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Don't know if this is what you were thinking, but you did get my imagination going.
Attached is a file that has 4 Dimensions that are projected/skewed into formZs 3D space.
Axes, planes and Tetrans (made up name) Negative portions of the axes are not considered here.
in the Skewed 3D space Negative values of any 4D axis can be regarded as positive values in the other 3 axes. i.e.: -1a = 1b,1c,1d (shown in a layer named as such.) and Visa Versa. Which is why I chose not to show the negative values Meaning, mathematically, orthogonal in 4D is not Orthogonal when projected to 3D
Scale is 1unit(4D) = 1foot(3D)
This is the conceptual leap to consider that 90º gets skewed into 3D space at 109.47º (no negative values) or 70.53º (with negative values)
Not considering the negative values, expands the space instead of compressing it.
1 degree of 4space is equivalent to 1.216333333333333 in 3D or (109.47/90)
If negative values are to be considered, this compresses to
70.53/90 1degree in 4space is equivalent to 0.783666666666667º in 3
I could be WAY off base, but this is the result.
How to script any of this, haven't a clue... Not smart enough to handle the translations needed.
Time Travel any one?
Nice job on the 4d space! I imported a star-tetrahedron with a VE, (vector equilibrium), frame object, but was confounded by how to play with alignments, (as I usually had the square profile in top, front & side views). I would have to sleep on it a while as I have been thinking of other geometries lately.
I followed a Synergetics forum many years ago and one of the guys had figured out some kind of tetrahedral coordinates, but it was never demonstrated visually. I'll try writing one of the guys to see if he is still interested, but I was more of a non-math head to those guys, so he might not answer. ^__^
Edit: oops, there seems to be a variety of approaches like this:
But I like your model much more because it is objective! I do wonder if a dual type of grid-work might also add useful snapping, with a triangular grid-work, (like optional grids , not necessarily displayed at the same time. Plus saving views along each axis would be useful.
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