In the three-dimensional world, there are five regular solids -- tetrahedron, cube, octahedron, dodecahedron, and icosahedron -- whose faces are composed of triangles, squares or pentagons. In four dimensions, there are six regular solids, which can be built based on the symmetries of the three-dimensional solids. Unfortunately, humans cannot process information in four dimensions directly because we don't see the universe that way. Although mathematicians can work with a fourth dimension abstractly by adding a fourth coordinate to the three that we use to describe a point in space, a fourth spatial dimension is difficult to visualize. For that, models are needed.In my head, when I put those three-dimensional shapes together, I just get another three-dimensional shape. Like building with blocks. What, exactly, is being built? Does each three-dimensional shape represent a world or dimension or moment in time?
I remember doing arrays back in high school computer science. Mrs. Murphy told us to think of fourth dimensional arrays as putting a "pocket" into the third dimension. I always hated that, because I didn't think that was accurate, and I didn't want to base my understanding on a fallacy. I didn't want to skip the difficult part so I could get my work done. I wanted to understand it.
Am I capable of that?
4 comments:
Let me try, though I don't promise to be successful :)
You've seen pictures of cubes or tetrahedrons or the other regular solids in books, on websites etc?
Well, if you think about it, those pictures are actually two dimensional, and are just a two dimnensional representation of a three dimensional object.
This sculpture is a three dimensional representation of a four dimensional object. It's been projects onto three dimensions, in the same way the cube is projected onto two dimensions.
As far as mathematicians go, we don't really see a whole lot of difference between 2 dimensions, 3 dimensions or 4 dimensions. We really think of N dimensions, where N could be some arbitrary number. The concept of width, depth, height and time are just constructs for dealing with a particular type of physics.
The software I write with is really an N-dimensional space where the dimensions are colors, but because computer displays are flat, we let the user work with one or two dimensional projections of that space. Kind of like plan views and side elevations.
So...these dimensions don't represent anything "real"? It's all intellectual? Like adding a layer of complexity to something that already exists? Or what?
Yeah. They're totally imaginary.
Math is like a chain of islands with little bridges linking them in ways you'd never expect. Geometry and Algebra are linked together, so there are a bunch of algebraic results that for example tell us how many possible kinds of wallpaper symmetry there are. Any wallpaper pattern can be classified into one of 17 different symnetries.
Dude, that's like the second time I've heard of that this week!
Must read.
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