For the paper & source code (in Mathematica notebook file format), please click here. Single-Variable calculus is commonly taught from a graphic standpoint, using the concepts of area and tangent lines to describe the world of integrals and derivatives. However, when we move to multivariable calculus, most students ask the question: what does a 4D surface look like? Without this idea, it is difficult to draw the connection between techniques learned in elementary calculus and techniques in the 4th dimension and beyond. We certainly cannot pull out a hypersphere from our textbooks. Here's one method: This simple technique projects the 5D hypersphere into a systematic set of 3D subspaces (i.e. all possible combinations of the 5 dimensions, taken 3 at a time). This is the equivalent of taking a sphere inside a box and looking at the shadow a sphere casts on the walls of the box. Each 3D subspace gives a unique perspective of what the 5D hypersphere looks like. Visualizing a 5D hypersphereIf you cannot see the slide show, click here. posted by Jeffrey Wang |
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IMUC 2009 >
5D Hypersphere Visualization On 3D subspaces |