They visited the trigonometric valleys where sine and cosine danced with exponential functions, their waves forming patterns that seemed to breathe with mathematical life. The valleys resonated with harmonic frequencies, creating a symphony of pure mathematical relationship.
At the peaks of the logarithmic mountains, the view revealed the entire complex plane spread below them like a vast mathematical continent, with branch cuts forming the mysterious pathways between different sheets of mathematical reality. Here, every complex number possessed infinitely many logarithms, creating a landscape of endless possibility that would challenge even the most adventurous natural philosopher.
"Do you perceive the profound truth?" asked i, gesturing across the mathematical vista with the enthusiasm of an explorer who has discovered new lands. "Every formula in your real world is but a shadow of richer truths that exist here in the complex realm. What you took for the whole of mathematical reality was merely one cross-section of a far grander universe."
One marveled at how e^(iθ) traced perfect circles, how the most complex exponential expressions could equal the simplest results, and how every mathematical relationship in the real world found its fuller expression in this two-dimensional realm. It was as if they had been studying the shadows on the wall of a cave and suddenly stepped outside to see the objects that cast those shadows.
The mathematical landscapes were not metaphors but realities—spaces where abstract concepts took on tangible form, where every equation became geography and every function a living ecosystem. In the exponential gardens, growth spiralled outward in golden ratios. In the trigonometric valleys, waves of possibility rippled through dimensional space. And from the logarithmic peaks, all mathematics spread below like a vast city of interconnected meanings.
Plate VI The mathematical Landscapes.
Gardens, Valleys, and mountains, where complex functions take on the aspect of living geographical features.