Geometry of Călugăreanu's theorem

MR Dennis, JH Hannay

Research output: Contribution to journalArticle (Academic Journal)peer-review

44 Citations (Scopus)


A central result in the space geometry of closed twisted ribbons is Călugăreanu's theorem (also known as White's formula, or the Călugăreanu–White–Fuller theorem). This enables the integer linking number of the two edges of the ribbon to be written as the sum of the ribbon twist (the rate of rotation of the ribbon about its axis) and its writhe. We show that twice the twist is the average, over all projection directions, of the number of places where the ribbon appears edge-on (signed appropriately)—the ‘local’ crossing number of the ribbon edges. This complements the common interpretation of writhe as the average number of signed self-crossings of the ribbon axis curve. Using the formalism we develop, we also construct a geometrically natural ribbon on any closed space curve—the ‘writhe framing’ ribbon. By definition, the twist of this ribbon compensates its writhe, so its linking number is always zero.
Translated title of the contributionGeometry of Călugăreanu's theorem
Original languageEnglish
Pages (from-to)3245 - 3254
Number of pages10
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume461 (2062)
Publication statusPublished - Oct 2005

Bibliographical note

Publisher: The Royal Society
Other: arXiv:math-ph/0503012


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