As in 4-dimensions, i.e. "see" a hyperplane. It's driving me nuts.

Here's an applet that helps the visualization. Or is supposed to. If anyone can use that to explain what the hell is going on, that person will be awesome.

Not impressed.

looks like it's rotating the inside to the outside through one side, and the outside to the inside through the opposite side.

Well, there's no reason that the human brain couldn't be trained to think in four dimensions, but it would be very hard to do with a 2-dimensional projection like this. Like trying to teach a 2-D person to think in 3-D by using a line.

What one could do, possibly, is have some kind of four-dimensional rendering system that's tied right into the optic nerve, such that we wouldn't just be looking at projections of the 4-space. That'd be weird. Though imagine how disappointed you'd be when you turned it off and you only had three dimensions? That would suck. Like if you woke up one day and you were flat.

I always wondered if 4-D objects would be any easier to understand if we were able to see them in actual 3-D space instead of a 2-D space. I want to hold a hypercube... >.>

Pfeh, I built a three-dimensional embedding of a hypercube using toothpicks and marshmallows once. It's not really that exciting.

Unless you meant a REAL hypercube. That would be harder to build.

EDIT: If you really want a challenge, try solving the 4-D Rubik's Cube

Too easy? Here's 5 dimensions

I don't get this at all. It's just an easily-reproducible 3D shape, and when you rotate it around it behaves exactly like a 3D shape.

It's still just 3 dimensions.

It's a four dimensional object being represented in 3D space.

^^^ That's a projection in math speak. A circle is just a sphere projected onto the lower dimension.

How I see objects in 4-D:

Let v be any vector in R[sup]4[/sup] (the vectorspace not "reals" to the 4th power). That is v = (v[sub]1[/sub],v[sub]2[/sub],v[sub]3[/sub],v[sub]4[/sub])

Let S = {v[sub]1[/sub],v[sub]2[/sub],...,v[sub]n[/sub]} be a set of vectors in R[sup]4[/sup]

And presto! You have an object!

There was a really interesting graph theory problem in my discrete book about a 4-D graph. I should rummage through and find said book.

You know, with the cross-stereogram view of the object, thus making an approximation of four dimensions in an illusion of three, I finally see what Lovecraft was talking about this whole time.

-Te-ke-li-li!

Reminds me of adjoins in JED.

The funny thing is, we don't even see in 3D. If we did, we could see what's behind objects.

If we only saw in 2d we'd have no depth perception.

Try this:

Draw a 2D representation of a cube on a piece of paper.
Turn the piece of paper.

You'll see that it's still an easily-reproducible 2D shape, and when you rotate it around it behaves exactly like a 2D shape.

The problem with that visualisation is that it doesn't allow you to rotate in the 4th dimension, just like the representation of the cube on paper doesn't allow you to rotate the representation in the 3rd dimension.

If it did allow you to rotate the hypercube in the 4th dimension, you'd see the 8 cubes changes shape and size relative to each other. (Yes, I said 8 cubes, not 2. There are 6 cubes between the cube in green and cube in blue. They are skewed because they are angled differently in the 4th dimension. They are each made of one blue square, one green square and 4 squares with one blue edge, one green edge and 2 red edges.

The reason they are skewed is exactly the same as the reason 4 of the squares in the 2d representation of the cube aren't square.


Actually, I tell a lie. You CAN rotate in the 4th dimension. Drag with the RIGHT mouse button and you'll see what I mean.

Not true. The eye can only produce a 2D image.

The reason we have depth perception is that the brain does a lot of processing of the data from both eyes. Each eye sees an object from a slightly different angle, and can tell whether things are infront of or behind each other based on the differences between the two images.

This is a huge simplification of course.

The pre-processing before we visualise the images is what is taken advantage of in "magic eye" pictures.

Wow, I learned something in massassi.. and it was useful =o
Thanks Giraffe

You need to hold down ALT and use the left mouse button.

What the hell isn't the 4th dimension, space time?

Time is often thought of as a fourth dimension, but this is dealing with a hypothetical (or perhaps not, see string theory) fourth space dimension.

Actually, I'm glad you brought that up. It's often helpful for me to visualized 4-D objects by letting time be the W-axis, in which case you get a three-dimensional shape which changes through a set amount of time. Think of it this way: imagine trying to represent a sphere to someone who only knows 2 dimensions. If you slice the sphere up into planes normal to the z-axis, then they could imagine the sphere as a circle that starts out infinitely small, then increases in size upwards to a maximum radius (the radius of the 3-sphere), then decreases back down to nothing again. Similarily, a 4-sphere would look to us like a ball that grows and shrinks. Other shapes are of course more difficult, but as long as you keep them aligned properly with the W-axis some are easy to see (a 4-cube is just a cube that pops into existence, sits there for a while, then disappears, for example).

EDIT: How coincidental: http://www.nzherald.co.nz/topic/story.cfm?c_id=325&objectid=10400645

Trying to represent a 4th dimension object via 2nd/3rd dimension physics just doesn't do it justice.

And trying to comprehend the 4th dimension is a feat in and of itself. We get really close, but even that is nearly impossible. Our astronomy professor once had us do an exercise where we tried to explain the 3rd dimension to a 2nd dimension being on a piece of paper. It's difficult when that being can't comprehend such basic terms to us such as depth and height. The closest we could get to explaining to these hypothetical beings what the 3rd dimension is was by touching their insides. He then translated this into how difficult it is to understand the 4th dimension, if not impossible for us, and it was linked to black holes and the expansion of space.

Hey, maybe God is a being in the 4th dimension. Hence when some people get certain "feelings" and being touched by God is being literally touched by a 4th dimension being. My head is about to explode.

And of course I went to calculus class today and we started doing triple integrals. "Sure, it's just integrating over a volume"

Wait... OVER a volume?

Actually I came up with a neat way to represent a four-dimensional "surface" that's a function of (x,y,z): have distance along the w-axis represented by different colours. Hard to draw, and not really useful for visualizing shapes, but it does make a pretty picture! Or it would, if I could get Maple to graph it... maybe I'll have to bite the bullet and learn Matlab.

Oh well, here's an example of a four-dimensional function where the fourth dimension is time. Or rather, the third dimension is time and the fourth is the z-axis. Or something. It's crazy.

It's dancing... pretty.

More pretty dancing; now with better looping! Gotta love those sinusoidal functions; easy to get them repeating.



EDIT: WHOA IT DID NOT HOM LIKE THAT BEFORE

For some reason this thread reminds me of Ezekial Rage in the TV show The Real Adventures of Jonny Quest. Not sure what the connection is though.

Imagine you have a point. It has no area or volume, just location. That is the zeroth dimension.
Now drag the point from one location to another. The line segment between the two locations has only length. That means it has one dimension.
Now drag your line segment to a parallel position. If you consider the boundary made by the end points of the two lines, you have formed a plane. This has two dimensions.
Finally, drag your plane in a normal direction away from its initial position so that you have two parallel planes. The boundary formed by the 8 corners of the two planes makes a volume. This has three dimensions.
Try to imagine what you do the volume to move from 3D to 4D and continue the sequence.

Pretty sure it'd just look like a cube to you.

How hard is multivariable integration? It like just single-variable integration or...

If I remember correctly, string theory suggested that successive dimensions wrapped around the previous kinda like a barber pole stripe. It was proven Mathamatically too...of course, so were the other 7-8+ string theories. Just goes to show you that you can prove almost anything with math...

Back to the realm of Normative Physics String Theory now be, yar!

Kleiner bottles disturb me.

-I fear that placing one on a desk would result in the destruction of the desk, but then, I fear science.

Yeah it's not really that hard, just more tedious. The hard part is trying to visualize it.

Anything made by a scientist who keeps a headcrab for a pet would scare me.

Different Kleiner. Though the HL character might be named after him.
A Kleiner Bottle, roughly speaking, is a 3d Mobius Strip (actually, the way it works requires four dimensions), and thus, if it were sitting on a desk, the desk would be both outside the bottle and inside the bottle simultaneously.

-And I've read enough Lovecraft to know that that sort of behavior is not only bad, it's Not Meant To Be.

Meh, isn't the universe a closed 4-dimensional surface anyways? Which would mean that anything could be inside and outside of anything else, at the same time.

Hey wait a minute, then according to Gauss's Law, the total charge in the universe has to be 0. Woot, I just proved that the universe has no charge.

Yeah, this is the hard part

I can totally understand HOW 4D works and what its principles are based on the 2D-3D example (like the Flatland example). I just can't "see" it. Which is duh, because of what someone else on this thread said with their 2D-being / flatland example.

Bit' o fixin.

Bah, semantics.

this was by far the most useful post to me. Thank you.

Awesome. What exactly are the implications of higher spatial dimensions? Does it mean our universe is contained in a marble that aliens in the 26th dimension are using to play jacks like in MiB? What, exactly? Would things like hyperspace be more plausible, or is it more like there are higher spatial dimensions we could never gain access to so it doesn't really matter outside of physics?


Oh, I feel much better now...

I would say hyperspace/wormholes would be more plausable if the experiment turns out to be good. A wormhole is just "burrowing" in spacetime. But there's still the problem of energy required being astronomical.