Ribbon (mathematics) | Wikipedia audio article

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- Socrates


SUMMARY
=======
In mathematics (differential geometry) by a ribbon (or strip)



(
X
,
U
)


{\displaystyle (X,U)}
is meant a smooth space curve



X


{\displaystyle X}
given by a three-dimensional vector



X
(
s
)


{\displaystyle X(s)}
, depending continuously on the curve arc-length



s


{\displaystyle s}
(



a
≤
s
≤
b


{\displaystyle a\leq s\leq b}
), together with a smoothly varying unit vector



U
(
s
)


{\displaystyle U(s)}
perpendicular to



X


{\displaystyle X}
at each point (Blaschke 1950).
The ribbon



(
X
,
U
)


{\displaystyle (X,U)}
is called simple and closed if



X


{\displaystyle X}
is simple (i.e. without self-intersections) and closed and if



U


{\displaystyle U}
and all its derivatives agree at



a


{\displaystyle a}
and



b


{\displaystyle b}
.
For any simple closed ribbon the curves



X
+
ε
U


{\displaystyle X+\varepsilon U}
given parametrically by



X
(
s
)
+
ε
U
(
s
)


{\displaystyle X(s)+\varepsilon U(s)}
are, for all sufficiently small positive



ε


{\displaystyle \varepsilon }
, simple closed curves disjoint from



X


{\displaystyle X}
.
The ribbon concept plays an important role in the Cǎlugǎreǎnu-White-Fuller
formula (Fuller 1971), that states that




L
k
=
W
r
+
T
w

,


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



L
k


{\displaystyle Lk}
is the asymptotic (Gauss) linking number (a topological quantity),



W
r


{\displaystyle Wr}
denotes the total writhing number (or simply writhe) and



T
w


{\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.