In
differential geometry, a ribbon (or strip) is the combination of a
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
space curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
and its corresponding
normal vector. More formally, a ribbon denoted by
includes a curve
given by a three-dimensional
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
, depending continuously on the curve
arc-length (
), and a unit vector
perpendicular to
at each point. Ribbons have seen particular application as regards
DNA.
Properties and implications
The ribbon
is called ''simple'' if
is a
simple curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
(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,
that states that
:
where
is the asymptotic (Gauss) ''
linking number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In E ...
'', the integer number of turns of the ribbon around its axis;
denotes the total ''writhing number'' (or simply ''
writhe
In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amou ...
''), a measure of non-planarity of the ribbon's axis curve; and
is the total ''twist number'' (or simply ''
twist
Twist may refer to:
In arts and entertainment Film, television, and stage
* ''Twist'' (2003 film), a 2003 independent film loosely based on Charles Dickens's novel ''Oliver Twist''
* ''Twist'' (2021 film), a 2021 modern rendition of ''Olive ...
''), 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 Topological ideas are relevant to fluid dynamics (including magnetohydrodynamics) at the kinematic level, since any fluid flow involves continuous deformation of any transported scalar or vector field. Problems of stirring and mixing are particula ...
,
DNA modeling and in
material science.
See also
*
Bollobás–Riordan polynomial
*
Knots and graphs
*
Knot theory
*
DNA supercoil
DNA supercoiling refers to the amount of twist in a particular DNA strand, which determines the amount of strain on it. A given strand may be "positively supercoiled" or "negatively supercoiled" (more or less tightly wound). The amount of a st ...
*
Möbius strip
References
Bibliography
*
*
*
* {{Citation , last=White , first=James H. , title=Self-linking and the Gauss integral in higher dimensions , journal=
American Journal of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press.
History
The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United S ...
, volume=91 , issue=3 , pages=693–728 , year=1969 , doi=10.2307/2373348 , jstor=2373348 , mr=0253264
Differential geometry
Topology