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