1sphere

The 1-sphere, denoted S^1 and commonly called the unit circle, is the set of points in the plane at distance 1 from the origin. Equivalently, S^1 = { (x,y) ∈ R^2 : x^2 + y^2 = 1 }. It is a one-dimensional smooth manifold and a compact, connected submanifold of R^2. It can be parameterized by angle θ, with r(θ) = (cos θ, sin θ), and identifies with the unit complex numbers via e^{iθ}. As a topological space, S^1 is homeomorphic to the quotient R/2πZ and is a simple closed curve. In algebraic topology, its fundamental group is isomorphic to Z, and its first homology group is Z; its Euler characteristic is 0. The universal cover of S^1 is R, with covering map t ↦ e^{it}. Geometrically, the circle has constant curvature 1 and circumference 2π. S^1 embeds naturally as the boundary of the unit disk D^2 in R^2 and as a Lie group under complex multiplication, isomorphic to U(1). In mathematics, S^1 serves as a basic example of a compact Lie group, a homogeneous space, and a building block in Fourier analysis, complex analysis on the unit disk, and in the study of loops and homotopy. Generalizations include higher spheres S^n, defined as unit spheres in R^{n+1}.