# Ribbon (mathematics)

In mathematics (differential geometry) by a ribbon (or strip) ${\displaystyle (X,U)}$ is meant a smooth space curve ${\displaystyle X}$ given by a three-dimensional vector ${\displaystyle X(s)}$, depending continuously on the curve arc-length ${\displaystyle s}$ (${\displaystyle a\leq s\leq b}$), together with a smoothly varying unit vector ${\displaystyle U(s)}$ perpendicular to ${\displaystyle X}$ at each point (Blaschke 1950).

The ribbon ${\displaystyle (X,U)}$ is called simple and closed if ${\displaystyle X}$ is simple (i.e. without self-intersections) and closed and if ${\displaystyle U}$ and all its derivatives agree at ${\displaystyle a}$ and ${\displaystyle b}$. For any simple closed ribbon the curves ${\displaystyle X+\varepsilon U}$ given parametrically by ${\displaystyle X(s)+\varepsilon U(s)}$ are, for all sufficiently small positive ${\displaystyle \varepsilon }$, simple closed curves disjoint from ${\displaystyle X}$.

The ribbon concept plays an important role in the Cǎlugǎreǎnu-White-Fuller formula (Fuller 1971), that states that

${\displaystyle Lk=Wr+Tw\;,}$

where ${\displaystyle Lk}$ is the asymptotic (Gauss) linking number (a topological quantity), ${\displaystyle Wr}$ denotes the total writhing number (or simply writhe) and ${\displaystyle Tw}$ is the total twist number (or simply twist).

Ribbon theory investigates geometric and topological aspects of a mathematical reference ribbon associated with physical and biological properties, such as those arising in topological fluid dynamics, DNA modeling and in material science.