Eccentric Pieces: why DICTO's sharp-edged pieces keep growing while the round stars freeze

In Star Cores of DICTO's Solids (#79) a star was built on a star: every triangle of the surface gets a spike, mirrored or golden, and DICTO's valley rule shortens each spike just enough to clear the pieces already built and its neighbours. The symmetric shapes freeze after two steps, while seven of DICTO's pieces keep growing: Lahdeli, Chucky, the Triskelion Tetrahedron, the Dodeca-13 Wedge, Samson, Fang and Kagemusha. We explain this by a theorem on a single edge. Cut the solid into pyramids from its centre; at an edge e the two pyramids meet the faces at angles α and α′, and the solid's dihedral angle there is δ = α + α′. While the spikes at e are full, the spike of step k meets e at an angle γk with tan γk = (t − 1)k tan α, where t is the spike factor; the two spikes of step k can both be full only if δ and all the γ's on both sides fit in 360°; and once the two valley faces at e lie in one plane, nothing grows on them again. For mirrored spikes (t = 2) every γk equals α, so step k needs (k + 1)δ ≤ 360°: an edge keeps its valley open through step two only if δ ≤ 120°, through step three only if δ ≤ 90°. For golden spikes the angles shrink by 1/φ each step, and a valley with equal sides stays open for ever if δ ≤ 112.20°. Across the 50 shapes of the valley runs this one number, the sharpest dihedral angle, sorts the growth: each of the 14 convex shapes with an edge of 120° or less still adds 16 per cent or more of volume at step three, and each of the 30 with none adds at most 4.2 per cent (26 of them less than 0.1 per cent). Six of the seven pieces have edges of 90° or less. Kagemusha is non-convex: by a corollary of the law, the faces at its five reflex edges never take a spike, and it grows on its convex edges and ridges instead. DICTO's notch pieces fill the notches: with Maedate the star grows most, and with Kuwagata the star of mirrored spikes stops growing at step three. Whether a star keeps growing thus depends on its sharp edges and notches and on the rule used there. The formula is checked against the valley runs to 4.4 × 10−15 rad, and none of 676 closed valleys grows again. Put a spike on every face of a solid, then a spike on every face of the spikes, and again. Where two spikes lean towards each other, the valley between them fills up after a few steps. Whether it fills depends on one angle: how sharp the edge under the valley is. The ideas and names are DICTO's: the spiked stars, the valley rule and the eccentric pieces. Computation and proof with Claude (Anthropic). More at dictospheres.com.