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2008-12-25

"Dimensions"

I came across the "Dimensions" video via Reddit. It tries to help you visualise four-dimensional objects via their projections onto ordinary three-dimensional space. If you have even the slightest interest in mathematics, you should check it out. You can either view it on-line or order a reasonably-priced DVD. The video is about two hours long.

If you have never thought about the difficulty of n-dimensional beings perceiving (n+1)-dimensional objects, I would recommend that you first read the short but excellent book "Flatland" (an on-line version available here). For example, if you are a two-dimensional being constrained to live on a two-dimensional plane, you cannot perceive three-dimensional beings except as two-dimensional projections onto your plane.


If the projection used is the cross-section, if any, obtained by the intersection of the three-dimensional beings with the two-dimensional plane of your world, then the projection of a three-dimensional being can even spontaneously appear in, and disappear from, fully-enclosed areas in your world as the three-dimensional being moves around in three-dimensional space. The shape of the projection might also change depending on the position and the orientation of the three-dimensional being. These shape-shifting three-dimensional beings will then appear magical to you since they do not seem to obey the natural laws of your world.


The "Dimensions" video uses the stereographic projection, which preserves shapes, to help us visualise four-dimensional objects in three-dimensional space. It uses some amazing graphics created using POV-Ray to illustrate the ideas. Though the narrative could have been a bit faster and the background music a bit less jarring at times, this is an excellent effort by some very talented folks with a lot of love for mathematics.

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