Monday, August 14, 2006

Hyper-Dimensional Poetry?

When I was first introduced to hyper-dimensional geometry I was quite fascinated but really didn’t have any clear path to understand it. I had seen two dimensional images of a hypercube (four-dimensional cube) but really understood nothing about what I was looking at. With computer imagery we are able to see things a little better because we can simulate three-dimensions in a video or other moving imagery. The following link will take you to a polytope slicer which allows you to take three-dimensional slices through a four-dimensional object.
Let me expound upon this a little bit. Just about all of us have experienced slicing a near two-dimensional piece of paper with a pair of scissors. When we do this we experience seeing a near one-dimensional line at the edge of the paper where we just cut. Many of us have also experienced slicing a ‘three-dimensional’ orange in half and noticing a two-dimensional surface showing the cells inside the orange. However things get a little trickier when we slice a four-dimensional object. If you notice on our previous examples that the slice is a dimension less than the object we started with. That is a slice of a three dimensional object is two-dimensional and a slice of a two-dimensional object is one dimensional. Therefore, to imagine a slice of a four-dimensional object our result would be something that has three-dimensions. Our polytope slicer does just that! It gives us a three dimensional-section cut of a four-dimensional regular polytope. Your next question may be, “what is a polytope?” A polytopes are to four dimensions as polyhedrons are to three dimensions or what polygons are to two dimensions.

As you vary the parameters in the polytope slicer you will get three-dimensional slices of our four-dimensional polytope. (click here for the polytope slicer).

Now what does this have to do with mathematical poetry? All maths can be used as language for poetry. Use your imagination … I predict that someone will write a poem on a hypercube so that we can read it by projecting it down to the third dimension. This may have already been done but I am not aware of it. After one does it with a hypercube then try doing it on a 120 cell hyperdodecahedron or maybe an epic poem on a 600 cell hypericosahedron

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