DICTO Vacuus Solids

The Centre Pyramids: from Plato to Kepler, Part II (companion to #44 Aristotle's Pyramid). Every solid with a centre O inside it is filled by its centre pyramids, one over each face with apex O. DICTO's Vacuus of the solid keeps only the walls between these pyramids: for each edge AB the flat triangle O–A–B, its fin. It is built in two forms, SHEETS (the fins) and STRUTS (a rod from O to every corner, with the solid's edges). We lead with a theorem on a solid P and its polar P* about the same centre: the apex angle of the fin of each edge of P* is π minus the dihedral angle of P at the dual edge, the wedge between two neighbouring fins of P* is π minus the face angle of P at the dual corner, and the fins of an edge and of its dual edge lie in perpendicular planes. The proof is one line from the classical facts that the face fan of P* is the normal fan of P and that a spherical polygon and its polar have supplementary angles and sides. We then give a census of 100 solids: the 69 solids of DICTO Solids v7 and the 31 Platonic, Archimedean and Catalan solids, 92 of them in exact arithmetic. The Vacuus is well defined for 99; the one failure is Lotus Hexa, whose centre lies outside 24 of its face planes. All fins are congruent only for the Platonic and Archimedean solids, the rhombic dodecahedron, the rhombic triacontahedron and Delilah, an octahedron. Twelve of DICTO's solids have all their centre struts of one length, nine of them build their whole frame in at most three lengths, and the cuboctahedron is the only solid whose whole frame has one length. Kagemusha holds a flat central regular pentagon of five 72° fins, and Lion six flat central decagons of 36° fins. Finally, DICTO's origami: each centre pyramid, folded open from one flat fan, nests with the others so that every fin is a double wall. Seven solids were netted, each from one to three nets, and every net folds back to its pyramid to 3.1 × 10−15 of the longest strut. The face fan itself is classical; Leonardo's vacuus bodies for Pacioli (1509) are edge frames with no centre. We found no source that builds the centre-to-edge fins or the centre struts as solids. View in the app: the DICTO Vacuus lens in Polyhedraverse (any solid as pyramids, exploded, Star, sheets and struts). The idea, the family and its names, the two built forms, the cardboard octahedron and the origami are DICTO's. Computation, proofs, exact checks and drafting with Claude (Anthropic).