Parametric Pasta: Relaunch of Nested Tori

Today we relaunch Nested Tori, and celebrate with a bit of history. Visualization has always been something that has run strong in my mathematical quests since declaring it as my major at UCLA, and is why I really chose to take up the field of differential geometry, and go to graduate school at UCSD. Of course, that subject is far more abstract than simply what can be pictured, so in effort to stay grounded in the motivations and keep it real, I took up some numerical analysis. That was a refreshing viewpoint, and in fact, what eventually enabled me to get that PhD finished. Well, that, and other things. Suffice to say, my journey has brought me to a very interesting intersection of geometry, topology, real analysis, and numerical analysis (and despite a joke from a colleague, that intersection is not empty!).

During that time, of course, I had to teach, and enjoyed showing several visualizations to, in essence, breathe a soul into some of the calculations that we had them doing. Not that it would be soulless without such pictures, but it was evident that such a soul became more "accessible" after some form of visualization. And one of the favorites, and in fact, apparently what I'm most famous for (according to Google), is my Pasta Parametrization Quiz: Match the following pasta (click to enlarge) [UPDATE: See here for higher-res photos and a more readable formulas]

to the following parametrizations ("equations", also click to enlarge):

(the numbers are given by De Cecco). Despite the media coverage, no, I didn't actually administer it as a "pop quiz" ... handling student rebellions isn't exactly being my favorite task. But I bring this back not just to provide a follow-up, or as a bid to get more attention after my supposed 15 minutes have expired (but it should be pretty scandalous that it hasn't been featured here on Nested Tori, right? Right??) but also to provide an answer key and talk about some of the mathematical ideas surrounding it. If you haven't tried the quiz, please try it before the spoiler.

Really, I want explain a bit about the philosophy behind such a thing, because it motivates much of my work. Parametrizations are one of the building blocks of visualization, and indeed, a lot of other science, principally, anything that has to do with generating numerical data. This is because the numerical data often satisfies certain constraints: it lies on some high-dimensional surface possibly in some higher-dimensional space. The power of geometry is not (just) in its literal descriptions, but rather, some form of metaphorical descriptions. Parametrizations are a way of translating numbers roaming around in easy, planar (flat) sets, into more complicated warped versions. In each of the above parametrizations, one sees the domain of the parameters are things like rectangles, or in one case, the union of three distinct rectangles. The formulas contort the rectangles using those smooth formulas. More complicated objects could be made by simply taking a bunch together, with adjustments such as rotation and translation.

The other common way to describe such sets is to start with a high dimensional space and add in constraints by requiring the coordinates to satisfy certain equations. We've covered an example of this, actually (a way to generate nested tori).

Now on to the answer key (Spoiler Alert!):

Cavatappi: Equation (0.3)
I thought of this parametrization by first considering a torus (of course) which differs from this example only in how the z-coordinate is treated: if we imagine a torus continually wrapping around itself multiple times (hence the greater range of the parameter t), but at the same time, start changing the height, then it doesn't wrap around itself anymore, but rather, moves "out of its own way", vertically. The "β + α cos(s)" part is the horizontal component of the cross-sectional circle, and "α sin(s)" is the vertical component.

Fusilli: Equation (0.5)
This was conceived of as a variation of a helicoid, which in turn is like parametrizing a flat disk by polar coordinates—think of a rotating ray sweeping out a disk uniformly in time. Except, now, we add in a variation in the same way as we did for Cavatappi: move the entire ray vertically as it sweeps. The s²/2 term adds some additional curviness to the pasta surface as we move out. Finally, n=0,1,2 means that we take three of these separate helicoids, rotate them so that they're spaced evenly, and put all three of them together (this increases the density of the spirals).

Conchiglie Rigate: Equation (0.2)
This graphs a parabola in the radial direction, varying with respect to height. In addition, variation in the angle is used to moderate how sharp the parabola should be. The "rigate" part simply comes from the coloring scheme which uses stripes.

Penne Rigate: Equation (0.1)
The simplest parametrization: simply a skewed cylinder, which says radial distance from the z-axis is a constant, and angle and height are free to vary. The "rigate" part is like previous example.

Farfalle: Equation (0.4)
My students provided this to me as a challenge! "But what about bow-tie?" Each one of the three coordinates has a different tweak to make things work. The easiest is the x-coordinate, which yields the end ruffles. For the y-coordinate, we wanted to model the "pinch" as an inverted bell-like curve, giving y as a function of t (no, not the Gaussian curve; this is only quadratic instead of hyperexponential decay). As the parameter s varies, this rescales the curve until it is flat (s = 0), and then inverts it on the other side (s < 0). Finally, the z-coordinate should be wavy and most pronounced in the center. This is the perfect job of that bugbear of beginning calculus students everywhere: sin(θ)/θ.

All in all, this shows that putting together a lot of simple concepts can generate some cool-looking results, even if they are not exact (and indeed, dealing with less easily-modeled objects is why I got into numerical analysis, so approximation is possible, even without a precise formula).