I’m trying to understand a concept from a sci fi flick, Cube 2: Hypercube.
This is a kinda helpful java-diagram to describe forth dimensional space. Or is it space? To me it seems the java-thing shows what would happen if you had a cube that was leaking ink, so where ever you drag the cube it leaks…err thats a bad explanation but basically I am trying to understand what a hypercube is?
It’s pretty simple, in principle. A normal three-dimensional cube is made up of six two-dimensional squares. A four-dimensional hypercube is made up of 6 three-dimensional cubes. Imagining how they are arranged in four-dimensional space is the hard part.
I thought it was 8. Wouldn’t it have a center cube, generally recognizable as a cube, with 6 cube faces surrounding that one, all tweaked into truncated pyramidal solids and then another cube, imperceivable in a 3-d projection connecting the large bases of the trapezoidal side cubes, sharing an “inside-out” face on those bases?
this is the shiznit
http://dogfeathers.com/java/hyprcube.html
especially the one where you cross your eyes and do the sausage finger thing and watch it pass through the third dimension
Oh yeah, it is eight. Here is a good description with diagrams.
but why would a 3d cube with a buncha little 3d cubes inside it be considered to have a 4th dimension?
I’d heard the six(well, seven, really)-cube explanation. An easy way to envision it: take a cube, and stick a smaller cube in the exact center. Draw lines connecting the corresponding corners of the cubes. Now, you have the center cube. The 6 frustums (chopped 4-sided pyramids) are also cubes, but in the 4th space. Each side of one of these cubes shares a side with another cube, which is why we have to squish them to make them connect when we make a 3d picture of them. I don’t know if the 8th cube effusant describes that encompasses all of these is actually part of the 4d tesseract or if it’s a byproduct of our 3d representation. But realize that all these cubes are the same size in 4d space.
I had one instructor tell me that the reason it’s a "hyper"cube is because the angles of the lines connecting the cubes are at 90[sup]o[/sup] to the lines making up the cubes. This makes it a cube in 4 dimensions (more cube-ness than a mere 3d cube = hypercube).
Okay, from QED’s link I see that it’s been explained as 8. I stand corrected.
A 1D cube (line) has 2 0D cube (point) endpoints.
A 2D cube (square) has 4 1D cube (line) sides
A 3D cube has 6 2D faces.
A 4D cube would have 8 3D cube faces.
This continues with a 5D cube having 10 tesseract faces.
The image you are looking at is not a tesseract. It is a 3D projection of a 4D object – Just like a drawing of a cube is not a cube, but a 2D projection.
hooray for kata and ana
Related thread I posted a while back in IMHO.
Personally I find all the animated stuff and most cites a bit baffling for a layman - but obviously that is my problem for being a bit of a geometic moron;)
The way it was once described to me and which - as I still remember twenty years later - must have had some merit to it was to use the analogy of describing a 3D cube to somebody who lives in a 2D universe.
The describer cannot show the 2D entity a cube but what he can show him is the shadow of the 3D cube projected into a 2D plan - think holding a cube above a sheet of paper with a light source above it and drawing around the edges of the shadow with a pen. The look at the result - note lack of right angles between edges etc.
Now imagine making a 3D model of the shadow of a 4D (or more but then it get quickly gets very complicated) hypercube projected into a 3D space - you can build one which is where some of the links above come into use.
The worked for me at least…
If you haven’t worked it out from the preceding posts, it’s not just that there are a bunch of cubes. You can’t have 8 cubes in that configuration in 3 space, just like you can’t have 6 squares all joined on all sides in 2 space. It’s hard to envision but think about making a cube as a wire frame, then projecting its shadow on a piece of paper. The cubes edges all meet at right angles, but there are all kinds of angles on the paper. Now think of the tesseract–it is the “shadow” of a 4 space object made of 8 cubes.
Oh, sorry notquitekarpov, somehow I missed your post.
I think this is a case where the math may be more helpful than the layman’s explanation.
The unit square S[sup]2[/sup] is the set {(x, y)| x = 0, y = 0, x = 1, or y = 1}. Those with some math background may recognize this as [0, 1][sup]2[/sup], the Cartesian product of the unit interval with itself.
Every square in the plane is the image of S[sup]2[/sup] under some angle-preserving transformation. A transformation is just some function that maps points in R[sup]2[/sup] to points in R[sup]2[/sup] subject to the following conditions:[ol][li]For every (c, d), there is an (a, b) which maps onto (c, d).If (a, b) maps onto (c, d), and (e, f) maps onto (c, d), then a = e and b = f.[/ol]Some of you may recognize that as a bijection.[/li]
The fact that it’s angle-preserving just means that the angles in the image of a shape will match up with the angles in the shape. Squares map onto squares, triangles onto triangles, and so on and so forth.
The unit n-cube S[sup]n[/sup] is the set {(x[sub]1[/sub], …, x[sub]n[/sub])| x[sub]i[/sub] = 0 or x[sub]i[/sub] = 1, 1 < i < n}. And the generalized n-cube is the image of S[sup]n[/sup] under some angle-preserving transformation.
The tesseract is just an image of S[sup]4[/sup] under an angle-preserving transformation.
I hope this actually helped, and didn’t just leave people more confused.
Hey raisinbread: Ramooom says hi.
Read The Boy Who Reversed Himself by William Sleator. It’s children’s sci-fi about 4-dimensional space. I read it a few years ago when I was probably about 10, so it should make some sense, hopefully.
Conclusions about tesseracts, and about four dimensional phenomena generally, can be reasoned by analogy. Because of the limits of our experience and senses, however, they can never be envisioned perfectly.
A good primer for thinking about higher dimensions is the short novel Flatland by George Abbott Abbott, a 19th Century British mathematician. This fantasy work (which is readily available in large bookstores in editions by various publishers)describes a universe where people only perceive two dimensions.
People and other objects there are a bit like spills of liquid on a flat table top. Their world is “sliced” so thin that people there have no perception that that they are physically removed from the surface they glide on, or that they, or or anything else in their universe has any dimension behind length and width. A sphere passes through Flatland one day, and is perceived as a circle which continually changes in size.
Some grasp of how four dimensional phenomena would behave in our three dimensional universe can be gotten by making Flatland analogies and considering the behavior and appearance of three dimensional objects there.
Another very good book is The Fourth Dimension by Rudy Rucker, which is a good nontechnical discussion of the subject.
For some reason speculation about a fourth dimension of space seems to have especially popular in The United States and Great Britain. In addition to the writings of Abbott Abbott, there was even a discussion of the subject over a couple of issues in the 1880s of St. Nicholas, a popular American children’s magazine.
And then there was Charles Hinton, a mathematician from England who taught at Princeton around the turn of the last century. He is said to have probably been able to picture and understand the existence of a fourth spacial dimension better than anyone else in history. A thorough-going eccentric, he developed a memory and visual imagery system which involved picturing objects as being occupied by arranged cubes to which identifying names were assigned. Using his method he was able to memorize exactly how objects were constructed, and to estimate their volume instantly. A man who studied Hinton’s system later said that he had condemned himself to a living hell.
Also of interest is a very entertaining story by Robert Heinlein called And He Built a Crooked House. In it a visionary architect tries incorporating theories about the fourth dimension into the design of a radically new house. He succeeds better than he intended, and he and the new owners have no end of trouble after they get stuck in the house after an earthquake.
While I don’t have much grasp of what exactly is under discussion, I have read that many physicists believe that the existence of additional spacial dimensions are not only possible but probable. Lately there has been talk that the span of some of these dimensions, at least as they intersect with the known universe, may be a measurable, if very small, interval. So nearly as I can follow, that could mean we are living in our own version of Flatland, occupying and passing through dimensions to which we are oblivious
Read the Time Quartet books by Madeline L’Engle. She explains it pretty well. And with illustrative matter, too.
Another way to see the analogy is to unfold and refold a cube a few times, getting a cross, then take a look at the cube-in-a-cube and mentally unfold that. Note that the contacts between squares are edges, while those between cubes are faces. Heinlein’s architect, mentioned by slipster, did the same for the house.