How does this 4D cube work?

http://www.mathematik.com/4DCube/4DCubePovray.html

quite by accident and it has fair freaked me out. I always thought time was the fourth dimension, or something. So how does it work? Is it just an optical illusion, or has it got another physical dimension of some kind?

While time is the fourth dimension, there are various ways to indicate a fourth physical dimension. What you see is a three-dimensional representation of a hypercube – a hypothetical geometric figure in four dimensions. It is actually made up of nothing but cubes, but the cubes in the example are distorted, much like representing a cube in two dimensions shows something other than right angles.

Time is not the fourth dimension. One could have a three dimensional representaion with time as one of the dimensions. One could have a four dimensional representation that did not consider time explicitly.

The ‘extra’ dimension, I think, is the hyperplane.

A very cool site all told!

Time can be treated as a fourth spatial dimension (it’s much easier following Hawking’s lead of manipulating its axis so as to chart so-called “imaginary time”), but that’s not really relevant here. We 3 (spatial) D beings with our 3-D brains and 3-D evolutionary history cannot “see in 4-D”, and so I guess this is an optical illusion of sorts.

Something on this page makes that animation more understandable. Click the link called “Multidimensional math”, and imagine the 3-D cube you created in the third step rotating, then the 4-D hypercube you subsequently create rotating also.

Yea, I agree.

From what I can tell, the image shows something that could be constructed in real life.

It could be constructed, but it’ll only be a 3D representation of the hypercube. Much like you can draw a 2D represenation of a 3D cube on a sheet of paper.

BTW, “dimensions” specifies how many numbers are required to locate a point given your own location.

Thus, on a line (one dimension), you can locate any point by giving only a single number of units (assuming positives and negatives can be given). Thus, the second point can be 3 units from you and you can direct people to it without any additional numbers.

On a plane (two dimensions), you need two numbers. These can be x and y coordinates (or length and width), but also an angle and a length (5 inches at 30 degrees).

In three-dimensional space, three numbers are required. They can be measured on the x, y, and z axes, or by other means (using angles, for instance).

Often not all the dimensions are needed; thus in 3D space, you can locate a town by saying “It’s five miles down the road.” The other dimensions are there, but are too trivial to mention, so you can give one-dimensional instructions.

Time becomes a dimension if you’re trying to locate a moving object. Thus when trying to locate an enemy aircraft, you need to know length, height, and width, plus the time when the aircraft is in a particular location. If any one of those factors is left out, you can’t locate the aircraft.

It looks more like 16 points plotted on a donut, or a 16 point line drawing representation of a donut.
Take four cubes side by side and stick them on a string (axis). Now attach the two ends of the string so the two cubes on the end now share a side.
Rotate the cubes around the string axis and thats what the picture shows.

There’s nothing hypothetical about it. A hypercube exists in exactly the same sense that a cube exists–that is, while neither exists in the real world, they both exist as abstractions.

A square has four lines defining it - all on the outside.

A cube has six squares defining it - all on the outside.

A hypercube has eight cubes defining it - all on the outside.

That figure shows a hypercube. All the cubes are the same size, and they are all on the outside. It doesn’t look like that because it’s distorted through the projection onto a 3D surface. Some of the cubes look bigger than the others, and one always looks like it’s inside the others. But that’s just the distortion.

What’s really tricky about it is that in the 2-D image, it seems that only 7 of the 8 cubic “faces” are visible.

You’ve got the six that make up the apparent outside of the hypercube. Then you’ve got the one extra cube that always appears to be in the “middle” of the hypercube. But for some reason, the eighth cubic “face” doesn’t seem visible.

I don’t see how the animation is fundamentally different then a toroid whose surface is rotating through the axis?

(er towards the axis)

I first came across the concept of a 4D cube in Heinlein’s story ‘And He Built a Crooked House.’ Great story. Another name for the 4D cube is a tesseract.

To elaborate on what RealityChuck said, and as Bricker indicated, you can take 4 lines (1D) and draw on paper (2D) a square and further take six squares and form in a space (3D) a cube. If we could see and manipulate 4 dimensions, you could take 8 cubes and form a tesseract in your 4D space. Here is another non-moving depiction of a 4D cube. The problem with perceiving it is that it is a 2 dimensional image (on the screen) of a 3 dimensional representation. If you could see in 4D, the 8 cubes would all be the same size and the six that we perceive as going from the ‘outer’ one to the ‘inner’ one are not slanted but instead have 90° angles in the inner corners. (This is the problem with viewing from an oblique angle to the 4th dimension. It is the same problem with viewing a representation of a cube drawn on a piece of paper. Some of the sides are shown as a parallelogram*, but our experience tells us they are actually squares.)

While the link in the OP is cool, it really doesn’t indicate anything. How does a tesseract rotate? Not being able to experience 4D (yet), I don’t know. The lines along edges lengthen and contract as the thing moves. Is this just a perception thing? Or is it just an optical illusion?

bordelond, the eighth cubic “face” is the bigger cube that encompasses the whole thing, “parallel” with the inner cube, but looking bigger.
(* - In preview, not a parallelogram, but another shape I’m spacing on.)

Aaaah … now I see.

Next conceptual question – does the tesseract have an “interior” bounded by the eight cubes that we can more or less observe? This would be analogous to the interior of a six-sided cube. My understanding is that the cube that seems to be more or less always apparent in the middle of the rotating tesseract is not an “interior”, but one of the eight bounding cubes. I know the animation linked in the OP is not a perfect reflection of a tesseract, but I would expect to be able to make out an interior distinct from the eight bounding cubes.

Instead, the tesseract’s eight bounding cubes seem to basically “boil” upon each other over and over as the hypercube “rotates”.

I think that animation is BS. That isn’t a “rotating” hypercube. A true rotating hypercube would not have some of the component cubes passing completely through each other. Imagine a normal cube undergoing an analogous transformation. You start with a square inside a larger square, with the corners connected by diagonals. Now repeatedly squish the larger square through the hole in the smaller square rotating the diagonal connectors. Is that a rotating cube? Hardly. That animation is just eye candy.

Ooh, ooh - the fourth dimension, one of my favorite topics.

Firstly, for Firefox users, if you click on the rotating cube and start dragging, it will freeze the image, which is useful for inspecting it.

There are some fascinating results involving the 4th dimension, some of which are not obvious unless you’ve given it a lot of thought.

1. A 4-dimensional being could perceive all of you at once - every side of you, including your insides (with his “3-dimensional retinas” !) Just like you can look at a hexagon on a piece of paper and see all of it, inside and outside, so a 4-D being could do the same with you. (This would require photons to be 4D-savvy, I guess.)

2. If the same being were to pick you up, take you into his world and “rotate” you in 4-space, then put you back here, you’d be a mirror image of yourself - right and left swapped, yet perfectly healthy (except rather traumatized). The being could turn a left shoe into a right shoe. This really blows me away.

I still can’t get the image of a tesseract in my head, even though I understand it intellectually. I can’t get the visceral understanding.

So far, my favorite book for understanding things 4D has been this one:

http://www.amazon.com/exec/obidos/tg/detail/-/0395344204/qid=1124819157/sr=8-13/ref=pd_bbs_13/102-5835682-1789726?v=glance&s=books&n=507846

I also started reading Hinton’s “Fourth Dimension” but the archaic language puts me off a little, and I’m not about to memorize his pile of cubes in order to visualize higher-space objects. It’s still a great book, though.

But that real-life object would only be the projection into three-space of the four-dimensional hypercube.

I can see that it’s somewhat analogous to the way a three-dimensional cube can be projected into two-space, as in a drawing on a sheet of paper. However, I can’t quite figure out whether our view of the hypercube’s representation in three-space is similar to our view of a drawing of a ordinary cube on paper. Is it? Or is it more akin to how a Flatlander would perceive a the drawing, trapped on the plane and not able to rise above or below it.

Two other great reads for helping to understand n-dimensional spaces are Flatland by Edwin A. Abbott and its companion, Sphereland by Dionys Berger.

I’ve done a lot of reading on hyperspace, but I think it was in Flatland that I read the following analogy:

A point has zero dimensions. Slide it in any direction to create a line.

A line has one dimension and two terminal points. Slide it perpendicular to that dimension to create a square.

A square has two dimensions, four lines, and four terminal points. Move it perpendicular to those two dimensions to create a cube.

A cube has three dimensions, six squares, twelve lines, and eight terminal points. Move it perpendicular to those three dimensions to create a tesseract (hypercube, 4-cube).

A hypercube has four dimensions, eight cubes, 24 squares (I think; here the math fails me and I don’t have time to look it up) . . . and 16 terminal points. Move it perpendicular to those four dimensions to creat a 5-cube.

. . . and so on. The number of each element proceeds in a logical progression as you add dimensions.

Another illustration goes something like this:

The cross-section of a line is a point.
The cross-section of a cube (assuming perpendicularity) is a square.
The cross-section of a hypercube (4-cube) is a cube.
The cross-section of a 5-cube is a 4-cube.
etc.

One of these books describes “trapping” a higher-dimension creature in the next-lower-dimension world and how the lower-dimension inhabitants would see it. For example, in Flatland (2-D), a 3-D being could step into Flatland from “above” (which Flatlanders cannot understand, because they have only two dimensions), and Flatlanders would see it as a suddenly appearing point that would “grow” into an irregularly shaped 2-D blob (really a cross-section of its body). They might surround it with a 2-D string and think that they have trapped it, but the 3-D creature could simply lift itself free out of Flatland and “disappear.”

Similarly, a 4-D creature could “drop” into our world, appearing as a slowly growing 3-D blob (a cross-section of its 4-D body). We could put the blob in a bag and tie it shut, but the 4-D creature could simply pull itself “out” of our space and “disappear” to our eyes.

But the 4-D creature would be similarly stymied by a 5-dimensional creature visiting its 4-D world.

Freaky stuff!

The animation is correct. An analagous situation in 3-D would be the following: start with the square inside a larger square like you describe, but move the inner square down through the bottom face of the larger square. The inner square becomes the bottommost of the four squished quadrilaterals, while the outer square flips and become the topmost quadrilateral.

You can see that in the animation linked to in the OP. Ignore everything except the 8 vertices at the front of the picture and the edges which connect them. Those vertices and edges form a plain old cube, which is rotating in place. You can clearly see the back face of that cube (the “inner square”) appearing to pass through the bottom edge of the front face of that cube (the “outer square”). Of course it’s not really passing through anything; it just appears that way because of the projection. The same thing is happening in the 4-D case.

The bolded part is what I really can’t visualize. The concept of “perpendicular” seems to break down here.