One Move Each: Classical Cut-and-Glue Space-Fillers Built Exactly

In short. Books describe several classical solids that fill space by how to make them from other solids: shave a belt off one, stick two pyramids onto another, glue a few slanted bricks together. Kaleidohedra makes the same solids in one go, by tilting and stretching a lattice of points, and so gives every corner and every volume exactly. The moves themselves are Fedorov's (1885); what this paper adds is the exact data, the settings and the proofs. Main theorem. Five space-fillers that the literature gives by cutting, gluing, zone removal or a drawing are each obtained exactly by one setting of Kaleidohedra's three moves: a shear of the lattice, the equal-edged cell (Cell = 1) and a stretch along a free direction. With edge 1: the regular-hexagon elongated dodecahedron, volume 4√2, symmetry D2h; the Bain rhombic dodecahedron, 2√2, D4h; the classical elongated dodecahedron, 8/3 + 16√3/9, D4h; the Bilinski dodecahedron, (4/5)√(5 + 2√5), D2h; and the 60° rhombohedron, 1/√2, D3d. Each tiles space by translation and is congruent, corner by corner, to its classical construction. Theorem 2: every rhombic-dodecahedron-type cell has exactly three free directions. The Bilinski dodecahedron is glued exactly from two acute and two obtuse golden rhombohedra, and DICTO's skewed rhombic dodecahedron (F1) is a linear image of it. The idea, that Kaleidohedra's lattice moves rebuild exactly the space-fillers the literature gives by cutting and gluing, is DICTO's, as are his skewed rhombic dodecahedron (F1) and his route to the regular-hexagon elongated dodecahedron (F5); computation, proofs and drafting with Claude (Anthropic). More at dictospheres.com.