Tuesday, March 4, 2008

Hypercubes and hypercube nets

Question: What is a hypercube? What are hypercube nets?

To understand the answer, we must first look at cubes and cube nets.

Cube nets

You all know what a cube is, right? In case you are in doubt, I give you the cube.
Ok, so I admit that I actually gave you two cubes. So I like drawing cubes. But if you think about it, I didn't really give you any cubes (sorry, no refunds). A cube is a 3-dimensional object, and all I gave you was a 2-dimensional image. I work with what I have.

The green image is how you would normally draw a cube. The blue image is a little more unusual. In the blue image, the outer square represents the face that is closest to you, while the inner square represents the face furthest from you (as we all know, things look smaller when further away). In both images, I drew all edges, even the ones behind the cube that you wouldn't normally see.

Let's say we want to build a cube by folding some paper. We need one square for each of the six faces of the cube. Now, we could cut out six little squares, and paste them all together. But then, we'd sometimes be cutting squares apart only to glue them back together in the same place. So what we should do is leave some of the squares connected together. Like so!

If it's unclear how this can be folded into a cube, I suggest cutting out this shape yourself, and folding along the dotted lines.

The shape I just showed you is called a "net." A cube net is basically a cube that has been unfolded into a bunch of connected squares. (See MathWorld for a more technical definition.) There are other ways to unfold a cube to form a net. How many ways? Eleven. (Obviously?)

The Hypercube

A hypercube is a generalization of the cube. A cube is 3-dimensional. A n-hypercube is n-dimensional. That means a 3-hypercube is, well, a cube. A 2-hypercube is a square. A 1-hypercube is a line segment. A 0-hypercube is a point. You can also have higher-dimensional hypercubes, such as a 4-hypercube, a 5-hypercube, or a 100-hypercube. Usually when people just say "hypercube", they refer exclusively to the 4-hypercube, which is our focus. The 4-hypercube is also sometimes called the Tesseract.

So I give you the hypercube.

Well, it's not really a hypercube. Again, I can only show a 2-dimensional image. In this case, even if you were to imagine it in 3-d, it would still just be a 3-dimensional image of a 4-dimensional object. The hypercube is hard to draw! I often try to draw hypercubes in the margins of my notebooks when I should be listening to lectures (true story!).

The hypercube image is directly analogous to the blue image of the regular cube at the top. To create a cube, you take the outer, larger square and connect it to the inner, smaller square. To create a hypercube, you take the outer, larger cube, and connect it to the inner, smaller cube. These two cubes are not actually different sizes. The smaller cube only appears smaller because it's in the "back" of the hypercube. Remember how I said that objects further away appear smaller?

I put "back" in quotes because that's not really what it is. It's really in a new direction that we have no word for. See, in 2 dimensions, there are two different directions. One direction is North/South. Another direction is East/West. If we enter the 3rd dimension, we get a third direction: up/down. If we enter the 4th dimension, we get a fourth direction. That's what it means to have an extra dimension.

Hypercube nets

Just like 3-dimensional polyhedrons, 4-dimensional shapes also have nets. To make a net from a 3-d object, we must unfold it and flatten it into 2-d. To make a net from a 4-d object, we must unfold and "flatten" into 3-d. That's right--hypercube nets are 3-d objects! A cube is made up of 6 faces, each one a square. A hypercube is made of 8 "hyperfaces," each one a cube. Finding all 8 in the above image is a difficult exercise.

It's really hard to visualize how exactly you can fold cubes into each other, but thankfully, we have the internet for that. I strongly recommend this site, which shows the folding of a cube and hypercube. Study it carefully.

By now, your mind might be spinning, but there is one more layer of complexity to this. The video linked above only showed one hypercube net. But there are many more, just as there are many nets for the cube. How many more are there? This is a very difficult question, not one I could have figured out myself. The answer, it turns out, is 261. The explanation of how to get there is shown here. Summary of the solution: 50% inspiration, 50% perspiration, 26100% awesomeness.

Concluding remarks

The reason I decided to write this up was primarily because I found out that by merely mentioning "hypercube nets," I became the first hit on Google--not that many people are searching for hypercube nets. My above explanation is very brief, and I just know many people will react with, "Wait, what?"

If you don't understand, I recommend doing some googling, since there are just a ton of people on the internet who have tried to explain the hypercube, with all sorts of approaches. I particularly like this site, which includes excellent videos. You may also drop me a question, though I do request that you be specific. (If you just say, "I don't understand!" I won't know where to begin.)

Anonymous said...

Very nice! I'm familiar with hypercubes, but I hadn't seen the number of cube nets before. Since they are basically a subset of the hexominos, one could figure out what fraction of the hexominos are 3-hypercube nets, and then attempt to generalize this to higher dimensions. I'll have to play with this…

Linda said...

ummm.. I don't understand! ;-)

Miller, geesh... you lost me at about paragraph 6. I do remember seeing some hypercubes when I was googling "four dimensional" a while back. I didn't understand it back then either.

Anyway, are there more than four? Theoretically, could there be additional dimensions that exist parellel or alongside what we can see? Or is that a stupid question? :-)

Am I completely off point?

miller said...

Susan,
Unfortunately, the number of nets for hypercubes above the 4th dimension are not known. We only have the sequence 1, 11, 261. It's difficult to determine any patterns from only three numbers.

Linda,
Theoretically, there can be any number of dimensions, be it 3, 4, 10, or 100. In reality, there seem to be only 3 spatial dimensions (plus one dimension of time). But there might be more hidden dimensions. Perhaps we're like drawings on a flat sheet of paper, and we simply can't see the other dimensions. Perhaps there are other parallel sheets of paper that we will never be able to see.

These are all real scientific speculations in advanced physics. If String Theory is correct, then there are actually 11 dimensions. Why we can't see these extra dimensions is the subject of much research.