数源几何:数字生成图形单值化原理

The relationship between numbers and geometry has been a fundamental pursuit of mathematics ever since the discovery of irrational numbers sparked the first mathematical crisis, challenging the Pythagoreans' doctrine that "all is number." This paper proposes a novel geometric framework—"Shuyuan Geometry." Departing from axioms and coordinates, this framework starts from numbers themselves to answer a fundamental question: Can a single number directly generate a geometric figure? Taking the circle as a special case, it demonstrates the process: number → the infinity formed by the diffusion and convergence of numbers → "pointization" → specific generation rules → geometric figure, making it possible to redefine geometry within this framework. In the generation of a circle, a fixed source-number and the radius r uniquely determine the circle. Based on this, the univalent mappings and theorems of "number → circle" (each number uniquely corresponds to the circle it generates) and "circle → number" (each circle uniquely corresponds to the number that generates it) are defined. The paper also discusses the unlocatability of points, the randomness of numbers, and the natural extension of the framework to figures such as disks, spheres, and cylinders. This work provides a number-driven new perspective for geometry, inheriting the spirit of the intrinsic tradition of geometry—freeing geometric objects from the constraints of external coordinates and moving towards a self-sufficient world.