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arXiv · 2608.13592

Origametry and antagonistic heptagon graphs: the braced {35,1} and the complete {21,2}

Haroldo Costa Silva Filho
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Abstract

We study two planar unit-distance graphs (UDGs) that both look like heptagons but have opposite rigidity yet kindred arithmetic: the braced heptagon {35,1} (19 vertices, 35 edges) and the complete heptagon {21,2} (21 vertices, 42 edges). Neither is straightedge-and-compass constructible -- already the regular 7-gon is not, by Gauss-Wantzel -- so the natural setting is origametry: the construction of each realization by Huzita-Hatori folds, with the quadratic fold O5 (straightedge-and-compass strength) versus the cubic Beloch fold O6. The complete heptagon is over-braced and globally rigid, with a unique realization whose 42 edges are certified unit exactly over Q(cos 2pi/7), a cyclic cubic field: every vertex is an origami number, reached by one Beloch fold. The braced heptagon is isostatic (minimally rigid) with thousands of non-congruent realizations and a large generic realization count N = 3869504 = 2^6 * 103 * 587; yet we prove that its unit realization is again origami-constructible. In Pegg's explicit construction the pinning angle is a root of an irreducible degree-12 palindromic polynomial whose Chebyshev reduction, an irreducible sextic, factors into quadratics over the heptagon cubic Q(cos 2pi/7); hence the coordinate field is a {2,3}-tower of degree 3*2^k, reached by one trisection followed by quadratics. The primes 103 and 587 therefore belong to the generic complex count, not to the folded coordinates. We solve the governing cubic explicitly by folding (Lill's method and the Beloch fold), place both graphs inside Alperin's field of origami numbers, and read the fold-branch signs of the construction as mountain-valley assignments with an empirical Maekawa-type balance. All claims are checked in exact arithmetic.

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Haroldo Costa Silva Filho. 2026-07-28. Origametry and antagonistic heptagon graphs: the braced {35,1} and the complete {21,2}. https://arxiv.org/abs/2608.13592

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