A and B modules by Richard Hawkins

Silicon Forest Andragogy

Kirby Urner
2 min readApr 12, 2023


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.


Established orthodox mathematics has shut its doors tightly against any talk of the BEAST modules. Hypercross dogmatists are frankly scared of the BEAST. Rattled. “But space is 3D” they mumble in prayer, proud of their brainwashing.

If you know your A modules, you see that angle between any two quadrays as 2 times arccos of 1/3 i.e. an easy fraction and a jump into trigonometry.

Schools that hold back about A and B modules are in my book earning the justified ire of future alumni. They’ll look back and think “why did our teachers keep us in the dark?” Answer: because the teachers themselves lived in darkness, at the bottom of some silo wherein As and Bs are not allowed.

My School of Tomorrow is helping Silicon Forest kids escape the vortex of ignorance that sucks them under and condemns them to premature dystopianism. I led an all sixth grade assembly on the A module. We folded them from plane nets and made tetrahedrons. That was decades ago. It’s not like we’ve been shy about sharing.

Hypercross dogmatists grow up fatefully block-headed, likely to never think outside their cube-shaped box again. They love their brain coffins.