The question

How can a convex polyhedron be encoded in a rational function so that its geometry and combinatorics stay recognizable, in the function and in its pictures?

There need not be one right answer, and this is not a story of one. The research compared several recipes and measured what each one preserves and what it costs. It also recorded where each one fails.

• The recipes: the construction in brief, the selected default (R2), its alternative (R4), and a complete worked example.
• Compare recipes: an interactive 3D comparison of where each recipe puts its zeros and poles, for any two solids and recipes side by side.
• The exploration: the full account. It covers how the question was set up and which experiments changed the picture. Every claim is labelled as established background, derived here, numerical observation, conjecture, or design preference.
• Experiments: the comparison atlas and the controlled experiments, with their figures.
• Limitations: how far each result reaches, what the construction gives up, the known weaknesses, and the open questions.
• Decisions: short records of each convention and choice, with the reasons for it and what would reopen it.