A friend on another network pointed out something which makes this seem obvious in retrospect: adding dimensions is just adding undefined variables to a point equation, linearly. The math/notation makes obvious what I was trying to visualize and failing.
So a point in 2D space is p(x, y), in 3D it’s p(x, y, z), and you add a previously undefined variable for each new dimension. That makes the fact that they’re linear super clear.
@a that's quite obvious, but I'm still thinking about the 4D counterpart of a plane, or a "plane" defined with 4 points?
@sirjofri 4 points defines a 3D space (like ours). But visualizing that requires being able to visualize 3D spaces (not really “spaces”; fields?) with a 4D space. I’ve practiced enough that I can do that (not easily), but next I’d need to visualize a 4D space within a 5D space, and there’s no way I was getting to that.
@a wdym 4 points describes a 3d space? A plane with 4 points in 3d space can not be visualized in only one way (that's why we use triangles in CGI). Or do you refer to the axes? (origin, x/y/z)?
And what should "visualizing 3d space in a 4d space" mean? I only know the projection of a 4d object into a 3d space (N to N-1).
@sirjofri Think about a single point (N=1). There are an infinite number of 1-dimensional spaces (that is, lines) which intersect that point. To define a single 1d space, you need a minimum of 2 points: then there is a single line running through them.
Given those two points (N=2), you have an infinite number of 2-dimensional spaces (planes) running through them; easy to visualize as planes rotating on a line. To define a single 2-dimensional space, you need at least one additional point.
@sirjofri So the pattern is always that to define a single N-dimensional space, you need at least N+1 points.
(Again, I think there’s a more precise/correct word than “space” here, which might be “field”, but you get the idea.)
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