Slashdot Mirror
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."
If you set the Stereo mode to "Cross-eyed," you can view the picture in 3-D using the Magic Eye technique.
My Blog Sucks.
That's nothing. I just took two Benedryl. I think I'll go to sleep now.
Here
Less pretty but more understandable
BTW: things like the famous Stereoscopic Animated Hypercube have been around for quite some while. There even is a game around to be played.
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.