Within his work: “On the principles of geometry,” (1830), the father of non-Euclidean geometry proved the independence of the 5th Euclid’s postulate from the other 4 axioms. That, by showing that in a plane with one line l and a point p, you can draw infinitely many lines s.t. their intersection with l is the empty set.
In 1884, one of the greatest math-fiction book was published: “Flatland: A Romance of Many Dimensions”, by Edwin A. Abbott. Back in those times, neither non-Euclidean geometries were as recognized as they are today, nor the theory of general relativity had emerged yet. Hence, it was not a coincidence that A. Square’s (Flatland’s main character) perspective on Spaceland ended up as limited as that of the confused Sphere.
Amongst the many sequels of Flatland that have been written, I have found “Flatterland: like flatland, only more so”, by Ian Stewart, to be the most beautiful one.
In Flatterland, Victoria line -Vicky-, whose great great grandfather is none other than A. Square, meets the Space Hoper (a curious being that can jump between subspaces).
Using a Virtual Unreality Engine (VUE) that changes Vicky’s “point of VUE”, Hoper guides Victoria line through several mathiverse’s subspaces. She even visits our Universe and travel’s backwards in time. We all know reality is just a particular case of the mathiverse.
Vicky and Hoper visit the Fractal Forest. There she meets the Sierpiński triangle and the Mandelbrot set, among other objects of fractional dimensions and self-similarity. In the forest, she gets to know fractal geometry.
At Topologica, Vicky learns that a double torus is homeomorphic to a tea pot. Also, she meets Moobius: an annoying, non-orientable surface cow.
Then questions related to the shape of the universe and the geometry of space-time arise: why is space-time hyperbolic? Do we live inside a torus? A sphere? Is the universe bounded? Why some Euclidean metrics are useful even though the space-time geometry is hyperbolic?
What is a symmetry? Why do humans keep searching for a unifying theory of everything?
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