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An animation of a geodesic in the Heisenberg group, showing the close connection between the Heisenberg group, isoperimetry, and the constant π. The cumulative height of the geodesic is equal to the area of the shaded portion of the unit circle, while the arc length (in the Carnot–Carathéodory metric) is equal to the circumference.

The constant π also appears as a critical spectral parameter in the Fourier transform. This is the integral transform, that takes a complex-valued integrable function f on the real line to the function defined as: