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Over the past few decades, topology—a branch of mathematics dealing with shapes that can be turned into other shapes by processes like bending and stretching—has evolved from an arcane pursuit into an increasingly powerful tool for analyzing the real world. The role of symmetry in the topological world has turned out to be particularly important.

A symmetry is any property that stays the same as the viewpoint changes. A square looks the same as it rotates through 90 degrees, for example. That form of invariance is called rotational symmetry.

But there are some topological structures that persist across scales. Mathematicians call these symmetries persistent homologies, and the study of them is turning out to be the key to a wide range of problems such as network analysis, data mining, and  understanding the brain’s wiring diagram.

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Category: Science