The White Queen 600-cell: an Exact Atlas of the Slices and Shadows of the 600-cell, and its Fifteen Cleo Shells

The 600-cell is the four-dimensional solid built from 600 regular tetrahedra, twenty round every vertex. DICTO asked for a definitive, proved picture of it. We give a complete exact atlas of its three-dimensional slices in its four main directions, computed in numbers a + b√5. Between two neighbouring vertex heights a slice keeps one pattern of faces, so the cuts at the vertex heights and one cut inside each gap give every slice pattern: 62 on one side of the centre. Exactly twelve are made only of triangles. Every four-sided face is explained: the 46 squares are the equators of the regular octahedra inside the cells, and the 104 kites come from cells sitting symmetrically but tilted across the cut. Each of the four shadows resolves exactly into two halves, its front cells and its back cells, each filling it once; along the six turnings between the main views the set of front cells changes at only 14 moments, one of them at the 600-cell's own fold angle. Round one vertex the 600 cells form 15 shells, each of one kind of piece, the innermost a core of 20 Cleo-Icos filling a regular icosahedron (DICTO's piece, #19): the 600-cell's original tetrahedron as it must look in flat space. DICTO's radial cell projection unfolds the same model in 16 steps from a cell. The most graceful slice, the 48-vertex egg, is Humpty Dumpty (DICTO's name); forced to equal edges it becomes Fallen Humpty, a side note. History from Aristotle's pyramid and Kepler's nested shells; Lewis Carroll's White Queen gives the title. Version 2 adds DICTO's observation that the fifteen Cleo shells read the same backwards (20, 20, 30, 60, 60, 60, 20, 60, 20, 60, 60, 60, 30, 20, 20), with mirrored shells' angles from the vertex adding up to exactly 180°, both from the 600-cell's symmetry through its centre. The ideas and names are DICTO's. Computation and proofs with Claude (Anthropic). Sources checked are listed in the paper; Alicia Boole Stott's section models (about 1900) are credited as the precedent. The study (exact atlas, source model, shells, halves, turnings, kites and squares checks, renders) is attached. Licence CC BY-SA 4.0. More at dictospheres.com.