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I found the pages on Linear Algebra very interesting! Particulary the bit about defining dimension as being able to draw in many directions. What about if it was defined as having 1, 2, or 3 axises (x,y,z)? :)
I might as well answer now, I don't know if it would notify you tho. The question is: what even is an axis? In Linear Algebra, this question is basically answered by the concept of the standard basis.