Cavatappo means corkscrew in Italian (used to open wine bottles!), which is the shape in common language (English) for this kind of spiral shape. It was originally created as a particular Italian pasta brand by Barilla now familiar in the USA, but quickly became popular in Italy and then named generically for its obvious shape. This surface shape has very interesting mathematical symmetry and differential geometry properties. The usual pasta shape is ridged (cavatappi rigati) to better hold pasta sauce, but the smooth version is nicer mathematically, although the differential geometry of both surfaces embodied in its so called geodesics (locally shortest distance curves) requires numerical solution of differential equations to describe.
The simplest cavatappo shape 1.0 takes a vertical circle centered on a point of a helix curve with a vertical axis of symmetry and translates and rotates it upwards along that helix to sweep out the surface for a certain low number of revolutions, a choice made by George Legendre in his pasta book Pasta by Design (2011). The details are described in the article Geodesics on Surfaces with Helical Symmetry: Cavatappi Geometry (2012). This class of surfaces exhibit the mathematical symmetry called screw symmetry, named for the simultaneous rotation and translation about and along an axis of symmetry, like a physical screw. The torus and in turn sphere are special cases of this symmetry in which the translation is removed. Indeed surfaces of revolution easily generalize to the screw symmetry case, with the parallels and meridian terminology generalizing the lines of longitude (parallels) and latitude (meridians) of the sphere.
One can tilt this circle backwards by a fixed angle before translating it upwards along the helix to get the larger 2.0 family of cavatappo surfaces. described in the article Cavatappi 2.0: More of the same but better (2014). When the tiltback angle equals the tilt angle of the helix tangent direction with respect to the plane orthogonal to its axis of symmetry, the circle is orthogonal to the helix tangent direction, giving the orthogonally tilted shape we will study here and which has a shape that is more pleasing than the 1.0 shape. This is of course a subjective observation! Various concrete parameter values model the actual pasta shapes one finds in most US supermarkets these days. Mathematically the orthogonal tiltback condition places these surfaces in the category of tubular surfaces, which are generated by sweeping along a curve a fixed (even variable) radius circle in the plane orthogonal to the tangent direction. This leads to a natural description in terms of the Frenet-Serret T-N-B frame along the curve (studied in multivariable calculus!). All of these surfaces (1.0 and 2.0) have what is called screw axis symmetry, since they are invariant under a simultaneous translation along the screw symmetry axis and a rotation about it.
Realistic cavatelli pasta one finds in the supermarket have their end cuts that can be either in the 1.0 shape parallel to the symmetry axis or the orthogonal 2.0 shape or simply not very precise, depending on the tilt back angle. The fact is that mathematical models do not always correspond 100 percent to an actual system we find in our world. Investigating the actual manufacturing process is not where we want to put our energy!
Investigating the geodesic structure of this family of surfaces has been a fun and fascinating experience, but would never have begun without a curious unintuitive bevavior of geodesics on the donut surface called a torus found by my friend and Villanova colleague of many decades Klaus Volpert. I reanalyzed the problem developing my own Maple procedure from his for numerically calculating those geodesics allowing it to apply to a whole family of rotationally symmetric surfaces and turned off the curvature to see the flat plane limit behavior where the geodesics should be straight lines. But they weren't straight because of a missing factor of two in the crucial formula. Restoring that factor led to straight geodesic lines, beginning my lengthy study of the geodesics on the torus summarized in the article Geodesics on the Torus and other Surfaces of Revolution Clarified Using Undergraduate Physics Tricks with Bonus: Nonrelativistic and Relativistic Kepler Problems (2010). This is a wonderful example of the value of academic collaboration. Maple is a powerful computer algebra system that allows sophisticated mathematical analysis combining symbolic, numerical and graphical tools. Its chief competitor Mathematica was used in the Pasta by Design book.
Extending the torus from its rotational symmetry to the screw axis symmetry of its helical generalization led to the combination of pasta and mathematics embodied in the cavatappo surface, the details of which are described in the article Geodesics on Surfaces with Helical Symmetry: Cavatappi Geometry (2012) with all the details. If that factor of two had not been missing, I never would have embarked on this journey. And this extension to pasta shapes arrived just in time to serve as a way to give a suggestive explanation of planetary orbits as spacetime geodesics in general relativity to a popular audience for a faculty research award talk I gave in 2012. Coincidentally Klaus and I got an award for an article On the Mathematics of Income Inequality: Splitting the Gini Index in Two (2012) measuring income inequality at a crucial point in American history when it led to the shortlived Occupy Wall Street movement of that year. This came out of another Maple collaboration in which absent either one of us, the key idea of the article would not have seen the light of day.
Lengthy Maple worksheets exploring the geometry of the cavatappo 1.0 and 2.0 surfaces is available here (but will be improved as time permits):
Archived articles explaining the details, all stemming from the discussion of geodesics on the torus. Numbered links point to abstracts and local PDF copies in my list of publications: