How can complex manifolds and complex numbers help us understand the structure of space?
Space is not necessarily something you see in day-to-day life. You can define geometry locally, but globally you cannot visualize the big picture, you can only imagine it and represent it through coordinates. We draw lines of latitude and longitude in a coordinate system for the continents. But that system doesn’t work well at the north or south poles, where all the lines converge. In order to get a more complete picture in those regions, we need another, more localized coordinate system for more detail. In the end, we need several such coordinate systems patched together to get a detailed picture of the entire globe. More generally, in describing any space, we are not restricted to the three dimensions we experience in our lives. Mathematically, we can suggest any number of dimensions: two, three, four, five, ten, just by drawing additional coordinates on a grid. In complex space, every number in a coordinate system describes not just one dimension but two. Most important, complex num
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