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How can a circle be continuously deformed into a square?
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topology
Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.
In topology, a circle can be continuously deformed into a square because they are topologically equivalent, meaning there exists a homeomorphism between them.
While properties like size and straightness are not preserved in topology, the fundamental structure of a simple closed curve is. Any simple polygon, such as a square, is homeomorphic to a circle. This means you can map every point on a circle to a point on a square using a continuous function that is one-to-one and has a continuous inverse. Intuitively, this deformation is possible because you can stretch or bend the circle into the shape of a square without tearing it or joining any new points, maintaining the same connectedness properties throughout the transformation.12