Geodesics and polyhedra

Apr 20, 2025

A geodesic on a convex surface is a curve on the surface that is “locally straight”. For example, a geodesic curve on a sphere is a great circle, meaning a circle whose radius is the same as the radius of the sphere. The shortest way to travel between two points on the surface of a perfectly spherical planet is to follow a great circle that contains the two points. The flight path between two distant airports appears curved when projected to a rectangular map because the most efficient flight path approximately follows a great circle.

A geodesic is called simple if it never intersects itself, and closed if it eventually comes back to its starting point. In the case of the cube, there are three types of simple closed geodesics. We still require a geodesic to be locally straight, which in this case means that when the polyhedron is folded flat, the geodesic needs to be a straight line. A perfectly horizontal or vertical equatorial band around the cube will satisfy these conditions, but it turns out that there are two other types of geodesic, as follows.

The picture above shows a straight line drawn on the net of a cube. The faces of the cube are labelled by the letters L, R, T, B, F, and K, which stand for Left, Right, Top, Bottom, Front, and bacK. If the line has a slope of 1, then it is possible to arrange for the line to form a geodesic once the shape has been folded back into a cube. The resulting geodesic is the boundary of a flat hexagon that is at right angles to one of the long diagonals connecting opposite points of the cube.

The third type of geodesic on a cube is the one shown above. In this case, the straight red line on the net of the cube has a slope of 2 (or ½), and the resulting geodesic is not the boundary of a flat two dimensional shape. It is a theorem that any other simple closed geodesic on the cube is equivalent to one of these three geodesics using a combination of reflectional and rotational symmetries, and parallelism. Pushing an equatorial band slightly to one side to form a different equatorial band is an example of parallelism.

In the case of the tetrahedron, there are infinitely many different possible angles that the straight red line can take. It turns out that any angle whose tangent is a rational multiple of √3, such as 30° or 60°, can give rise to a geodesic. The geodesics on a tetrahedron can be long compared to the side length of the shape, as can be seen from the picture above.

A geodesic has the property that there is an angle of 180° (π radians) on either side of the curve at any of its points. More generally, a curve that has an angle of at most 180° on either side at any point is called a quasigeodesic. Unlike a geodesic, a quasigeodesic can pass through a vertex of the shape; for example, the quasigeodesic shown in the picture above passes through four vertices. The vertex v₁ is surrounded by three right angled corners, which add up to only 270°, but the quasigeodesic condition is satisfied because the angles above the geodesic at v₁ add up to 180°, and the angles below the geodesic at v₁ add up to 90°, neither of which is more than 180°. In 1949, Aleksei Pogorelov proved that every convex surface has at least three simple closed quasigeodesics, although he did not give an algorithm to construct them.

The recent 12-author paper Quasigeodesics on the Cube by the MIT CompGeom Group proves that, up to equivalence, there are precisely 15 distinct quasigeodesics on the cube, in addition to the three geodesics mentioned previously. Their classification is shown in the diagram above. The authors expect it to be difficult to generalize this result to other shapes, and there is currently no known bound for the number of quasigeodesics that are not geodesics for a polyhedron with n vertices.

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