折纸几何学与对抗七边形图：带支撑的{35,1}图和完全的{21,2}图
Origametry and antagonistic heptagon graphs: the braced {35,1} and the complete {21,2}
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中文总结 AI 辅助
该研究探讨两个形似七边形但刚性与算术性质相反的平面单位距离图，证明二者无法尺规构造，需用折纸方法实现，分析其刚性、坐标域及构造关联，明确{35,1}由{21,2}删两相邻顶点得到。
中文摘要 AI 辅助
我们研究两个平面单位距离图（UDG），二者均形似七边形，但具有相反的刚性与算术性质：带支撑的七边形{35,1}（19个顶点，35条边）和完全七边形{21,2}（21个顶点，42条边）。二者均无法用尺规作图构造——正七边形本身就无法通过高斯-旺策尔定理构造——因此自然的构造场景是折纸几何学：通过Huzita-Hatori折叠构造每个实现，其中二次折叠O5（尺规强度）与三次Beloch折叠O6。完全七边形是超支撑的且全局刚性，具有唯一实现，其42条边被严格证明为单位长度，位于Q(cos2π/7, sin2π/7)上——该循环三次域包含Q(cos2π/7)的三次扩域，因此需要O6（一种循环类型的三次根，与提洛2^(1/3)不同，其伽罗瓦群为S₃）。带支撑的七边形是等静的（最小刚性，通过(2,3)- pebble游戏和通用刚性矩阵验证），但非全局刚性：其拉曼数为N=3869504=2⁶×103×587，素数103、587使其坐标域超出尺规和单次折纸的范围，存在数千种非全等的忠实实现。我们证明{35,1}可通过删除{21,2}的两个相邻顶点得到，并将构造的分支符号与折纸的山-谷分配对应，闭包系统充当平折条件。本研究借鉴了与Edward Pegg Jr.通信中提出的问题。
英文摘要
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.
