I think what the word “quadrays” has over “tetrays” is its close association with the word “quadrant”.

Those groomed to be hypercross dogmatists (starting with “space is 3D”) aren’t used to imagining dividing space into four symmetrically identical sectors, by means of four “fences” defined by any pair of quadrays.

Go back in your mind to K-12 and ask yourself at what point did any textbook ever suggest “four quadrants” of space. Instead, we go from a number line dividing space into two “half planes” to two planes intersecting at 90 degrees, while enclosing no volume.

In that sense we have four quadrants of space, two planes at 90 degrees with no well defined origin, as their intersection is a number line.

Once we establish an origin somewhere along that line, we get a corner where now three planes meet, but at 90 degrees to each other, and still with no volume enclosed. Space has now been divided into 8 octants and we’re locked in to worshipping the 90 degree angle. We’ve entered a rectilinear hell.

The four quadrays reach out as “spines” of the 24 A modules. The center of volume of a regular tetrahedron floats 1/3 of the way from each floor to the opposite apex. This gives us enough info to compute that central caltrop angle of 109.4712206 degrees.