From the time of Euclid and Archimedes, humankind has wondered about and explored the objects around them. It seems as if we have a natural tendency to question what we see and make puzzles out of what may already be puzzling to grasp.
From solving practical and engineering projects all the way to recreational mathematics, mathematicians constantly confront simple problems that often start with a spark of curiosity but go on to become major puzzles. This piece starts by asking: “Can every geometric shape pass through itself?”
Tunnelling
Suppose we have an object in 3D space. What does it mean for the object to “go through itself”? Perhaps “tunnel through itself” would be a better description. Suppose we have two copies of the object: is it possible to create a tunnel through the other copy, at any angle, without disturbing or breaking the geometric coherence of the object, such that the first copy can, in any orientation, pass through the tunnel? Think of two cubes on a table; if we rotate one by an angle of less than or equal to 45 degrees, we can see with a little thought that one square can pass through the other. The question of whether or not all objects have this property has been a mystery for a long time.
“We are just humble mathematicians”, said Jakob Steininger and Sergey Yurkevich, who first came across this question at university and one day settled this question.
The Rupert property
Unlike Steininger and Yurkevich, the beginning of this problem was not that humble after all. It was Prince Rupert of the Rhine, later Duke of Cumberland, who first wondered about this problem and even went so far as to place a bet with an unknown person, presumably a mathematician, that a cube can tunnel through itself. Hence the name “Rupert property” became a common term for any 3D shape that satisfies the tunnelling procedure described.
Mathematicians and puzzle enthusiasts were on a mission ever since to construct, test, and verify the Rupert property for various shapes. The particularly intriguing and somewhat mysterious observation was that no one seemed to find a convex polyhedron that was not Rupert. A convex polyhedron is a 3D object with flat faces, no concavity, and no internal angles greater than 180 degrees. With a bit of thought, you can observe that you cannot find any two points, either on the surface or in the interior volume of the shape, such that the straight line connecting them leaves the domain covered by the object. With newer polyhedra proven to satisfy the property, it was conjectured that the Rupert property holds for all convex polyhedra – until August 27th, 2025.
Counter example
Given how widespread Rupert property appeared to be, many mathematicians, including Steininger and Yurkevich, decided to try and aim at finding a counterexample. If the conjecture were false, as hypothesised by a few mathematicians, finding a counterexample seemed like a more approachable direction. For this purpose, algorithms were devised that ran on a large sample of possible shapes and orientations to find a counterexample. The first problem with most implementations was that the algorithm was not sufficiently encoded with rigorous constraints to reliably test the Rupert property. As a result, it would accept a shape simply because it could not find an orientation, within its limited search space, that violated the Rupert property. Note that given an object, there can be infinitely many orientations for testing the tunnelling procedure and no computable algorithm can possibly search through these combinations. The second problem was that, since the algorithms can only search through and verify a finite set of orientations and relevant parameters, finding a counterexample is never enough – one still has to prove that the shape does not satisfy Rupert’s property. Hence, it appeared to most mathematicians, that new constraints should provide tighter bounds for the property.
Proof
The original work of Steininger and Yurkevich consists of two important results: the global theorem and the local theorem. Given any orientation of an object, described by a set of rotations, if we (or the computer) can establish that the orientation does not satisfy Ruperts property by a specific margin, we can then reject the property for an entire neighbourhood of nearby orientations. This is known as the global theorem, since it deals with a relatively large, continuous space of points in the higher-dimensional set of all possible rotations. The theorem also quantifies how large the rejection margin can be.
The second theorem, known as the local theorem, deals with cases where three vertices of the shape lie in the overlapping region of a given orientation that satisfy certain geometric conditions related to the triangle formed by connecting them. When these conditions are met, any sufficiently small rotation of the shape will fail to maintain the Rupert property.
Rather than going through the exhaustive process of verifying these conditions for large databases of known shapes, the two mathematicians decided to write an algorithm that constructed convex polyhedra according to the specified constraints and then tested them according to the local and global theorems to verify their properties. Eventually, the “Noperthedron” with 90 vertices, 240 edges, and 152 faces was discovered and proved, using their methods, to not satisfy the Rupert property.
With the first shape proven not to be Rupert, many more conjectured shapes can also be verified. This leads to new doors in this research direction and possibly new problems for the future. Furthermore, this discovery once again affirms the growing role of newly developed algorithms in the ever-growing area of computer-assisted proof construction. The deep ocean of mathematical puzzles, from recreational to professional, seems to create new problems that require human creativity and carefully curated computing machinery.
