Before today’s 3D printing technology allowed us to make objects of virtually any shape, humanity was limited in the kind and precision of shapes we could produce. Driven to overcome these limitations, we gradually invented a series of machines that could create ever-more-complicated types of shapes, culminating (just before the 3D printer) in machines like this multiaxis computer-numerical-controlled (CNC) mill:

The first of this series of machines was perhaps the potter’s wheel, which allowed us to make precisely round objects of any profile. To me, it still seems a bit magical to watch as potters trace out a curve with their hands, and seeing, as the wheel spins, that curve get copied all the way around the vase:

A simple generalization of the potter’s wheel is the lathe. It’s basically a potter’s wheel for wood or metal. Lathes are similar to potter’s wheels, but the work is laid horizontally and a firmly braced, sharp cutting tool is used. The cutting tool can also be attached to a curved guide specifically built to help carve out a certain precise shape repeatedly.

Both the lathe and potter’s wheel create only a certain class of shapes, called solids of revolution. The surface of a solid of revolution is called a surface of revolution.

Mathematically, a surface of revolution is the result of taking a curve in the two-dimensional plane (like the guide on the lathe) and revolving it about an axis. The path swept out by the curve is a surface in three dimensions. A solid of revolution is the result of taking a two-dimensional region of the plane and revolving it about an axis; the path swept out is then a three-dimensional object.

Surfaces and solids of revolution are traditionally taught in the second semester of calculus, as an application of integration. I found them useful for teaching because they provide the next-most intuitive example of the integration concept after the area under a curve, while still only requiring a single integral.

With 3D printing and the Wolfram Language, calculus students can now have imagination and design experience added to the theory and intuition they already get. And as I’ll show, you don’t even need to understand calculus or coding to use Wolfram|Alpha’s virtual potter’s wheel! So, in the spirit of the potter’s wheel, let’s make a cup!

After playing around for a while looking at functions in Wolfram|Alpha (OK, I admit I used the Wolfram Language so I could push and pull on the curve interactively), I found a good candidate “guide” curve, perhaps for a fancy cocktail glass:

Now, I can just change “plot” to “revolve,” and Wolfram|Alpha tells me all about my cup and shows a picture:

In addition to the equations defining the surface and solid here, I get the “Volume of solid” and the “Area of surface.” The volume is about 120—this means that if my unit of measurement were centimeters, my cup would hold 120 mL (120 cc).

But this includes all the volume inside of the stem and base. Some of this space needs to be used by the actual glass (or whatever the cup is made of). So let’s add an inside curve, which I’ll create by taking the part of our first curve from 3 to 7 (using a step function) and subtracting 1/6:

Now, I could either print out the plot and use it as a pattern on my own lathe (if I had one), or continue to use Wolfram|Alpha’s virtual potter’s wheel:

I can now see how much raw material we need to buy for each cup: about 20 cc, which leaves about 100 cc of space for the liquid. I can also calculate how much paint is needed to cover the cup, since the “Surface area of solid” is about 233 cm², and paint typically says on the container how many gallons are needed to cover a certain area.

To print this cup using the Shapeways website, I followed a simplified version of the process described in Vitaliy Kaurov’s post last year.

Although I can produce any surface of revolution in the Wolfram Language using RevolutionPlot3D, I began instead by copying the “Parametric representation of surfaces” and just sticking them into ParametricPlot3D (or if you’re using the Wolfram|Alpha website, the “Wolfram Language plaintext inputs” from the “Parametric representation of surfaces” part of the Wolfram|Alpha result). It turned out that I had to change a couple things to meet Shapeways’s requirements: since they accept millimeters but not centimeters, I had to scale up the plot by 10 in every dimension (using ScalingTransform), and I had to make it 1.6 mm thick, which I did with PlotStyle:

The only other step was to upload the cup in an .stl file to the website:

And… voilà! Here is an image of the finished product, created in plastic for $23:

Wolfram|Alpha can create a lot more kinds of surfaces and solids of revolution, by using different kinds of functions, axes, and amounts of rotation:

I can even rotate an infinitely long part of a curve, which leads to a mathematical “paradox” called Gabriel’s horn (AKA, Torricelli’s trumpet):

Wolfram|Alpha says that the volume is π, but the surface area is infinite! This seems to imply that I could fill the horn with a finite amount of paint, yet this would not be enough to cover the inside surface, which is clearly absurd. Of course, paint is not really a two-dimensional surface coverer, and at some point even a single molecule of paint will not be able to fit into the ever-narrowing tube, so there is no physical paradox here.

Soon Wolfram|Alpha Pro users will be able to access step-by-step solutions for the surface area and volume, showing how to compute each part of the surface area or volume and add them up. I’d love to hear your ideas for other ways to use this technology.

With my new cup now in hand (partly because I forgot to analyze its propensity to tip), I would like to propose a toast to creativity, and the tools that enable us to realize our visions.

Cheers!

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