Abstract. We study two elementary periodic functions — the scaled fractional part spp x (a pure ramp of period p) and its complement cpp x = p−spp x — together with the single algebraic identity spp x + cpp x = p. Because their derivatives are the constants +1 and −1, they turn a range of nonlinear differential equations into closed one-line solutions, they solve first-order transport and inviscid Burgers equations exactly, and they let one write intricate periodic curves — half-circle chains, staircases, asymmetric arcs, modulated ramps — each from a single algebraic expression. A complex generator Z(x) = sp x+i cp x opens a second, geometric part: its powers Z n are explicit algebraic curves of degree n, and the normalized family {Z α/pα} covers an eye-shaped region with an exactly empty core bounded by the unit circle. The aim is not to compete with Fourier analysis but to offer a light, exact notation for the angular, piecewise-affine patterns where Fourier series are heavy and only approximate. All identities have been verified symbolically and numerically (sympy, machine precision).