for the International Mathematical Union logo. Its length is , while the best proven lower bound on the length is . For a discrete analogue of ropelength, the shortest representation using only edges of the integer lattice, the minimum length for the Borromean rings is exactly . This is the length of a representation using three integer rectangles, inscribed in Jessen's icosahedron in the same way that the representation by golden rectangles is inscribed in the regular icosahedron. Hyperbolic geometry The Borromean rings are a hyperbolic link: the space surrounding the Borromean rings (their link complement) admits a complete hyperbolic metric of finite volume. Although hyperbolic links are now considered plentiful, the Borromean rings were one of the earliest examples to be proved hyperbolic, in the 1970s, and this link complement was a central example in the video Not Knot, produced in 1991 by the Geometry Center. Hyperbolic manifolds can be decomposed in a canonical way into gluings of hyperbolic polyhedra (the Epstein–Penner decomposition) and for the Borromean complement this decomposition consists of two ideal regular octahedra. The volume of the Borromean complement is where is the Lobachevsky function and is Catalan's constant. The complement of the Borromean rings is universal, in the sense that every closed 3-manifold is a branched cover over this space. Number theory In arithmetic topology, there is an analogy between knots and prime numbers in which one considers links between primes. The triple of primes are linked modulo 2 (the Rédei symbol is −1) but are pairwise unlinked modulo 2 (the Legendre symbols are all 1). Therefore, these primes have been called a "proper Borromean triple modulo 2" or "mod 2 Borromean primes". Physical realizations A monkey's fist knot is essentially a 3-dimensional representation of the Borromean rings, albeit with three layers, in most cases. Sculptor John Robinson has made artworks with three equilateral triangles made out of sheet metal, linked to form Borromean rings and resembling a three-dimensional version of the valknut. A common design for a folding wooden tripod consists of three pieces carved from a single piece of wood, with each piece consisting of two lengths of wood, the legs and upper sides of the tripod, connected by two segments of wood that surround an elongated central hole in the piece. Another of the three pieces passes through each of these holes, linking the three pieces together in the Borromean rings pattern.