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faul_sname comments on Should correlation coefficients be expressed as angles? - Less Wrong
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All's well and good, until you get to adding angles together, unless you visualize A,B,C existing on a sphere and the angles between them being from the center of the sphere- are the possible angles of the sphere identical to the inverse cosines of the possible correlations? (If A and B are .707 correlated, and B and C are .707 correlated, are A and C necessarily somewhere between 0 and 1 correlated?)
For four variables (six angles), you would require visualizing them on a four-dimensional surface analogous to a sphere. I can do some of the math involved in calculating the possible angles there, but I can't visualize it beyond 'there are four points on that surface such that all of them are 90 degrees apart.' I know that any three of those points uniquely define a sphere, and each of those three spheres is a cross-section of the four-dimensional surface.
Apparently, some people can visualize more than 3 dimensions fairly easily. As for me, I use a little trick that engages the ability of my visual-spatial mind to visualize more than one object at a time.
To visualize a 6-sphere, I usually visualize a sphere of fixed size with three axes going through it. This sphere and the axes represent the higher-order dimensions. I then imagine myself somewhere in this three-dimensional space (specifically, somewhere inside the fixed sphere). I note the distance directly from the x, y, and z axes directly through me to the edge of the sphere. Each of these distances defines the radius of the 'visible surface' of the 6-sphere from that point in higher-order space looking from the x, y, or z axis respectively. By rotating the axes, the relative sizes of these surfaces change, and I'm guessing you can already visualize a rotating normal sphere in your mind. So you can rotate the 6-sphere in two sets of 3 dimensions fairly easily. Rotating between higher and lower dimensions is a bit more challenging, but still doable. For a 4- or 5- sphere, replace the 3-dimensional higher-order sphere with a circle, or just ignore the z axis.
If you can figure out how to do that, you can get a feeling for the possible orientations of the angles to one another. And that actually is fairly interesting, and the first time I really understood on a gut level why r^2 is used instead of r for correlations.
Glad that works for you. I lose sight of where (0,0,1,0,0,0) is as soon as I rotate around any axis other than the z-axis, and I can never find (0,0,0,0,.707,.707) or any other point not on the reference plane for a higher-order dimension.