One trick I thought of for thinking about high-dimensional spaces is to put multiple dimensions on the same axis: Consider the vectors in R² from the origin onto the unit circle. Lengthen each into a axis, each going infinitely forward and backward, each sharing all its points of R² with one other, all of them intersecting at 0. Embed this in R³, then continuously rotate the tip of each axis into the new dimension, forming a double cone centered at 0. Rotate them further until all tips touch, forming a single axis that contains the information of two dimensions.
You can now have an axis contain the information of any R-vector space, and visualize up to 3 at a time. Of course, not all mental operations that worked in R³ still work.
One trick I thought of for thinking about high-dimensional spaces is to put multiple dimensions on the same axis: Consider the vectors in R² from the origin onto the unit circle. Lengthen each into a axis, each going infinitely forward and backward, each sharing all its points of R² with one other, all of them intersecting at 0. Embed this in R³, then continuously rotate the tip of each axis into the new dimension, forming a double cone centered at 0. Rotate them further until all tips touch, forming a single axis that contains the information of two dimensions.
You can now have an axis contain the information of any R-vector space, and visualize up to 3 at a time. Of course, not all mental operations that worked in R³ still work.
I don’t get what you’re saying. Do you mean, you define a map f: R² → R by f(v) = abs(||v||) (and then you map this to the z-axis)?
“each sharing all its points of R² with one other”
I don’t know what this means; I think you’re saying each axis overlaps with the axis from its antipodal point.
Could you give an example of something that’s difficult to visualize, but it’s easier with this method?