One tetrahedron: what collapses, what is kept
A separate construction from the assembly workbench. Start with step 1. No host attachment or particle interpretation is shown here.
Loading the geometry…
The page needs WebGL and access to jsDelivr for Three.js.
One colored sample per sector. There are continuously many possible directions, not just 24.
Tetrahedron
Face detail
What exactly is being drawn?
Step 1. A flag is a nested choice of vertex, edge, face, and body. The barycenters of those four objects form one small tetrahedron. There are 4 × 3 × 2 = 24 flags. This is a subdivision count.
Step 2. If b is the body center and v is a boundary point, the visible point is y = (1 − t)b + tv. Sending t to zero erases v from the visible point. The right panel explicitly stores v separately. It does not claim that an ordinary geometric point contains a visible miniature tetrahedron. The direction boundary is a tetrahedral surface, topologically a sphere.
Step 3. The right triangle is a coordinate chart for the selected coefficient image, not a host spatial triangle. For tetrahedron barycentric coordinates v, its coordinates are u = (vA + vD/3, vB + vD/3, vC + vD/3). This represents the kernel quotient of the retained projector in the declared ABCD alignment. It is not a display of the three complex coefficients themselves.
For m = min(uA,uB,uC), the two boundary preimages are upper = (uA,uB,uC,0) and lower = (uA−m,uB−m,uC−m,3m). On the boundary m = 0 and the points coincide. The dashed segment joins equal-readout endpoints through the solid interior; only its endpoints belong to the direction boundary.
The animated loop follows these explicit global lifts. It does not demonstrate every possible transport law. The host's six-panel belt is deliberately omitted: it is a different object.