Ribbon (mathematics) | Wikipedia audio article
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Date: 2019-10-07

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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.