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    Post Made Community Wiki by whuber
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One prebuilt tool for visualizing high dimensional data is ggobi. It lets you color the points to represent groups and then has a few options for reducing the high dimensions to a 2 dimensional representation. One particularly nice tool is the 2D grand tour that basically rotates the data cloud in multiple dimensions and shows you an animation of the 2D projection of the rotation. You can slow down or pause the rotation when you see interesting patterns.