from planespace to realmspace may seem overly simple, it is necessary to understand exactly what is happening so that you can understand the process of going from realmspace to tetraspace.
Just as tetraspace is mind-boggling to us realmic beings, realmspace is mind-boggling to planar beings. Things we take for granted are extremely difficult for planar beings to conceptualize. The only thing a planar being can see of our realmic objects is their planar cross sections in his planespace, and the effects of them rotating. At first, he will only be able to conceptualize the effects in planespace of realmic objects. He will only see planar objects in his mind. Eventually, with much deep thought and introspection, he may be able to conceptualize the realmic object itself. Similarly, when starting to think about tetraspace, we will only be able to conceptualize the effects of tetral (four-dimensional) objects in realmspace. We will only be able to see realmic objects in our mind at first. But, by the effects we see, we can carve a sculpture in our minds of how the full object really looks like, and conceptualize the whole of the tetral object.
The name of our planar subject is "Fred". He lives in a plane called Flatland. The only directions that Fred knows of are forward, backward, up, and down. Fred has no concept of right or left. His viewing area is merely a vertical line. Here is a picture of Fred in his native habitat:
Since Fred's viewing area is only a line, he can't see as much at once as we can. Here is the view that Fred would see if he was in the picture above:
Unfortunately for Fred, he can't see backwards - he can only see forwards. It is impossible for him to simply spin around and face the opposite direction. It is easy for beings in realmspace to face a different direction because they can just spin around, left or right. The only way for beings in planespace to face a different direction though is to turn their head down and heels up. Fred either has to stand upside down to be able to see what is behind him, or he has to walk backwards blind because he only has an eye on his front side.
Fred's writing system is very rudimentary, since a linear surface doesn't allow much detail. His writing system looks somewhat like Morse code. Here is what a page from one of his books looks like, in his view:
Fred understands the concept of a square. It can be imagined as a series of lines stacked side by side, so that the resulting shape has four sides. He has no concept of what a cube would look like, though. The best he could do is imagine a succession of squares next to each other. As you can see, this is far from the actual representation of a cube. A cube is a series of squares stacked on on top of the other:
Fred also understands what a circle looks like. It can be imagined as a series of lines stacked side by side in the way that a square is formed, but the lines start small and expand to a certain size, then shrink down again at other side.
Fred can't understand what a sphere really looks like, though. He can imagine it as a series of circles, starting small and expanding, then shrinking down again. Here is how he might picture it:
Or, he might decide to picture it as a group of concentric circles (circles with the same center):
But, as you can see, that's not really what a sphere looks like. Now imagine Fred one day is sitting in his living room, and a sphere from realmspace passes through the air right before his eyes. To him, it will appear as if a circle appeared out of nowhere in front of him. It expands rapidly, and Fred fears that it is going to devour him and his house with him. But to his relief, its growth slows down and stops. Then, the circle starts shrinking again, but slowly. Then it starts to shrink faster and faster until it quickly disappears. Here is what the event looks like:
the third dimension to the fourth dimension
In realmspace, there is a character named Bob. He is basically your average guy - two legs, two eyes, a brain, all the usual body parts. His view of his world is a 2 dimensional plane, one dimension more than Fred has. Here is Bob in his native habitat, and his view of that habitat:
Since Bob has one dimension more than Fred has, his writing system can be more elaborate. He can write on a 2 dimensional surface. Instead of only one line of text on a page, there can be multiple lines. This allows a lot more text to me stored in a realmic book. Here is what a page from one of his books looks like:
Bob fully understands the concept of a cube. A cube is a sequence of squares laying top to bottom flat on top of each other. For any one square in this cube, every point on one surface of the square touches every other point on the surface of the square above it, and on the other side, touches every point on surface of the square below it. Here is an illustration:
Bob also understands the concept of a sphere. A sphere is a sequence of concentric circles laying top to bottom on each other, as with a square. Like a square, adjacent slices of the circle touch each other at every point. Here is an illustration:
Bob has troubles imagining a tetracube, however. Here is the best he can do to imagine a tetracube:
A tetracube is actually a sequence of cubes, where every part of a cube in this tetracube (all the insides of it) touch all the insides of the cube to its right and to its left (or its up and down, or its front and back). In actuality, the direction of the adjacent cubes are not imaginable in terms of 3d directions. I will call one of the directions into tetraspace Upsilon (for up) and the other Delta (for down).
If Bob was viewing Fred's planar world, he would see that there are two sides to it. Bob could either be on top of Flatland or below it, but in order to get to the other side, Bob must go through Flatland. There is no other way to get to the other side. Planespace separates realmspace into two separate parts. In the same way, a tetral character (Let's call her Emily) would see that she is on one side of realmspace, and she'd have to pass through Bob's world to get to the other side of it. Realmspace separates tetraspace into two separate parts.
In the first illustration below, an arrow shows the path from one side of Flatland to the other. If the arrow was an object, Fred would only see a small point. In the second illustration, a tetral arrow that passes from one side of realmspace to the other would intersect realmspace with a small point. Bob doesn't see any other part of the arrow but this:
One day, Bob is innocently sitting in his living room (as Fred was that one day), and all of a sudden, in the middle of the air, a sphere appears and grows rather rapidly. Its size levels off at a point, then starts shrinking slowly. Its shrinking quickens until it completely disappears. Bob just experienced the passage of a tetrasphere (four dimensional sphere) through realmspace.
Flatness and Levitation
Below is a table showing the relation between the dimensions:
persondimensionviewaxis of rotation
--- 1st point(0D) no rotation
Fred 2nd line(1D) point(0D)
Bob 3rd plane(2D) line(1D)
Emily 4th realm(3D) plane(2D)
I'm now going to describe in more detail the worlds of Fred, Bob, and Emily. Fred doesn't actually have much of an area to live in. His little 2D plane is actually hanging on the wall of Bob's living room. Bob can walk over and observe Fred any time he likes. Fred's world has a ceiling, a front, a back, and a floor. Bob's world, actually, isn't very big either. He has a ceiling, 4 walls, and a floor. His 3-D world is also hanging on a wall: in Emily's living room. Emily's world could also be hanging on someone else's wall, and so on. Here is Fred's world on Bob's wall:
If a square is rotated in fred's world, it rotates around a single point. This means that only a single point on the square stays at its original location as the square spins around. Thus, the axis of an object in planespace is zero-dimensional.
Now imagine that Bob has Fred's world resting flat on a table. Bob has placed a cube on Fred's 2d world, as if the cube is a paperweight to hold down Fred's world from fluttering away in the wind. Fred's world is sandwiched between the table and the cube paperweight. The bottom surface of the cube, which is intersected with Fred's world, appears to Fred to be a simple square. If this cube is rotated, it appears to Fred as though a square is rotating around a point. In reality, it is a cube rotating around a line, whose axis extends perpendicular to Fred's world from the point of rotation. This means that all of the points along the axis will stay at their original location as the cube spins, but every other point in the cube will move in a circle. Thus, the axis of an object in realmspace is one-dimensional.
Now imagine that Emily lays Bob's world flat on a tetral table. The direction towards the table from Bob's world is delta (down in tetraspace), and the opposite direction into the air above the table is upsilon (up in tetraspace). Emily places a tetracube on top of Bob's world on the upsilon side of it. Bob's world is now sandwiched between the tetral table and the tetracube. The tetracube is intersected with Bob's world, so Bob sees a cube floating in mid-air. Now let's say that Emily rotates her tetracube around one of it's edges. Bob will only see a cube rotating, but the tetracube it's attached to is rotating with it. The axis of rotation for the tetracube isn't a line, though. It's a plane formed by extending the linear edge of the cube into Emily's world, perpendicular to realmspace. All of the points on this square stay in place as the tetracube rotates in tetraspace. Thus, the axis of an object in tetraspace is two-dimensional.
Imagining how the linear axis of a cube could extend into tetraspace is hard. We can try to picture it by imagining its axis shifting off in a random direction as pictured below, even though this isn't what is truly happening:
Fred is sitting in his living room the next day, and there is a square object floating in front of him in mid-air. Bob decides to play a trick on Fred. He grabs an edge of the square and pulls it into realmspace, leaving a linear edge in Fred's universe remaining. Fred only sees a line left of the original square. Here is what Bob just did:
Bob completes the rotation and places the square fully back into Fred's world. But, now the edge that had disappeared is on the opposite side. Fred is now astonished to see that the square has been completely inverted. Such an action would be impossible in a two dimensional world. Fred tries to invert the square back to its normal state, but no matter how many times he turns the square around, it doesn't work:
The square has become its mirror image, with its left and right sides swapped. Fred has difficulty understanding how this could have happened to his square. The first way he tries to imagine it is to imagine that one of the sides has been pushed through the middle of the square and to the opposite side, making everything inverted. This is how he imagines it:
Fred contemplates what happened some more, but can't quite wrap his mind around it. He thinks about another way it might have happened. One side could have been shifted upwards and towards the other one until one side was above the other, then shifted further until it came into place on the other side. The moved side would have made a half circle around the side that stayed. Here is how he imagines it this time:
But, this is not what really happened. It is merely a futile attempt of Fred's to understand what's happened in the third dimension using only what he knows of the second dimension. The effects of Bob's actions are easy to see, but explaining them to Fred is not.
Emily sees what Bob did to Fred, and decides to play a similar trick on him. Bob is sitting on his floor staring at a box that he has just filled with his favorite sentimental possessions, when all of a sudden the whole box except one of its sides disappears. Here is what he sees:
He stares at it astonished, and is even more surprised when all of a sudden the rest of the box appears again. But, the box has just appeared on the opposite side of the face that had remained. All of the lettering on the box has been turned into its mirror image. Here is what the box looks like afterwards:
Bob looks into the box, and finds that something has happened to all the letters in the books that were in the box - they are all their mirror images! He takes one of the books and puts it up to a mirror, and finds that the mirror reverses the letters back to their original state. As Fred did, Bob tries to understand how this could possibly occur. He tries to imagine one side of the box being squished into its opposite side, but being pushed further until it comes into place on the other side of the box from where it was before. All of the letters have been inversed:
This is analogous to Fred's solution with the disappearing side being squished through the remaining side. It isn't what really happened in tetraspace, though. Bob contemplates further, and again conceives of another idea like Fred did. Maybe one face of the box was shifted in a half circle around the side that remained. All the letters would end up being reversed just like in the previous theory:
As we know by looking at Fred's example, the size and shape of the square don't actually change as it's rotated through realmspace. The only ways that Fred has thought of to explain what happened involve the square changing shape. The real event doesn't require the square to change shape at all. It is the same situation with the cube. As it is rotated through tetraspace, the actual shape of it isn't changed at all; it remains a perfect cube. The only way we can imagine the result, however, is by distortions in realmspace.
Flatness and Levitation
When Bob watches Fred's world, he can see every point in his world at once. He can see Fred's insides, what's inside his safe, what's underground, where the quarter he lost is, everything. Fred is basically naked to Bob, even though he has clothes on. Bob would be able to touch Fred's insides without having to go through Fred's skin, and there's nothing Fred would be able to do about it to stop him.
Bob doesn't realize it yet, but he is also exposed. Emily can see every part of his world at once. She can see inside his tightly locked safe, into his refridgerator, into his desk drawers, and she can see his insides. Bob is basically naked to Emily, even though he feels fully clothed. She can touch any part of his insides without having to go through his skin or his clothes.
One day, Bob is feeling particularly devious and decides to remove the contents out of Fred's safe. He walks over to Fred's world, removes Fred's money from inside the safe and sticks it onto Fred's table. Bob doesn't have to open the safe because he can just bring it out of planespace into realmspace, move it, and then put it back into planespace. The money doesn't have to go through the walls of the safe because it can just pass "over" its walls in realmspace.
Fred comes in from a leisurely walk in the woods and is astonished to see all his money lying on the table. He was sure he had put it in the safe the night before. He checks out his safe, but nothing has happened to it. He opens it up, and it's empty. He sticks the money back into the safe, and makes sure it's closed securely. As soon as Fred closes the safe, Bob grabs the money again and puts it on the table behind Fred. Fred turns around to see the money he had just secured now lying behind him. Now Fred thinks he is going crazy.
Emily has seen what Bob did to Fred, so she decides to teach him a lesson by doing the same thing to him. Bob has his journal locked up in his safe, which has both a combination lock and a key lock on it. Bob knows his safe is secure because he bought the best safe on the market and only he knows the combination to it. Emily removes the book from the safe by pulling it into tetraspace. She moves it over to Bob's desk and reinserts it back into realmspace. The book didn't have to go through the walls of the safe because it could go "over" its walls in tetraspace.
Bob is reading another book and looks up to see his journal sitting right in front of him. He is quite stunned and confused, because he was sure he had put the journal into his safe. He checks out the safe, and there seems to be nothing wrong with it. He takes the journal and puts it back into his safe, making sure it's definitely locked. He even ties a rope around the safe so that he can see if anyone tampers with it. He returns to his desk and resumes his reading. Emily grabs the journal again, moving it to tetraspace and inserting it back on the table in front of him. When Bob looks down and sees the journal in front of him again, he thinks he is going crazy. He looks back to see the rope still tied around the safe, exactly the way he left it. Then he remembers what he did to Fred, and realizes that someone else may be doing the same thing to him.
Bob decides to have some more fun and tease Fred again. Fred's world is hanging on Bob's wall, but not tightly. Bob takes a nail, and right in front of Fred's view, he hammers the nail right through Fred's world. All of a sudden Fred sees a circle appear in midair. He decides to investigate, so he grabs it and tries to move it, but it won't move. It is thoroughly fastened in mid air.
This is rather bizarre and he decides to try some things with the floating circle. He sits against a wall of his house and pushes against the circle with his feet. It requires a lot of effort, but the circle moves. It feels like he is holding up the whole world, and in reality that is what he is doing. When Fred lets go of the circle, it swings back to its original position and Fred falls to the ground.
Bob removes the nail, so Fred sees the circle suddenly vanish. Bob raises Fred's safe into the air and drives a nail through the middle of it. He lets it go, and the safe hangs in mid air on the nail. The nail didn't have to go through any of the walls of the safe in order for the safe to be hanging. The nail just went through the air through the inside of the safe. This is similar to hanging a loop with a nail on the wall in realmspace - the loop itself doesn't have to be pierced, but the nail has to go through the area in the middle of the loop.
To Fred, the circle was just a strange object hanging in mid-air, but now Fred's own safe is hovering. Fred opens his safe and looks inside, and to his surprise he sees that the circle has reappeared, but now his safe is hanging on it. Fred removes the safe from the "hook" and puts it back on the ground. This has been enough bizarre events for one day, so he goes outside to get a breath of fresh air.
Emily sees that Bob hasn't learned his lesson. She does the same type of thing to Bob with the hope that Bob will realize what's happening to him. She has Bob's world hanging on her wall and she drives a "tetranail" (four-dimensional nail) through his world. Bob sees a sphere suddenly appear in midair near a wall. He remembers what he just did to Fred, and to test out his hypothesis he does the same thing that Fred did. He sits back against the wall and pushes against the sphere with all his strength. He succeeds in pushing it a few inches, but it requires a lot of effort. When he lets go, the sphere swings back to its original position and he falls on the floor. Bob is starting to realize that something else is out there.
Emily also performs the "Levitating Safe" trick. She raises Bob's safe into the air and drives the tetranail through the air in the middle of it. Emily didn't have to pierce any of the sides of the safe in order to hang the Bob's safe on her wall. Bob opens the safe and sees that the safe is hanging on the sphere, just like Fred's safe was hanging on the circle. He pulls the safe off of the levitating sphere and places it back on the floor.
Emily decides to play another trick on Bob. Bob has a plant hanging from his ceiling by a chain. Emily holds the plant in the air while she pulls one of the chain links into tetraspace. She puts the chain link back into realmspace at a different location and lets both the plant and the chain link drop to the floor. Bob sees this and walks over cautiously to inspect the scene. He sees a plant on the ground with dirt everywhere, a single chain link intact sitting next to the plant, and the rest of the chain hanging from the ceiling, intact. None of the chains had to be bent or broken open for Emily to perform her trick. Bob couldn't have played this particular trick on Fred because chains aren't possible in planespace.
Bob decides to continue playing tricks on Fred so that he can see what the equivalent is in his own world. Bob decides to try rocking Fred's world back and forth along the wall. Since Fred's world is swaying back and forth, he can't hold his balance and falls over. One surface of Fred's world scrapes against the wall, but on the other side is only in contact with the air in Bob's world. Fred's world can only move two directions and still be in contact with the wall. If Fred's world is moved in a third direction (outward), it is no longer in contact with Bob's wall.
Emily watches what Bob has done and does the same thing to Bob. She has 4 directions she can push Bob's world to and still have it scrape against her wall. First she pushes it front and back, then left and right. To Bob it feels like there's an earthquake happening. Emily stops Bob's world from moving, then starts it moving around in circles. This whole time, all of Bob's world is scraping against Emily's wall, not falling off. If Emily moves Bob's world in a fifth direction (outward), then it is no longer in contact with her wall.
Wheels in the different dimensions
Imagine Bob has a tire (which is a type of cylinder) on a piece of paper. The tire is on one of the corners, and Bob lifts up the corner. The tire will travel to the opposite corner of the piece of paper, leaving a rectangular track behind it. Here is what it would look like:
Emily could do a similar thing in tetraspace. She has a swock and a tetratire. A swock is a four-dimensional piece of paper, which would be a cube in realmspace. Her tetratire is a type of cubinder, which is confined to movement along a line. If she sticks her tetratire on one corner of her swock, at lifts the corner "upsilon" (4D up), it would roll to the opposite corner of the swock. If her swock is laid flat onto realmspace, here is how the track would look across it:
Bob can have his tire pointing in any direction to another side of his piece of paper. No matter how he lifts his piece of paper, the tire would either travel straight to the side it's pointing to or just fall over.
Emily can also point her tetratire in any direction she wants to. She has a lot more directions to point it to than Bob does:
If a tire drove straight over the middle of a piece of paper, parallel to two of its sides, its track would look like this:
In the tetraspace, if a tetratire drove straight over a swock parallel to two of its sides, its track would go right through the middle. This means that there are three paths a tetratire can take, as opposed to Bob's realmic piece of paper, where there were only two. There is an extra parallel path for each dimension.
If you shine a light over the top of a wheel, its shadow would look like this:
Notice that the shadow is two dimensional. The tire doesn't actually touch the paper except at the contact point, but it lies above all the points where the shadow is. Similarly, the shadow of a tetratire would look like this in realmspace:
The tetratire only touches realmspace at the contact point, and the rest is a shadow of the tetratire in tetraspace. If a realmic tire is flat, it touches more of the paper at once, spreading out; if it is filled with air and very tight, it touches less of the paper at once:
Similarly, a tetratire would cover more "volume" if it was flat, and less if it was full of air:
If a tire has some sort of pattern on it, and you watch only the contact area of the tire, it's the same as taking binoculars and sweeping them across a scene to see a little bit of the scene at a time. If you focus on the area at the bottom of the tire and track the binoculars as the tire rolls, you know that the tire doesn't "drag" along the surface it's rolling on, it "rolls" along the surface.
The same type of thing will happen with a tetratire. If you watch only the contact point, you will see the dark area "passing", but it isn't actually dragging, it's rolling along the "flat" 3D surface:
When falling, as a tire tips to its non rolling side, its contact surface will lift from the ground and only a line will be in contact with the ground. When it hits, a circle will suddenly appear two-dimension wise. Here is how it looks:
A tetratire would have a total of four directions to fall to. Here is a picture of all the possible ways a tetratire could fall:
A spherical tetratire (which is a type of spherinder) has four directions to move to and two directions to tip to. A spherinder is less likely to fall over, but it will roll around a lot more. Here is a picture of its roll directions and the spots it can fall to:
The track shape for a tetratire or cubinder is a cube, while the track for a spherinder or spherical tetratire is a cylinder. For a glome, the track is a sphere. The duocylinder, which is not pictured, has a cylindrical track, but it's direction of movement is along the symmetry axis. Here is a table and illustration for each:
tetrashapetrack on 3D surface
Picture Bob holding a book in the air. He has a ball on the surface of the book. If Bob moves the book around in a circle, the ball will travel in a circle in the opposite direction. The ball is basically moving in a circle around a linear axis:
Emily would be able to do the same thing with a tetrabook. Her ball is called a gongyl, the solid counterpart of the glome. The gongyl would have 3 axes to circle around, instead of just one for Bob's ball.
War in the dimensions
There is an ancient story in a 2d world named Myrandia of a great battle that was fought over an island called Yulomar that separated two oceans between the land of the Washtaks and the land of the Bicks. On any 2d planet, a land mass in a body of water separates the body of water completely in two. This particular island was of great strategic importance because it was halfway between the two main continents that the two countries inhabited.
The Washtaks were from the west, and the Bicks were from the east. The Washtaks caught wind of the Bicks' plan to take over the island and create a fortress there. They had to act fast. Their leader, a tyrannical type in the likes of Saddam Hussein, ordered a large number of boats to be constructed. He had each household in the city that was closest to the sea construct one boat. The thing you must understand is that since the surface of a 2d planet is linear, to pass anything while traveling, you have to go either over it or under it. To store in one place the number of boats that was needed would have required an extremely deep hole. Deep holes are problematic in 2d because of numerous reasons. Air pockets can form extremely easily, easily suffocating anyone trapped inside. Corkscrew drilling is impossible (3 dimensions are required). Limitations in 2d make extracting dirt from the deep quite difficult. The best way to store things is to spread them out across the surface.
The residents of the city slaved away for weeks, constructing the boats in the least time possible. Numerous skipped meals and lost hours of sleep later, the army's water transportation had been completed. The leader ordered the advance to begin. The residents of the city descended into their underground homes as the army lumbered overhead. The leading regiment of the army hopped into its boat and pushed into the water; subsequently, one boat after another was carried to the shore, tossed in, and the regiment hopped in to follow the boat in front of it. Once in the water, the boats were not able to change their order.
Traveling in a boat in 2d isn't on any top ten lists for exciting multidimensional tourist activities. It consists of staring at the back of the head of the person in front of you for the duration of the trip. Many great philosophical realizations have occurred during boat trips, because that's about the only to do with yourself. After the army hit ground on the shore of Yulomar, the Washtaks charged across the land in an attempt to secure the island before the Bicks arrived. Unfortunately, the Bicks had also arrived on the opposite shore just before them. Like the Washtaks, they sprinted across the terrain, trampling plants and small animals in a desperate attempt to claim as much land as possible. In 2d, the single soldier at the front of the army is the only person that can fight. The battle consists of the two front line soldiers battling it out until one is taken out, then the next person steps up and goes at it. The battle proceeds until one of the armies has been annihilated.
So, the forces clashed head on near the middle of the island and the battle began. If traveling in a boat in 2d is the most boring thing to do, waiting in the middle of an army while the two people on the front lines fight it out is the most nerve-wreaking. It would be like standing in a tight ditch so thin you had to walk through it sideways, with the sun's rays pounding on your skin. All you can do is stare at the back of the head of the soldier in front of you, study every bump and imperfection for several hours as your legs became sorer and sorer. Sweat drips into your eyes and stings like salt on an open wound. You know that as soon as your comrade falls, you are next in line, and that you have to kill an entire army just to survive. Here on earth we at least get the comfort of the possibility of survival. In 2d, the only warning you get is your buffer man dropping in a bloody mess in front of you with an ugly face staring at you from the other side as he jabs his sword at you. The average soldier has fought in approximately 0.0 battles. Soldiers are ordered to kill the person in front of them if they try to retreat. Pure havoc would result if this rule wasn't in effect.
Soldier after soldier in the Washtak army fell at the hands of the Bick army. The situation wasn't looking good for the Washtaks (even though they couldn't know given their viewpoint). The battle came to the last man surviving in the whole Washtak army, a young soldier named Dosha. He had waited in the far rear of the army for hours as the battle raged on, far away on the opposite side of the army. He considered ditching the battle since it was apparent that it would never come to him. But then he realized that if the rest of the army in front of him was killed, he'd be the only one protecting his country. Late afternoon, when his hunger had reduced him to considering passing insects as a source of food, the man in front of him suddenly dropped to the ground and an ugly Bick face stared at him from the other side.
After an intense duel with swords clanging and dirt flying, Dosha performed a quick feint and nailed his opponent hard. He proceeded to take out Bick after Bick as he fought on pure adrenaline. Feint, jab, feint, jab, he performed countless times, eating up the seemingly endless stream of them. One of them falls, another is waiting behind him to take his place. Just as twilight approached, he performed his trademark maneuver as usual on his opponent before him. But, the view he laid his eyes on was the most beautiful he had seen in his life. The sun was just flashing out of existence. He was the only man left, the last soldier standing. He was standing on top of the remains of hundreds of dead bodies, the remains of the two largest armies that had ever existed on Myrandia.
Battles in the 3rd dimension run quite differently than in the 2nd dimension. The meeting point of the two armies is a line instead of a point. Maneuvering, which was impossible in 2d, takes on a whole new importance. Lines can be flanked, a front line can retreat through the line behind it, rear lines can see through the front lines to launch stuff into the enemy, such as arrows, cannonballs, and other flaming objects. The element of surprise factors in a lot more. In 2d, you know your enemy is in front of you; in 3d, he can be in any direction, since he could have circled around to your rear. Soldier survival rate is a lot higher, but then there's hardly a possibility of a one-man army as is easily do-able in 2d. Overwhelming numbers, given other things equal, will win.
[ARMY MANEUVERING IN 3D]
If higher numbers help in the third dimension, they are even more of an advantage in 4d. A battle is fought on a "realm". A realm is the analog of the plane in 3d, line in 2d, and point in 1d. Our entire universe is an infinite realm, just as an entire 2d universe is an infinite plane. To picture a 4d battlefield, we use all of the x, y, and z coordinates. Maneuvering in a 4d ground battle would work the same as a space battle in 3d. The battle fronts of the armies are planes. An army with the amount of soldiers of a 3d ground army could all easily fit on the whole front of a 4d army. Individual targets can be destroyed much quicker because so much more firepower can simultaneously be focused on it. A much larger army is required to prevent passage; thus, it is much easier to circumvent or punch through enemy lines,
Bodies of Water
If one were to conduct a search for a missing person on a 2d world, all that would be required is to walk across it, because all of the surface of the island would be covered in one crossing. An island completely separates the bodies of water on either side of it.
In 3d, the surface of an island is planar, so one must cross the island multiple times at different points to cover the whole island. If one person follows the shore of a 3d island in a clockwise direction and another in a counterclockwise direction, they will meet each other at some point because they are following a one dimensional barrier between land and sea.
In 4d, the surface of an island is "realmar" (it is three dimensional). The shore is 2 dimensional, so if two people encircle an island in opposite directions, they won't necessarily meet up with each other. It's just as likely two people crossing a 3d island from random points will meet up with each other. Two large continents can be connected by a thin strip of land without actually blocking off water passage because boats can just circumvent it.
If there was a permanent river on a 2d world, nothing below the source of the river would be inhabitable. A river covers every inch of the mountain below where it starts. Only one river is possible on a particular slope (because two rivers would essentially be one river with a second source partway down the slope). A bridge to cross a river in 2d is like making a bridge to cross a flood in 3d.
In 3d, infinitely many rivers are possible on a mountain, so long as it is large enough to hold them all. A river in 3d separates the land on either side of it, so that a bridge is necessary to cross it. Rivers naturally snake back and forth.
In 4d, one can merely walk around a river. No bridges are needed. Making a bridge across a river in 4d is as pointless as us making a bridge over a pond instead of just going around it. A river in 4d would corkscrew.
There is only one possible form for a lake in 2d, and it looks like so:
In 3d, there are two types of lakes: linear and roundish. A linear lake can be crossed by a bridge easily, but a roundish lake is quite difficult to cross with a bridge:
In 4d, there are three types of lakes: linear, flat, and globular. For a linear lake, no bridge is necessary. For a flat lake, a bridge is easy to construct. For a globular lake, a bridge would be a difficult task to create.
The higher in the dimensions you go, the more porous things become because of the additional directions for things to slip through. In 2d, air can be trapped quite easily. Only two points of contact are require to seal it in. When water runs over the ground in 2d, water can only sink in so far because a seal is formed so easily at all points along the slope.
In 3d, a liquid or gas must be sealed in by a linear contact. When water runs over ground in 3d, some of it seeps into the ground because of the air gaps between the particles.
In 4d, a liquid or gas must be sealed in with a planar contact. Gasses and liquids can escape containers much easier in 4d. The ground can be much more porous in 4d and still have the same strength as 3d ground. Organic matter could be composed of a higher percentage of water than in 3d, i hope that clears that up as, standardz, hahahahaha, :) #edio