4-d

Factual Questions
1

I know this has probably been mentioned before, but I was sicerely wondering. A number of things, most notably reading about what Tesseracts are, but also working with 3-D vectors in Calc III and 3 dimensional arrays in my Data Structures Class. Easy enough, but I wondered what I would do if I had to go to the next step and do something in 4-D(I don’t think any of my teachers would be that sadistic, but I still wonder).

If we live in a 3-D world, what would a 4-D world be like? How does one even imagine a 4th dimension(assuming Time doesn’t count as the 4th dimension)? I imagine it’s just another layer added on, but it’s hard for me to really imagine how that extra layer would work.

If 1-D is a line, 2-D is a Square and 3-D is a Cube, would 4-D be a series of cubes to make a larger cube(or am I totally off here?)?

2

No, you’re pretty close actually.

A 4-D cube (sometimes referred to as a “tesseract”) is the collection of ordered quadruples (x,y,z,w) with all four variables between 0 and 1. Mathematically, any number of dimensions is the easiest thing in the world to deal with: just add more numbers.

As far as what living in a 4-D world would be like, it might be easier to first consider lower-dimensional worlds. I’d recommend reading Edwin Abbott’s Flatland: A Romance of Many Dimensions. Getting an idea of the difference between 2-D and 3-D should give you ideas for what would change from 3-D to 4-D.

On the other hand, there are a lot of very special things about 3-D space (or 4-D spacetime) that wouldn’t hold on 4-D (5-D) space. Some of them are rather quirky, such as the lack of shoelaces (knots can’t stay tied in 4-D) and some of them suggest that physics itself wouldn’t work in the extended setting (no instanton solutions of Yang-Mills equations in 5-D spacetime). Still, for a rough idea, Flatland is the way to go.

3

Thinking in 4-d is fun, but does involve a little mental gymnastics. It helps to look at patterns in two and three dimensions and extrapolate. The platonic hypersolids are a good place to start.

Consider the “triangles”
(This, by the way, is not supposed to be mathematically rigourous. Just my way of visualising.)

A zero dimensional triangle would be a single point – one vertex.
A 1-D triangle takes that point and extends it to another point. Now we have two vertices and one edge.
A 2-D triangle takes both of these points and extends them back to a third. Now we have three vertices, three edges and one face.
A 3-D triangle, or tetrahedron, extends these three points back to join a fourth. Now it’s four vertices, six edges, four faces, and one cell.
Trying 4-D, you simply extend the four points back into another dimension. I visualise something like a java applet that shrinks a tetrahedron down to a single point as you slide a scroll bar. The scroll bar is merely a way of taking a cross-section at a certain point. Now you have five vertices, ten edges, ten faces (all triangular), five cells (all tetrahedra) and one hypercell.
It’s not hard to see a familiar pattern to all of this.



1
2  1
3  3  1
4  6  4  1
5  10 10 5 1

(I’m not sure what happened to the first column. Maybe someone intelligent can figure it out and tell me.)
Symmetry in this pattern indicates that these are self-dual.

You can do the same with “squares”
0-D – a single vertex
1-D – duplicate the vertex and connect with a line. So two vertices and one edge.2-D – duplicate those two vertices and conect with two more lines. Four vertices, four lines and one face.
3-D – duplicate the four vertices and connect with four new lines. Eight vertices, 12 edges, 6 faces, and one cell.
4-D – duplicate the cube and connect the vertices with eight new lines. Now we have 16 vertices, 32 edges, 24 faces (draw a diagram and count them), 8 cells and one hypercell.
The pattern this time is:



1
2   1
4   4   1
8   12  6   1
16  32  24  8   1

(Working out the number patterns here is left as an exercise for the reader.)
There isn’t the same symmetry here. But they are dual with a family that includes the octohedron.

Here we go: the “octohedra”
0-D – I’m not sure how to do this one, so I’ll leave it out.
1-D – two points separated by a line. I like to think of them as +1 and -1 on a number line with a line segment between.
2-D – here is where you begin to see a pattern emerging. Take the mid-point of the two vertices so far. bump it forwards and backwards in a new dimension, and stretch a skin around it. You should have in your mind a picture of vertices at (1,0) (-1,0) and two new ones at (0,1) and (0,-1) A closed polygon (square) connects the four points.
3-D – Do the same again. This time the midpoint is bumped out negative and positive in the third dimension to form an octohedron with vertices at (1,0,0) (-1,0,0) (0,1,0) (0,-1,0) and two new points at (0,0,1) and (0,0,-1).
4-D – this is a wee bit more difficult to visualise. But essentially you take the six existing points and connect them woth two new points positive and negative in the fourth dimension. I like to visualise a slider bar again that grows an octohedron from a point until it reaches maximum size and then shrinks it back to a point on the other side.
The pattern this time is:



2   1
4   4   1
6   12  8   1
8   24  32  16  1

You can see that this is just the reverse of the “squares”. And sure enough, these are dual with the squares. For example, if you take the midpoint of each face on a cube and join these up, you have an octohedron. The same is true for all other members of these two families – up to as many dimensions as you wish. Although it does fall apart a little in 1-D and below.

I’ll post this, and then continue with the spheres – assuming there is anyone who is interested.

4

Ok then spheres.

Zero dimensions – All you need are two points at +1 and -1 on a number line. Think of them as being equidistant from the zero point. The equation if you want one is x^2=1 The space inside a sphere is called a ball, and is one dimension higher than the sphere. In this case it is the line segment connecting the two points.

One dimension – Now you have the set of points equidistant from a central point on a plane. It looks like the familiar closed curve we call a circle. The equation is x^2+y^2=1. The ball in this case is the round flat thing inside the closed curve.

Two dimensions – The equation this time is x^2+y^2+z^2=1. All the points equidistant from a central point in 3-D space. THis forms a closed surface like the shell of a ping-pong ball. Although the space inside (fill it with sand) is more correctly called a ball.

For three dimensions I pull out my scroll bar again. Imagine the shell of a ping-pong ball that grows from a point until it reaches a maximum, and then shrinks again. The growth is not linear – rather the radius is given by the cross-section of a semicircle sitting next to the scroll bar. (Got that? Oh, never mind.) The equation this time is W^2+x^2+y^2+z^2=1. The four dimensional ball is all the stuff inside.

Four dimensions. If you like the scroll bars, here’s an extension on that idea. Visualise a circle on a flat surface and a mouse pointer able to move anywhere inside it. Now imagine a hemisphere sitting above the circle. Wherever the mouse pointer is, project a line directly up until it touches the hemisphere. The length of that line gives a radius. The four dimensional sphere can be thought of as the set of two dimensional spheres that can be produced as the mouse cursor is moved around inside the circle.

I can go one higher, but that’s about my limit. And I do have difficulty visualising two dimensional spheres linking to form a chain in 4-D, but maybe that’s just me.

5

One more visualisation. This I borrow from playing around with fractals (Julia and Mandelbrot sets), although any function on the complex numbers will work the same way.
If you imagine a complex number plane – real axis at right angles to imaginary axis. Amouse cursor on this surface represents a complex number. This is the argument of the function.
Imagine a second complex number plane. It has a dot marked on it – the value of this dot is the result of putting the mouse cursor through the function. As the mouse cursor moves, the dot on the second number plane also moves.
Now imagine using the mouse to draw a line (straight for now) on the plane. The dot will trace out some weird curve on the second plane. This curve may or may not cross over itself.
Final step – on the mouse plane map out some area – a portion of the plane. This will be represented on the second plane by some shape that is a transformation of the area mapped out. It may be warped, stretched or twisted, or even have rips in it or double back on itself. (Think of a wrinkled sheet that has been ironed wrinkled with bits overlapping.)
The interplay of these two planes is a four dimensional system, creating shapes that cannot properly be viewed in a 3-D world. The idea is not too difficult, but it takes a bit of practice to become familiar with any given function. And then something of an art form to describe it adequately to anyone else.

That’s it from me.
j_sum1

6

A classic fictional depiction of such a thing is in the Robert A. Heinlein short story “And He Built a Crooked House” (available in several anthologies, including this one). For other imaginative depictions of “the fourth dimension” there are several books by Rudy Rucker and the YA novel The Boy Who Reversed Himself by William Sleator.

7

All these are interesting things you can do with higher-dimensional mathematical spaces, but none of them respond to the OP, which was asking about what life would be like if space were a 4-manifold and spacetime were a 5-manifold.

That said: nitpicks

The first bit is simply combinatorial. For “triangles” (all the geometric names you use become rather undescriptive as one generalizes), you’re getting at an “n-simplex”. This is a collection of n+1 linearly independant points v[sub]i[/sub] (0<=i<=n) and all points v such that

v = ?[sub]i=0[/sub][sup]n[/sup] x[sub]i[/sub]v[sub]i[/sub]
1 = ?[sub]i=0[/sub][sup]n[/sup] x[sub]i[/sub]

A subset described by n-k of the x[sub]i[/sub] vanishing is called a k-face, and is a k-simplex in its own right. Now, how many k-faces are there? [sub]n+1[/sub]C[sub]n-k[/sub], which works out to (n+1)!/(k+1)!(n-k)!, which is n+1 for k=0. Nothing “happened” to the first column. It’s not supposed to show up. This is what happens when you trade on mathematical rigor.

Cubes can be considered similarly, although usually one just thinks of it as a sequence of numbers between 0 and 1. There’s really no good reason to call the third collection of objects “octohedra”, except that the “duality” (which, by the way, you never defined except vaguely in passing) gives an octohedron in three dimensions. This whole section is unmotivated numerology and probably confusing to people who don’t already know what you’re talking about.

Your treatment of spheres also mysteriously jumps from the formal “things one unit from the origin” definition to a description of what one would see if a 3-sphere fell through our space. You know: this is exactly why I suggested Flatland. If you give the description of what a 2-sphere falling through a plane looks like from within the plane (a point growing to a circle of maximum size and then shrinking back to a point) then the analogical train is jumpstarted to figure out what a 3-sphere falling through 3-space should look like. By the time you get to 4-spheres, you’ve completely lost the definition and only give a rough idea of how 3-slices behave. Again, if I didn’t already know what you were talking about, I wouldn’t know what you were talking about.

In the last one, I really don’t know why you muddied the water mentioning complex numbers; you never use the fact that they’re complex numbers once. In general, the graph of a function f from a set A to a set B is the subset of AxB of points of the form (a,f(a)). If A is R[sup]m[/sup] and B is R[sup]n[/sup], the graph is a surface in R[sup]m+n[/sup]. Now, letting m and n both be 2 (which is all you really used), the graph sits inside R[sup]4[/sup]. Voila.
[/nitpicks]

In practice, most mathematicians don’t even try to visualize higher dimensional spaces. The closest that they come is more like visual shorthand for giving lectures than anything else. Once you get up to speed it’s a lot easier to just work formally and don’t even think what these things “look like”, which ultimately tells you about as much as knowing what a 2-D or 3-D object in mathematics looks like: very little.

8

You’re absolutely right Mathochist. It was a rather muddled sort of an explanation and definitely not something I’d like to read in a textbook. That said, it is a bit of a story on how I came to have some sort of grasp of the situation one afternoon. And if someone likes to drag their way through a screenful of guff because they happen to visualise things in a similar way, then they are welcome to it.
I like the nice tidy world that mathematicians work in. But things are never that tidy when first learned. They are always a sort of hodge-podge of half-made connections with a few haphazard associations and mental images thrown in and maybe a couple of misconceptions as well.
Let me add that my background is in engineering and not mathematics. And now I am a teacher. I am always interested in what goes on inside kids’ heads and interested in ways of steering and directing that muddle that is there. My brief foray into 4-D and trying to visualise it was an exercise for me in stepping onto the unfamiliar and trying to make some sense of it – in much the same way as some of my students as they learn about trigonometry or algebra or the central limit theorem.
Let me also endorse Flatland. It is available online. Also good is Flatterland. I forget the author. It looks a bit more in detail at modern geometry. It probably helps to have a bit of background first. It is written well enough, but is not a primer for this sort of thing.

[Aside]

Maybe not. But I was recently reading a book on the Riemann Hypothesis. (Prime Obsession by John Derbyshire. Recommended.) In it a statement was made concerning the degree of familiarity with the Zeta function held by those working in this field. Beginning with Riemann himself who was very visual on such matters. It’s clear that they have a far deeper mental construct of what is happening than I do and at least part of it is visual.
[/Aside]

9

I think my first experience with thinking in 4 dimensions was from a book called “The Boy Who Reversed Himself” which is children’s or young adult fiction. It addresses this exact question, but I can remember at least one thing that happens in the book which is probably inaccurate.

First, some explanation. Imagine a creature living in 2-space, who probably looks like a paper doll. Now imagine that it is peeled off the surface of 2-space (ok, so 2-space IS just a surface) and flipped around, then put back into 2-space. Now, they look like their own mirror image, and they see the world as the mirror image of itself. This is what happens to two of the 3-space characters in the book. They reverse themselves by entering and leaving the 4th dimension. When they come back, the hot and cold water faucets are in the wrong place, cars seem to drive the wrong way, buildings are built backwards, signs read as mirror images, etc.

The mistake is that several foods also taste different, because their molecules aren’t symmetrical.

Also, IIRC, there is some talk of 5, 6, 7 and higher dimensions, but very intelligent life only develops in the odd-numbered dimensions.