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Tangential question: What are good ways to visualize multidimensional data? Excel etc can chart 3D data, but even that is hard because stuff overlaps. And years ago, I had some Excel code that generated 3D views of 4D data.

But with five or more dimensions, it's too tedious and unworkable. So I end up guessing a lot about what projections are useful. It's like that story about blind people trying to perceive an elephant.

Anyway, just thought I'd ask.

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Use non spatial dimensions as well, like color, size, and shape. You can also use clever projections like t-sne to reduce the number of dimensions while retaining some of the structure of the data (https://www.youtube.com/watch?v=RJVL80Gg3lA)

There's an improvement to T-SNE called UMAP

I've used them both recently and couldn't really tell which one is visually better; UMAP is supposed to be a bit faster though.

Thanks, that's very useful.

I know someone with aphantasia who is an excellent mathematician, I asked him what he does instead of visualising and he says that he does mathematics with the kinesthetic feeling of arbitary imaginary limbs. This may possibly explain why he is one of the very few people I know who seems to experience no conceptual jump in difficulty when solving problems that have 4 or more dimensions when compared to solving those with 3 or less.

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