The use of the golden ratio in simulations on a torus reveals a fascinating mathematical property: the generation of non-repeating, non-intersecting paths that densely cover a surface without retracing any point... This behavior arises from the golden ratio's irrational nature… Unlike rational numbers, which can be expressed as fractions and eventually cycle through patterns when used in modular arithmetic, irrational numbers like the golden ratio never resolve into repeating decimals… When used

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The torus is a three-dimensional surface shaped like a donut, formed by rotating a circle around an axis in its plane that does not intersect the circle. It is a classic example of a surface with a hole, making it topologically distinct from a sphere or a cube. In mathematics, the torus is a fundamental object in topology, where it serves as a basic model of a surface with genus one. Beyond pure math, the torus appears in physics, computer graphics, and engineering—modeling anything from magneti