A mathematical knot is just a string knotted up in 3-dimensional space with its ends tied together. These play an extremely important role in low-dimensional topology. Usually, one considers two knots to be the same if one can deform one into the other without cutting or tearing the string. However, in order to understand the topology of 4-dimensional manifolds (ones that look locally like 4-dimensional Euclidean space), one should consider a 4-dimensional equivalence relation on knots, called concordance. It turns out that the set of knots up to concordance forms an (non finitely generated) abelian group, called the knot concordance group. Even though this group was defined in the 60's it is still far from being understood. Until recently, most people have just been interested in the group structure. We put a real valued metric on the group and use this to give evidence that the space has self-similarities making it into a fractal set.