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 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.

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@a ah, yeah, the term "space" here is very misleading... But I guess "field" is also similar. Maybe something like "system" (from coordinate system) or just "object"?

In 3d CGI, we know the term "object space" which doesn't refer to the object itself, but the coordinate system around that object (from the perspective of that object). So in the end, "space" might as well be just right

sirjofri@sirjofri@mastodon.sdf.org@a btw following your explanation it also refers to the amount of "axes", basically. To build a 3d coordinate system you have three axes. To build these three axes you need (at a minimum) 4 points, etc. It also fits the explanation 100% and is even easier to explain, imo