i / i² and the Hyperbola: A Fascinating Link Among i, π, e(x), and ln(x)

① This GeoGebra example dynamically demonstrates the ratio of the imaginary unit i to i², constructing a hyperbola while integrating constants and functions such as π, the exponential function e(x), and the natural logarithm ln(x), revealing their deep interconnections. ② Core mathematical concepts include complex number operations (i and i²), hyperbola equations, trigonometric functions, exponential and logarithmic functions, as well as piecewise or stochastic process visualizations. ③ Construction highlights: Based on the construction summary, commands such as Point, Circle, Angle, CircleArc, Area, PointIn, Segment, Textfield, and Intersect are used, creating objects a, b, v, u, w, A, B, c, α, C, s, d, Textc, Pointc, i_x, m, r, β, γ, text1, InputBox1, text2, text3, f, g, D, s_°, Area_∜, h, text4, E, F, G, H, text6 through text11, involving points, circles, angles, arcs, areas, segments, text input boxes, and intersections, with interactive input boxes for dynamic parameter adjustment. ④ Suitable for high school to university-level mathematics learners, this example is ideal for exploring the intersection of complex numbers and analytic geometry, and can serve as a demonstration tool for mathematical modeling or interdisciplinary teaching.