Paradoxical Cohabitation and Emergence in a Universal Structure—A Geometric Model

The unit circle is the universal basis for the trigonometric functions on the two-dimensional plane. It is applied to analyze the paradoxical internal structure of a universal state, the 1-D geometric model. The geometry includes an inner circumference to the unit circle and is analyzed by applying two mathematical frameworks to the right triangle. The first is a formal interpretation, and the second is nonformal, in which the dimensional complexity is outwardly emergent from the inner to the outer circumference. A novel format of vector structure, each with a unit identity of 1, replaces the magnitudes for the hypotenuse and its sides. Although the two formats of the unit circle are paradoxical, with the second having no rational basis in geometry, the cosine squared calculations for the right triangle agree in both frameworks. The study concludes that two paradoxical frameworks cohabitate within a universal state, and the mechanism of paradox is validated as the basis of the relationship between them. The geometric model explores the structural basis for the emergence of complexity across dimensional boundaries. Paradoxical cohabitation is conjectured not as an anomaly but to represent the static format between segments in the process of emergence. The 1-D geometry defines its universal state as an infinity. Beyond the two-dimensional basis of the toy model, it broadly conjectures the role of paradox as a fundamental relationship mechanism in the Universe.