AI 中文总结
研究邻接正方形的矩形游戏，从正方形或非正方形矩形开始，证明两种首步类型由相同斐波那契型递归控制，长宽比收敛到黄金比例，还指出直角三角形的HAI构造能确定这两个首步类。
AI 中文摘要
一个邻接正方形的矩形游戏通过简单几何规则生成斐波那契数和斐波那契螺旋。从正方形开始，四个可能的首步通过旋转是等价的。从非正方形矩形开始，第一个正方形有四种几何放置，分为长边优先和短边优先两个等价类。我们证明这两个类由具有不同初始条件的相同斐波那契型递归控制，且两种情况下连续的长宽比都收敛到黄金比例（φ）。然后添加一个简短的几何注释：直角三角形的斜边轴截距（HAI）构造产生一个自然有序种子，其向外和向内的分支恰好确定了这两个首步类。
英文摘要
A square-adjoining rectangle game generates the Fibonacci numbers and the Fibonacci spiral from a simple geometric rule. If one starts from a square, the four possible first moves are all equivalent by rotation. If one starts instead from a non-square rectangle, there are still four geometric placements for the first square, but they split into exactly two equivalence classes: long-side-first and short-side-first. We show that both classes are governed by the same Fibonacci-type recursion with different initial conditions, and that in both cases the successive aspect ratios converge to the golden ratio (phi). We then add a brief geometric remark: the Hypotenuse-Axis Intercept (HAI) construction from a right triangle produces a natural ordered seed whose outward and inward branches determine precisely those two first-move classes.
Comments9 pages, 14 figures
