Ribbon (mathematics) explained
In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector. More formally, a ribbon denoted by
includes a curve
given by a three-dimensional
vector
, depending continuously on the curve
arc-length
(
), and a unit vector
perpendicular to
at each point.
[1] Ribbons have seen particular application as regards
DNA.
[2] Properties and implications
The ribbon
is called
simple if
is a simple curve (i.e. without self-intersections) and
closed and if
and all its derivatives agree at
and
. For any simple closed ribbon the curves
given parametrically by
are, for all sufficiently small positive
, simple closed curves disjoint from
.
The ribbon concept plays an important role in the Călugăreanu-White-Fuller formula,[3] that states that
where
is the asymptotic (Gauss)
linking number, the integer number of turns of the ribbon around its axis;
denotes the total
writhing number (or simply
writhe), a measure of non-planarity of the ribbon's axis curve; and
is the total
twist number (or simply
twist), the rate of rotation of the ribbon around its axis.
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.
See also
Notes and References
- Blaschke, W. (1950) Einführung in die Differentialgeometrie. Springer-Verlag.
- Book: Vologodskiǐ, Aleksandr Vadimovich . Topology and Physics of Circular DNA . 1992 . 978-1138105058 . First . Boca Raton, FL . 49 . 1014356603.
- Fuller. F. Brock. The writhing number of a space curve. Proceedings of the National Academy of Sciences of the United States of America. 1971. 68. 4. 815–819. 10.1073/pnas.68.4.815. 0278197. 5279522. 389050. 1971PNAS...68..815B. free.
