Twist (mathematics)
In differential geometry, the twist of a "ribbon" is its rate of axial rotation. Let a ribbon formula_1 be composted of space curve formula_2, where formula_3 is the arc length of formula_4, and formula_5 the a unit normal vector, perpendicular at each point to formula_4. Since the ribbon formula_1 has edges formula_4 and formula_9, the twist (or "total twist number") formula_10 measures the average winding of the edge curve formula_11 around and along the axial curve formula_4. According to Love (1944) twist is defined by where formula_14 is the unit tangent vector to formula_4. The total twist number  formula_10 can be decomposed (Moffatt & Ricca 1992) into "normalized total torsion" formula_17 and "intrinsic twist" formula_18 as where formula_20 is the torsion of the space curve formula_4, and formula_22 denotes the total rotation angle of formula_23 along formula_4. Neither formula_25 nor formula_10 are independent of the ribbon field formula_23. Instead, only the normalized torsion formula_28 is an invariant of the curve formula_4 (Banchoff & White 1975). When the ribbon is deformed so as to pass through an "inflectional state" (i.e. formula_4 has a point of inflection), the torsion formula_31 becomes singular. The total torsion formula_28 jumps by formula_33 and the total  angle formula_25 simultaneously makes an equal and opposite jump of formula_35 (Moffatt & Ricca 1992) and formula_10 remains continuous. This behavior has many important consequences for energy considerations in many fields of science. Together with the writhe formula_37 of formula_4, twist is a geometric quantity that plays an important role in the application of the Călugăreanu–White–Fuller formula formula_39 in topological fluid dynamics (for its close relation to kinetic and magnetic helicity of a vector field), physical knot theory, and structural complexity analysis.