Limitations and open questions¶

What the research shows, how far each result reaches, and what it does not show. Labels follow the exploration: [E] established, [D] derived, [N] numerical, [C] conjecture, [P] design preference.

How far the evidence reaches¶

The numerical evidence comes from:

eleven solids: the five Platonic solids, the rhombicuboctahedron and its dual, the triakis tetrahedron, a hexagonal pyramid, and two irregular solids;
five one-parameter families: three transitions on the cube, a pyramid height, and a box shear.

A third irregular solid was added for printing. Results fall into three groups.

Derived, so they hold for every convex input with the origin inside:

the incidence identity \(D_{R2} = D_{R4ve} - D_{R4fe}\), for unreduced divisors;
the polarity relations: R1 and R2 give the reciprocal function, R4ve and R4fe swap, and flag is self-dual;
symmetry of the relief under every symmetry of the solid;
the phase-character form \(f(gx) = e^{i\theta_g} f(x)\);
continuity under fixed-combinatorics deformation;
R4ve's continuity, and R2's jump, under truncating any vertex.

Shown only for the cases tested:

the edge-bevel and raised-face results, on the cube;
the values of the phase characters;
every correlation, degree ratio and gap measurement.

Observed in the corpus, not proved:

that R2 makes no close zero–pole pairs;
that R4 is the only recipe with any transition continuity.

What the construction gives up¶

Distances. Points are placed by direction from the centre, so a solid and a radially rescaled copy with the same directions give the same function. Nothing here reconstructs the solid from the function.
The choice of centre. Every result assumes the origin is strictly inside the solid. Moving it changes the polar dual and every placement.
Some information under reduction. The gcd-reduced function is a root of the recipe's function. Identities and continuity statements refer to the unreduced one.

Known weaknesses of the selected recipes¶

R2 jumps at combinatorial transitions. A vanishing face changes the function by a fixed divisor. This is proved for vertex truncation and observed at the tested edge bevel and raised face.
No incidence recipe is continuous at the tested edge bevel.
A pole can leave its face. On strongly oblique solids the polar direction of a face falls outside the face (on a sheared box, from shear 1). Centroid placement avoids this, but breaks duality.
The phase is not as symmetric as the relief. Under a symmetry the function picks up a factor \(e^{i\theta_g}\), so the colors of a symmetric relief can rotate.
R4 makes tight zero–pole pairs at short edges and small faces. They are what makes R4 continuous, but a print shows them as a spike beside a pit.
The flag recipe gives the same function for a solid and its dual.

Choices that are conventions, not findings [P]¶

The constant: chordal normalization, with geometric mean 1.
The phase: the stereographic chart with a positive leading constant.
The display: relief sharpness. The order-tuned rule equalizes only the highest-order features.
Placement: polar faces and edge feet, chosen for exact duality.
The printed shapes: depth 0.2, the logistic transfer, and the sizes and cut planes.

Printing¶

The printability reference is narrow. It is the smallest feature gaps among four pieces already printed at 130 mm, scaled linearly to 80 mm. Nothing has yet been printed at 80 mm.
The gap measure is a proxy. It is an arc length at sea level, and does not capture the ridge profile between a spike and a pit.
New-recipe pieces are not yet tested. Until they are printed, their settings are proposals.
"Identical" halves are congruent only up to mesh resolution. The latitude–longitude mesh is not exactly symmetric.

Open questions¶

Is any natural recipe continuous under every class of transition, or at least under edge bevels?
What determines the phase characters \(\theta_g\)? Why do R1 and R2 pick up cube roots of unity on tetrahedral inputs, while R4 picks up signs? Can a recipe be chosen to make \(f\) itself invariant when a symmetric phase coloring matters?
How should phases be compared across rotations of the solid? The phase depends on the chart.
When an edge's foot falls off the edge, should the point fall back to the midpoint, at the cost of duality?
What about solids beyond this corpus: non-simple solids with many small faces, or near-regular solids close to several transitions at once?
Prior work. No literature review has been done, so nothing here claims novelty.