See 4-D Space With 3-D Glasses

purpleant writes: "A hyperplane is a 3-dimensional space that slices through the 4-dimensional space, the same way a 2-dimensional plane can slice through our 3-dimensional space. The bounding hyperplanes can be extended infinitely so that they criss-cross through each other, chopping up hyperspace into many 4-dimensional 'chunks.' Again the inner chunks are finite, and they are distributed in shells around the core polytope. The HyperStar applet displays those finite chunks, one shell at a time. The inner shells are complete -- each shell completely encases the previous shell. The outermost shells have holes in them."

Comments

hold up...

[circletimessquare]

so it's 5 am monday morning on the east coast and i decide to check out slashdot before going to bed for 4 hours of shuteye before work tomorrow (don't ask, crazy weekend). i am bleary eyed, brain dead, exhausted.

and i read "The bounding hyperplanes can be extended infinitely so that they criss-cross through each other, chopping up hyperspace into many 4-dimensional 'chunks.' Again the inner chunks are finite, and they are distributed in shells around the core polytope."

dudes! my functional iq right now is about 50! if you are going to post these kind of stories on slashdot, could you PLEASE post them around, say 3pm on a thursday? thanks ;-P

i should be awake by then, and i promise i will come back and try to wrap my mind around this story at that time... grumble, grumble

You don't need 3-D glasses...

[AaronBaker2000]

If you set the Stereo mode to "Cross-eyed," you can view the picture in 3-D using the Magic Eye technique.

Re:You don't need 3-D glasses...

[flonker]

Simple solution. Stick a crayon in your left eye. That'll fix things real good.

It is difficult, but...

[tanveer1979]

the thing about opening our minds is right.
We have always lived in three dimensions, so visualizing 4 dimensions Per Se is almost impossible coz our nuerons have been hardwired for 3 dimensions. So we can observe 4 dimensions in transit. For example if youwere a 2 dimensional being(thats not possible coz 3 is the minumum number of dimensions to sustain life) and a 3D sphere passed through your space, you will see a point, growing into a circle and then again into a point.
So if a 4D object came it would look like a morphing 3D object.
If mankind were able to create and use 4D's travel would be a whole new frontier. Esp since space-time is curved, Just imagine traveling a million miles instantaniosly
Confused! Go through stephen hawkings works! you will be even more so :-)

This was an old college argument...

[Thalia]

We used to argue this in the computer science lab at college. Can the human mind gain visualization skills in four dimensional geometry? We came up with the following interesting answers:

1. It's hard. We never see four diminsions. The brain would keep wanting to make one dimension some known continuim such as time, a color sequence, tone, or intensity. Only after this intermediate step would you get a true four dimensional geometry in your head.

2. You would need to have a true 3D display. Current rendering of three dimensional pictures flattened onto simple two dimensional screens would never work. Imagine using a laser pointer as a point source, and imagine that you had never seen a three dimensional object; now draw a three dimensional picture of a pick-up truck using the laser pointer. At the time, we were trying to get a simple three dimensional output, like <a href="http://www.stereographics.com/frames/frame-p rod.html">Crystal Eyes</a>. Now there are liquid crystal on silicon solutions that are much cleaner, if not cheaper.

We were students once, and poor.

I have a large collection of 4D media

[guttentag]

Each one takes a 3D image and progresses along a fourth dimensional plane I like to call:

<fingerquotes>TIME<fingerquotes>

This combination is stored in a media format I call a:

<fingerquotes>MO-VIE<fingerquotes>

4D is very subjective, so long as you are allowed to define the four dimensions involved.

Re:Reminds me of a story...

[JohnsonJohnson]

No.

That's not a question in general relativity. The curvature of a spacetime can be measured without considering it as being embedded in manifold of higher dimension. How? Here is the common demonstration of the idea without using the language of math which makes the idea harder to convey. If you want a more rigourous explanation see here. Note that the curvature we are concerned with here is the Gaussian curvature which is intrinsic, ie it can be measured without considering directions outside of the dimension of interest.

Consider the surface of sphere, any ball is a reasonable approximation. Now consider the following path. Starting at the equator while facing west (these are all well defined directions if you use the right hand rule and call north the direction of your thumb then east follows the curvature of your fingers and west is opposite east). Now go 1/4 of the circumference of the circle west, turn to face north. This is a 90 degree turn. Go to the north pole. Now turn 90 degrees again (again this is a well defined operation, when facing any direction a 90 degree turn is accomplished by orienting yourself such that the direction previously over your right shoulder is now the direction you are facing). Now continue to the equator. You should be at the original starting point, and another 90 degree turn will leave you facing west, your original direction of departure.

So you've traced out a triangle: a closed path with three vertices, but you've made 3 90 degree turns so the sum of the interior angles is greater than 180 degrees in violation of euclidean geometry. Therefore you know your 2 dimensional world: which is the surface of the sphere, is curved. Note that is is not necessary for this surface to actually be curved "into" anything. If the sphere is the spacetime of a universe then there is by definition nothing outside of the surface of the sphere to consider, all of space and all of time are contained on the surface. The surface is still curved, but it doesn't "curve" into anything, that's just a property of the spacetime.

I'm not sure I can make it any clearer, but if you consider Occam's razor you'll see that it doesn't make sense to thing about curved spacetimes as being embedded in some higher dimension. Since it is possible to measure curvature without appealing to a higher dimension (remember we never left the surface of the sphere in the above example) then you don't need the higher dimension, all the information required is contained in your local spacetime.

Tic tac toe

[tuxedo-steve]

A fun four-dimensional exercise is 4d tic-tac-toe. It's easiest if you build yourself up to it:

1. Start with your typical 3x3 tic-tac-toe, on a piece of paper.
2. Now add two more grids. Visualise each grid on top of the one before. It's not difficult to see how this is played. You can get three in a row on a single grid, just like normal. Or you can get three in a row by getting the middle square of each grid (3 in a row, vertically). And so on. This is basically tic-tac-toe in 3D. 3 sets of 3x3 grids. 3x3x3.
3. Now, add another two sets of three grids. So now you've got 3x3 3x3 grids (still with me?). You can still win just like in the 3x3x3 version. But you've got another 3x3 ways in which to do it. The tricky part is, to visualise each possible `3 in a row', you've got to mentally `rotate in' any one (and only one) 3x3x3 cubic plane.

Sorry if that's difficult to follow. If you work it through on paper, you'll see what I mean. This is what being bored in math class will lead you to think of, when plain old tic-tac-toe just doesn't seem challenging anymore. :)