Ever since I built a Tetra Pak® model of the Campanus Sphere, I’ve been dreaming about building one of the mazzocchio using the same method. Upcycling Tetra Paks is not just good for the environment; it is a free source of a rugged material with has an elegant silvery sheen.

The mazzocchio is a donut-shape headdress worn by men in Italy in the 1400’s. Like the Campanus Sphere, the mazzocchio has been the subject of many Renaissance studies in perspective in various media.

I quickly found out that making a mazzocchio in the medium of snapology origami was much more difficult than making a Campanus Sphere. Both shapes require “custom made” quadrilateral “faces” — as opposed to traditional snapology origami, in which faces are typically regular polygons. Other than the quadrilateral faces, the Campanus sphere is quite straightforward to make. The hinged connectors all open away from the center of the “sphere”, and the hinges approximately span a spherical surface. In mathematical sense, the Campanus sphere approximates a surface with constant positive curvature.

The mazzocchio is an entirely different animal since it is a torus, or donut, and therefore has areas with positive, zero, and negative curvature. Areas with positive curvature are shaped like mountains, those with zero curvature are flat, and those with negative curvature are roughly saddle-shaped.

What I mean by curvature here is officially called *Gaussian curvature*. Roughly speaking, it measures how a surface bends in space.

Let’s keep in mind these shapes as we have a closer look at Leonardo’s Mazzocchio.

The way to construct surfaces with variable curvatures in snapology origami is to carefully choose the orientation of the hinged connectors. Here is a mock up of representatives of the two families of circles of the mazzocchio.

The above picture illustrates a very simple principle: in order to create a polygon in snapology origami, the hinges must open *away* from the center of the polygon. In the case of the mazzocchio, this means that some of the hinges will open into the torus (like the pink hinges) and some hinges will open out of the torus (like the yellow hinges). Differential geometers will recognize that the orientation of these hinges exactly corresponds to the sign of principal curvatures at the various “points” of the mazzocchio.

Here is another illustration of the hinge orientations in a partially constructed mazzocchio.

All this is well and good, but it creates significant challenges in determining the dimensions of the various polygons that make up the mazzocchio. With the various orientations of the hinges, there is not a clearly designated “inside” and “outside” (in contrast, with, say, the Campanus Sphere, where all the hinges point away from the center). Because of this, it is never quite clear whether the goal is to model the inner or the outer surface.

There were calculations, measurements, calculations based on the measurements, and many unsuccessful mazzocchi before I arrived at the one in the main image of this post. The name “Full Circles” has several references. It refers to the two topologically distinct and mutually perpendicular families of circles which comprise the piece—the large “circles” (32-gons), and the smaller “circles” (octagons). The name also hints at the fact that the Tetra Pak® containers have gone full circle from being a valuable container material to potential trash to being used again to make an object from Leonardo da Vinci’s canon.

*This is my submission to the Bridges Exhibition of Mathematical Art at the 2025 Joint Mathematics Meetings in Seattle.*