Narcissus and Covenant: One Half, Two Ways

At the end of the HexOct walk (#75, λ ≥ 3/8) the octahedral pieces vanish and each layer tile is two copies of one piece: a cube in a c layer, and in an h layer a ten-cornered piece we call Narcissus. Narcissus is its own mirror image (three mirror planes), so copies of one shape fill space. Two copies joined on their hexagonal faces give two solids. Put on straight they make the twisted cube of #86; turned by 60° they make Covenant, a convex dodecahedron of twelve parallelograms, which is a parallelohedron and fills space by translation along the fcc lattice. The same law of two ways, straight or twisted, one free choice per layer, holds for the classical half of the rhombic dodecahedron (whose two ways are the RD and the TRD), as it holds for the octet pyramids in #92. We also give the Dehn invariants of every HexOct piece exactly: 36(5/24 − λ) ⊗ β for the h pieces (zero from λ = 5/24 on) and 24(3/8 − λ) ⊗ β for the c pieces, with β = arccos(1/√3). Cut a solid in half, and you can put the halves back two ways: as they were, or with one turned. For most solids the turned version has a dent. For a few it does not, and for fewer still it fills space. This chapter is about the few. Figure Also a chapter of HexOct: The Hexagonality Law. The ideas are DICTO's: that the h-tile's piece is one piece, not a pair (and its name, Narcissus); that it might be halved and turned; the name Covenant; the RD/TRD half put on straight or twisted. Computation, proofs and drafting with Claude (Anthropic).