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Maybe this is more in the spirit of the physicist, but I've found it useful for n= 4, 5,... 6? Basically any

Any kind of varying visual property of a surface (e.g. color, texture, opacity) can be used to describe one extra dimension.

For This really only helps in low dimensions, but it's quite effective!

To give an example, Hatcher describes visualizing the embedding of the Klein bottle into four space by letting most of the bottle be blue, but having it "blush" as it passes through itself.Quite effective!
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Maybe this is more in the spirit of the physicist, but I've found it useful for n= 4, 5,... 6? Basically any kind of varying visual property of a surface (e.g. color, texture, opacity) can be used to describe one extra dimension.

For example, Hatcher describes visualizing the embedding of the Klein bottle into four space by letting most of the bottle be blue, but having it "blush" as it passes through itself. Quite effective!