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

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.

Re:It is difficult, but...

[psych031337]

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.

Breaking up these limitations is not as hard as it might seem. The traditional length X width X depth is just an example of a 3d room. I understood multi-dimensionality with this simple analogy:

Imagine the "room of cookies"

1st dim: color (red, green, blue,...)
2nd dim: shape (round, square, triangular,...)
3rd dim: consistency (very hard, hard, soft,...)
4th dim: size (from very small to very large)

There you have it. A 4dim room that can be used to express any kind of cookie in a mathematical vector. For adding more dimensions all you have to make sure is that the new dimension os orthogonal, which means that the new component/unit has to be linear independent of all the other components/unit (which could for instance be the 5th dimension of texture (like smooth, rugged, etc.)

(Not an native english speaker, so please excuse me for using incorrect/half correct words.)

Re:It is difficult, but...

[vjzuylen]

Actually, a sphere passing through a 2D world would appear to be a point, then a line of increasing length, then decreasing length, then a point again.

If 2D lifeforms did exist (Planiverse suggests they would need zipper-like 2D organ structures) it's quite likely they would have some form of 'depth' perception - along a plane, of course. A sphere intersecting with their world could indeed be recognized as a circle, much like our brains can recognize the depth difference between a ball and a flat disc. The interesting thing about a book like Planiverse is that it inspires you to think about higher dimensions, using the simple 2D to 3D examples in the book as analogies. In my opinion, this serves as a better starting point for learning about the fourth dimension than any visualization applet.

Re:Reminds me of a story...

[kasperd]

Instead of being the 4th dimension, time could be the 5th dimension.

For some of that, there doesn't have to be more than 4 dimensions. The curving of space could happen in the time dimension, this just would require the curving only to change continuously over time, which seems quite reasonable. It might put some limitations to the geometrical structures of space, and I'm not sure if it would allow for wormholes, but it would still allow black holes to exist. The universe would be like a growing four dimensional ball with Big Bang in the middle at begining of time. Black holes would bend space backwards in time, all the way down to Big Bang. And the attraction in black holes could be coming from Big Bang itself which is presumably quite heavy. Anything entering a black hole could be traveling back in time to the very start of time, where they add to the mass that will in the past explode. (again?)

It might require quite some imagination to understand this, but it surely makes sense to me.