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Mitchell_Porter comments on Rationality Quotes April 2012 - Less Wrong
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I knew a guy who credibly claimed to be able to visualize 5 spacial dimensions. He is a genius math professor with 'autistic savant' tendencies.
I certainly couldn't pull it off and I suspect that at my age it is too late for me to be trained without artificial hardware changes.
The way I would do it for dimensions between d=4 and d=6 is to visualize a (d-3)-dimensional array of cubes. Then you remember that similarly positioned points, in the interior of cubes that are neighbors in the array, are near-neighbors in the extra dimensions (which correspond to the directions of the array). It's not a genuinely six-dimensional visualization, but it's a three-dimensional visualization onto which you can map six-dimensional properties. Then if you make an effort, you could learn how rotations, etc, map onto transformations of objects in the visualization. I would think that all claimed visualizations of four or more dimensions really amount to some comparable combinatorial scheme, backed up with some nonvisual rules of transformation and interpretation.
ETA: I see similar ideas in this subthread.