Life in 4d

My guess is that anyone who reads this has at some stage heard some nerdy person talking about stuff in 4d, and not been able to understand. Honestly, the ideas aren't that hard. The trick when dealing with these sorts of problems is to phrase them in terms that people understand.

So lets start simply. What does 0 dimensions look like? A 0 dimensional shape has no length, no width, nothing. It's just a point. My incredible paint skills allow you to see one of these for your self. Aren't I kind?

Aren't I kind. I guess the question to ask is this... Now that I've got 0 dimensions, how can I make 1? One way to do it is to make 2 0-dimensional points, and connect them.

And now we have a 1 dimensional thingamajiggy. I'm going to call it a 'line'. How do we make a 2 dimensional shape? Connect 2 lines, of course.

The red lines can be the one that are connected to the pre-existing blue ones.

I hope you can all see the pattern. More than that, I expect many of you have drawn cubes in this way... if not, you should try it. It's pretty cool.

So again, we've started with 2 2-dimensional shapes,joined them at the corners, and we get a 3d shape. Now that the pattern is firmly established, let's try using it to gain insight into what a 4d shape could look like

At this point I should probably explain why this is an insight, and not a full understanding. We can visualise shapes and bodies in dimensions up to 3 dimensions, but our world is not 4d. 1 way to see this is to imagine what a sphere would look like in a 2d world. The 2d version of a sphere is, predictably, a circle. When a sphere travels through a 2d plane, you can't really understand it as a sphere, rather as a circle whos radius increases and decreases.


This is my attempt at drawing a sphere passing downwards through a horizontal, flat plane. If you can't see the image in your head, get an orange and cut it anywhere. You'll always get a circle in the flat part. Now imagine cutting circles in it every 5cm upwards... you'll get bigger and smalle circles every time, depending how close to the middle you cut.

So imagine you lived in that 2d world. You're able to see the 2d equivalent of the 3d shape, but you'll never really understand what the sphere looks like because you can't imagine the extra dimension necessary. That's the basic issue we face. That said, we can still move our 4d cube (called a 4d hypercube) around in 3d (well, 2d actually. It's a computer screen, not a hologram... yet), to see what it looks like.

Here are 2 different rotations of the 4d hypercube, in 2d.


AAAH! These may look complex but when you look at it in terms of what you know, this becomes easy. I'll start with the one on the right. Can you see the cubes on the inside and outside? So it's all the same as the other ones. The one on the left looks more all over the place, but it's not so hard to visualise either. If you pick, say, 8 of the closer dots, you can probably make a slanted cube. Choose another few dots! There are actually 8 cubes to be made.

Sadly, this link is where my ability to draw ends, and my linking to other peoples' work begins. Imagine this as a pair of cubes, looking around eachother.

That's all i've got to say on this post. I'll leave you with a question. How many vertices (corners) does a 4d hypercube have? edges (lines)? faces? 3d chunks?! Look for a pattern that occurs when increasing your number of dimensions, and the problem becomes easy.

:)