Visualizing Hypercubes: A Beginner's Guide

[AJ_2010]

I guess this is my final 'starter' question that I wish to get clear in my head with me being a newbie on the forum.

How exactly do I visualise a hypercube in my head?
I've seen all the nice complex imagery on the internet about it...such as this:
http://www.google.co.uk/images?hl=e...tle&resnum=6&ved=0CEMQsAQwBQ&biw=1450&bih=735

But I still can not work out in my mind how a 4th spatial dimension 'works'.


I can fully understand that a 1D point can exist in a 3D world.
Similarly how a 2D line can exist in a 3D world.
Also a 3D shape can exist in a 3D world.
I can also visualise how these entities would 'pass through' 3D worlds...but I can not work out the 4th spatial element.

How do I visualise a 4D world because these nicely animated hypercubes that I see on the internet to me seem just like 2D representations of 3D shapes that have corner points and edges that are moving.

I'm sure I'm missing some vital piece of the equation that will make it become clear, but what is it?


Many thanks for any help on this.

 

[Danger]

I'm no expert by any means, but in my opinion no 4-dimensional beings such as ourselves are capable of envisioning a 5-dimensional structure whether it be a hypercube, a hypersphere, or anything else along that line.
I can't currently address your second question, but will try in future.

 

[Tac-Tics]

You don't.

You understand the general idea: a point is 0D, two points connected by a perpendicular line makes a 1D line, two lines connected point-wise by perpendicular lines makes a 2D square, two squares connected point-wise by perpendicular lines makes a cube).

You also know the mathematics behind it. An n-cube has 2^n corner points. Each corner point is touching a number of perpendicular edges equal to the dimension. It can be rotated around the center using 4 dimensional rotations (which still rotate around a single 2-dimensional plane). A point divides a line in two. A line divides a plane in two. A plane divides a 3-space in two, and in general, an (n-1) dimensional hyperplane divides an n-space in two.

Etc.

That is the best we can get. Sure we can imagine the 3 dimensional "shadow" of a 4 dimensional hypercube, but you aren't your own shadow and neither is a hypercube.

I think you need to take a step back and try to understand what it is exactly you don't understand. The human mind is hard-wired to comprehend shapes in three dimensions with a very strong grasp in two. But what is that "comprehension"? Simply put, we can make very fast judgments. "Is this object bigger than that?" "If I rotate the Rubick's cube this way, I can move this yellow square into the position I want it." You can already answer those same kinds of questions in any number of dimensions assuming you know the necessary math (and it's pretty easy stuff, as far as math goes).

The difference is, you can't do it in your head. You have to write it out. And it might take hours instead of milliseconds to figure out.

There might be shortcuts, but you can't always intuit everything. My favorite math is stuff you can easily visualize in your head (for example, the visual proof of the pythagorean theorem). But not all math is like that. Many times in higher-level math topics, you are required to think about infinite dimensional spaces that are almost "incomprehensible"... but they almost always follow a simple set of LOGICAL (as opposed to visual) rules that allow us to reason about them correctly without having to "see" them in our little ape brains.

 

[jgm340]

I think the best way to think about it is by thinking about how it is constructed.

You can think of a square as being constructed by translating a copy of a line segment in the perpendicular direction, and then connecting the corresponding vertices of the two copies of the line segment.

You can think of a 3d cube as being constructed by translating a copy of a square in the perpendicular direction, and then connecting the corresponding vertices of the two copies of the square.

You can think of a 4d cube as being constructed by translating a copy of a 3d cube in the perpendicular direction, and then connecting the corresponding vertices of the two copies of the 3d cube.

You can continue on like this forever, and it's actually not to hard to abstractly visualize the process (even though you can't exactly visualize all the geometry of the final result).

Another way to do it is as follows:

Start with a square. Now lay four more squares so that they each share an edge with the original square. You get what looks like a cross:

Now imagine you can somehow connect the adjacent edges (marked in blue) together. If you can do this, then you will be left with an object with exactly four edges left to be connected. Then simply place another square here, and you have a 3d cube!

Start with a 3d cube. Now lay six more squares so that they each share an face with the original cube. You get what looks like a cross:

Now imagine you can somehow connect the adjacent faces together. If you can do this, then you will be left with an object with exactly six faces left to be connected. Then simply place another 3d cube here, and you have a 4d cube!

 

[AJ_2010]

Thanks for the answers given so far. This is really helping me.

Tac-tics ... I can fully understand that in maths it is not about visualising anything, it is all about following the formula, calculus, equations and rules etc. So I can accept easily that a 4D spatial object can not be comprehended by the human mind.
What I find confusing is all the images that can be found on the internet relating to hypercubes.
I find myself looking at them and wondering where exactly is this 4th spatial dimension. And more importantly if they are not really to be visualised, why are they all over the internet?
But as you say, I don't visualise it, so maybe I should leave it down to the maths. :)
jgm340 - your answer tries to give the visualisation route which I have tried and tried. I follow your reply perfectly up to the point where you state in your 4th paragraph
"... translating a copy of a 3d cube in the perpendicular direction,..."

The problem is, is that I run out of perpendiculars to think of. Which direction is perpendicular to the already used x, y and z?

Do I visualise your point if I were to think of the next spatial dimension as a reflection in a mirror for example?

Or to think of it totally differently; could the next spatial dimension be that 'imagined space'...as in if I just visualise the actual 3D object in my mind then this mental image is possessing further dimensions in order to create the mental image?

Am I straying way off the path with this already?

 

[jgm340]

One thing you can try is to imagine that the fourth dimension is brightness/color/temperature.

First imagine two 3d cubes lying in the same 3d space, but having opposite colors. A space of middle-grey is provided as a reference (this greyness takes up an entire 3d space). You imagine that if two points are different colors, then they have a different 4th coordinate. They represent points far away from each other, even if the first three coordinates coincide. The white cube and the black cube are the extremes of the four dimensional cube (like the top face and bottom face of the 3d cube).

If the fourth dimensional axis were perpendicular to your monitor, then the two cubes would be overlaying each other completely, but for clarity, we have done a slight skewing. Our fourth axis connects the points (0, 0, 0, black) and (1, 1, 1, white), whereas normally, it would connect the points (0,0,0,black) and (0,0,0,white).

Now the image on the bottom right is showing the connecting cubes between the two extremal cubes. You can see that the cubes start out white, then pass through the 3d space of all grey, and then emerge again on the other side being all black.

So, what we have here is actually a projection of a 4d object (slightly skewed) into 3d space! The first three coordinates are projected into 2 dimensional space, but the fourth is preserved completely as color.

Does this make any sense?

 

[Redbelly98]

From this thread:

 

[Danger]

Redbelly... I love the inherent beauty of that clip, but it's giving me a freakin' headache.

 

[AJ_2010]

jgm340 - thanks again for a nicely written reply. This does make it clearer to me.
Thinking of extra spatial dimensions as some other form of infinite scale of property to represent what the maths is suggesting is in effect the result here.

Trying to think of spatial dimensions more than x,y,z in a length unit is not possible by the human mind, but using substitution for other properties that can be given any infinite value is a way around this matter.

Thanks guys.

 

[Born2bwire]

Don't have time to read through before I have to go, but Carl Sagan has a very nice interpretation.

 

[Redbelly98]

Redbelly... I love the inherent beauty of that clip, but it's giving me a freakin' headache.

I wish it gave me a headache, then I wouldn't spend so much time looking at it.