Ribbon (mathematics)

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 (X,U) includes a curve X given by a three-dimensional vector X(s), depending continuously on the curve arc-length s (a≤s≤b), and a unit vector U(s) perpendicular to X at each point.[1] Ribbons have seen particular application as regards DNA.[2]

## Contents

* 1 Properties and implications
* 2 See also
* 3 References
* 4 Bibliography

## Properties and implications

The ribbon (X,U) is called simple if X is a simple curve (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+εU given parametrically by X(s)+εU(s) are, for all sufficiently small positive ε, simple closed curves disjoint from X.

The ribbon concept plays an important role in the Călugăreanu-White-Fuller formula,[3] that states that

Lk=Wr+Tw,

where Lk is the asymptotic (Gauss) linking number, the integer number of turns of the ribbon around its axis; Wr denotes the total writhing number (or simply writhe), a measure of non-planarity of the ribbon's axis curve; and Tw 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

* Bollobás–Riordan polynomial
* Knots and graphs
* Knot theory
* DNA supercoil
* Möbius strip

## References

1. ↑ Blaschke, W. (1950) Einführung in die Differentialgeometrie. Springer-Verlag. ISBN 9783817115495
2. ↑ Vologodskiǐ, Aleksandr Vadimovich (1992). Topology and Physics of Circular DNA (First ed.). Boca Raton, FL. p. 49. ISBN 978-1138105058. OCLC 1014356603.
3. ↑ Fuller, F. Brock (1971). "The writhing number of a space curve". Proceedings of the National Academy of Sciences of the United States of America 68 (4): 815–819. doi:10.1073/pnas.68.4.815. PMID 5279522. PMC 389050. Bibcode: 1971PNAS...68..815B. http://www.pnas.org/content/68/4/815.full.pdf.

## Bibliography

* Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 0-8218-3678-1
* Călugăreanu, Gheorghe (1959), "L'intégrale de Gauss et l'analyse des nœuds tridimensionnels", Revue de Mathématiques Pure et Appliquées 4: 5–20
* Călugăreanu, Gheorghe (1961), "Sur les classes d'isotopie des noeuds tridimensionels et leurs invariants", Czechoslovak Mathematical Journal 11: 588–625, doi:10.21136/CMJ.1961.100486
* White, James H. (1969), "Self-linking and the Gauss integral in higher dimensions", American Journal of Mathematics 91 (3): 693–728, doi:10.2307/2373348

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Ribbon (mathematics). Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Ribbon_(mathematics)&oldid=193102"