Ribbon (mathematics)

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

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

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

$Lk = Wr + Tw \;,$

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

References

• Blaschke, W. (1950) Einführung in die Differentialgeometrie. Springer-Verlag. ISBN 9783817115495
• Fuller, F.B. (1971) The writhing number of a space curve. PNAS USA 68, 815-819.