It just seems interesting that the complex plane seems unreal -- yet as Tobias Dantzig says

For centuries [the concept of complex numbers figured as a sort of mystic bond between reason and imagination.” He quotes Leibniz to convey this turmoil of the intellect: “[T]he Divine Spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world, that amphibian between being and not-being, which we call the imaginary root of negative unity.” p.9

I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment.”

Could there be a relationship between "the twist" and the complex plane?

The twist is (at least for me) an inconceivable geometry.

What is the meaning of multiplication in an imaginary number

The interplay between algebra and geometry is one of the grand themes of mathematics. The magic of high school analytic geometry that allows you to describe geometrically intriguing curves by simple algebraic formulas and tease out hidden properties of geometry by solving simple equations has flowered—in modern mathematics—into a powerful intermingling of algebraic and geometric intuitions, each fortifying the other. René Descartes proclaimed: “I would borrow the best of geometry and of algebra and correct all the faults of the one by the other.” The contemporary mathematician Sir Michael Atiyah, in comparing the glories of geometric intuition with the extraordinary efficacy of algebraic methods, wrote recently:

What is the square root of minus 1?

This seems like a core clue. Maybe this works because the coordinate framework goes through the twist, and the twist is actually the correct and whole way to define numbers

Sat, Feb 20, 2021