Suppose you choose an orientation for your two shapes, and the computer tells you that the second shadow sticks out past the border of the first shadow. This rules out one point in the parameter space.
But you may be able to rule out much more than a single point. If the second shadow sticks out significantly, it would require a big change to move it inside the first shadow. In other words, you can rule out not just your initial orientation but also “nearby” orientations — an entire block of points in the parameter space. Steininger and Yurkevich came up with a result they called their global theorem, which quantifies precisely how large a block you can rule out in these cases. By testing many different points, you can potentially rule out block after block in the parameter space.
If these blocks cover the entire parameter space, you’ll have proved that your shape is a Nopert. But the size of each block depends on how far the second shadow sticks out beyond the first, and sometimes it doesn’t stick out very far. For instance, suppose you start with the two shapes in exactly the same position, and then you slightly rotate the second shape. Its shadow will at most stick out just a tiny bit past the first shadow, so the global theorem will only rule out a tiny box. These boxes are too small to cover the whole parameter space, leaving the possibility that some point you’ve missed might correspond to a Rupert tunnel.
I think of this problem as being quite canonical.… Aliens would have come to this one. Tom Murphy
To deal with these small reorientations, the pair came up with a complement to their global theorem that they called the local theorem. This result deals with cases where you can find three vertices (or corner points) on the boundary of the original shadow that satisfy some special requirements. For instance, if you connect those three vertices to form a triangle, it must contain the shadow’s center point. The researchers showed that if these requirements are met, then any small reorientation of the shape will create a shadow that pushes at least one of the three vertices further outward. So the new shadow can’t lie inside the original shadow, meaning it doesn’t create a Rupert tunnel.
If your shape casts a shadow that lacks three appropriate vertices, the local theorem won’t apply. And all the previously identified Nopert candidates have at least one shadow with this problem. Steininger and Yurkevich sifted through a database of hundreds of the most symmetric and beautiful convex polyhedra, but they couldn’t find any shape whose shadows all worked. So they decided to generate a suitable shape themselves.
They developed an algorithm to construct shapes and test them for the three-vertices property. Eventually, the algorithm produced the Noperthedron, which is made of 150 triangles and two regular 15-sided polygons. It looks like a rotund crystal vase with a wide base and top; one fan of the work has already 3D-printed a copy to use as a pencil holder.