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Hobosnake's avatar

Is there any interactive software useful for learning the basics of the fourth spatial dimension?

Asked by Hobosnake (796points) November 15th, 2010

The fourth spatial dimension has interested me for a long time and I’ve long wanted to understand it on a deeper level. Are there any games/simulations and such (preferably freeware) that could help increase my understanding of it? Some examples of software close to what I’m looking for are the upcoming puzzle platformer miegakure and magic cube 4d, but the former isn’t out and the latter isn’t beginner-friendly.

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6 Answers

gasman's avatar

4-dimensional objects can be described mathematically in detail. Equations speak for themselves to the mathematically trained. Actually visualizing 4 spatial dimensions is probably impossible for any human brain, yet you can develop intuition about 4 dimensions by considering an analogy with 3 dimensions vis-a-vis 2.

For instance, when a sphere “visits” Flatland by moving through its plane, the inhabitants first see a point, then a series of ever-enlarging circles, then a reverse of the process. The peak circle represents a cross-section at the sphere’s equator.

Analogously, a 4-dimensional hypersphere has 3-dimensional cross-sections. As it moves perpendicularly through our world, it appears as a point, then a series of ever-enlarging spheres, then a reverse of the process. A 4-dimensional hyper-human crossing through our space might appears as a series of blobs of flesh that seem to rapidly grow, shrink, move and jump around. This theme has been used in science fiction.

Note that here time is standing in for a spatial dimension. You can show the surface of any 4-dimensional object by considering a time series of 3-d cross-sections taken along its 4th axis.

Then there’s projection. Forget spheres and think cubes. An empty cube made of sticks can be projected to cast a shadow on a flat paper, which may be distorted in the process but still shows 6 faces, 12 edges, and 8 vertices no matter how you rotate it in the light. What you lose in projection, of course, is one dimension: 3 becomes 2.

Likewise a 4-dimensional hypercube (known as a tesseract) can be projected from 4 dimensions into 3 dimensions to show 8 cubes, 32 edges etc. (I had a cool animated gif of this but I can’t find it now. It looks like a small cube inside a large cube, connected at the corners by stretchy lines, constantly turning inside out as the the tesseract rotates in the 4th dimension.)

Similar to projection is folding & unfolding. A paper cube can be unfolded into 6 connected squares or folded back up into a compact cube. A tesseract can be unfolded into a cluster of 8 connected cubes. Salvador Dali rendered some of these in his paintings.

Another way of seeing 4-dimensional spatial structure arises in the mathematics of complex functions, where a complex variable z (which is is 2-dimensional) has a complex function value f(z), requiring a 4-dimensional graph to represent the function. Since you can’t actually make one, what’s commonly done instead is to make two separate 3-dimensional graphs (called the real and imaginary parts of the function) and somehow “imagine” that these are two different perpendicular projections of the same object.

Btw, cool links in your question!

Hobosnake's avatar

@gasman I assume you mean this gif? Yeah it’s incredibly helpful. The only thing about it that leaves me confused is trying to fathom something rotating about a plane, although I suppose it makes sense. Thanks for an intelligent response and summary.

Would you happen to know if a 3-dimensional method of projection (such as a hologram) would make 4d objects (such as a tesseract) significantly easier to grasp, or is that simply speculation on my part?

Zyx's avatar

Had to figure this out on my own when I was 14 and had never even heard of it. By drawing flat octeracts. It seems so unfair that I learned all there is to know about this on my own…

Miegakure doesn’t seem 4D at all. A real (and really cool) 4D game would make all four dimensions equal. This is just a stack of chessboards in an attempt to create 3D chess…

@Hobosnake THAT gif is actually really really bad if you want to learn anything. this is 5D but it’s a lot more useful.

@gasman I HATE that, I want to just be able to make 4D graphs, for the sake of sense.

Rarebear's avatar

I can recommend a good book that I’m actually reading right now.

But to read this book it does help to read Flatland first.

Shuttle128's avatar

Flatland is available for free online due to the copyright expiring. I highly recommend reading this to anyone who is interested in exploring higher dimensions. I also recommend Linear Algebra for the mathematically inclined.

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