Hacking Super Mario 64 using covering spaces

4 days ago

Covering spaces in topology involve a continuous surjective map from a space C to X, with local homeomorphism properties.
A homeomorphism is a bijective map between topological spaces where both the map and its inverse are continuous.
A universal cover is a simply connected covering space of X, unique up to homeomorphism, serving as a universal object.
The universal cover of a doughnut (torus) is visualized through stacking copies of a base world, illustrating covering space concepts.
Super Mario 64 exploits covering spaces via floating-point to integer casting, creating 'parallel universes' for speedrunning.
The fundamental group of a space is defined via loops and homotopy, capturing the space's topological structure.
Hyperbolic space models (Poincaré half-space and disk) are introduced as universal covers for certain topological spaces.
Practical exploits in SM64 and real-life scenarios (like banking) show the implications of improper data type casting.

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