The Icosaxes Hexa: Bloom, Bud and Seed

In short. The DICTO Hexa (#18) is eight DICTO Jewels that pass into each other until they close into one block. DICTO asked for the same move with icosahedral symmetry. Twenty Icosaxes, DICTO's golden truncated octahedra (#48), placed on the corners of a dodecahedron and slid towards the centre first touch as the Icosaxes Bloom, with twelve windows (at distance φ²√3/2); at exactly 1/φ of that distance the windows close into the Icosaxes Bud, sealed round a hollow; at exactly one third the hollow fills: the Icosaxes Seed, one solid (with Lotus Jewel Axes: at 1/φ³). Proved exactly. In the lattice, eight Icosaxes round an Unborn Axes make one solid (the same as nine Icosaxes passing through each other, along the whole golden bond from Kelvin onwards), and these blocks form a Kagome network in which every Icosaxes is shared by exactly two blocks; keyed with Tricaps, its piece has volume exactly φ².

The idea is DICTO's; the names are DICTO's working names. Computation and proof with Claude (Anthropic); a first draft by a local open model (gpt-oss-120b), checked line by line. Licence CC BY-SA 4.0. More at dictospheres.com.

View in the app: https://kaleidohedra.dictospheres.com/?open=bloom