Four circles of radius $1$ have their centers at the vertices of a square with side length $2$. The inward-facing quarter-circle arcs joining the midpoints of adjacent sides form the closed curve shown. What is the area enclosed by the curve?

1. The four arc endpoints form a diamond of area $2$. Each arc cuts off a circular segment with area $\frac{\pi}{4}-\frac12$.
2. Subtracting the four segments gives $2-4\left(\frac{\pi}{4}-\frac12\right)=\boxed{4-\pi}$.