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Ribbon (mathematics)

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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[edit]

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[edit]

References[edit]

  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.{{cite book}}: CS1 maint: location missing publisher (link)
  3. ^ Fuller, F. Brock (1971). "The writhing number of a space curve" (PDF). Proceedings of the National Academy of Sciences of the United States of America. 68 (4): 815–819. Bibcode:1971PNAS...68..815B. doi:10.1073/pnas.68.4.815. MR 0278197. PMC 389050. PMID 5279522.

Bibliography[edit]