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The Mathematician Who Will Make You Fall in Love With Numbers (
3 points by RiderOfGiraffes to math science culture 483 days ago | 2 comments

One thing I've wondered: How many different ways can you visualize 4D?

Suppose you wanted to communicate the nature of a 3D object to someone who exists in two dimensions (on a piece of paper, say).

The 2D person could only see a slice of the 3D object at any given time. But you could help them visualize it by "unwrapping" the 3D object in various ways and packing it into the 2D world. 3D objects have planar sides, whereas 2D objects only have lines; you could help the 2D person understand a 3D object by coloring each side of the shape differently, or even make the color become more or less intense depending on the side's orientation relative to the floor.

(And "the floor" in 3D would also be a hard concept to convey to a 2D person, too!)

So, stepping back to 4D: Suppose you wanted to make sense of a 4D object. Is there a "best" way of visualizing it? What are some good ways?

Klein bottles got me thinking about this:

(That whole video is great.)

I posted a video of 4D toys awhile back: and it seems like there are many more ways of visualizing the 4D shapes than what's shown in the video.

There have also been a few attempts at non-euclidian renderers:

It's tempting to believe that there are an infinite number of ways of visualizing 4D shapes, and so there is no "best" way. On the other hand, there are an infinite number of non-euclidian universes, yet they can all be analyzed with the same formulas. Maybe there is an equivalent unifying principle for visualization.

I've been meaning to post the history of non-euclidian geometry: That series is basically the whole history of math and physics, but the title is rather less assuming.


Partial answer:


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