a number of ways of seeing that the Borromean rings are linked. One is to use Fox -colorings, colorings of the arcs of a link diagram with the integers modulo so that at each crossing, the two colors at the undercrossing have the same average (modulo ) as the color of the overcrossing arc, and so that at least two colors are used. The number of colorings meeting these conditions is a knot invariant, independent of the diagram chosen for the link. A trivial link with three components has colorings, obtained from its standard diagram by choosing a color independently for each component and discarding the colorings that only use one color. For standard diagram of the Borromean rings, on the other hand, the same pairs of arcs meet at two undercrossings, forcing the arcs that cross over them to have the same color as each other, from which it follows that the only colorings that meet the crossing conditions violate the condition of using more than one color. Because the trivial link has many valid colorings and the Borromean rings have none, they cannot be equivalent. The Borromean rings are an alternating link, as their conventional link diagram has crossings that alternate between passing over and under each curve, in order along the curve. They are also an algebraic link, a link that can be decomposed by Conway spheres into 2-tangles. They are the simplest alternating algebraic link which does not have a diagram that is simultaneously alternating and algebraic. It follows from the Tait conjectures that the crossing number of the Borromean rings (the fewest crossings in any of their link diagrams) is 6, the number of crossings in their alternating diagram. Ring shape The Borromean rings are typically drawn with their rings projecting to circles in the plane of the drawing, but three-dimensional circular Borromean rings are an impossible object: it is not possible to form the Borromean rings from circles in three-dimensional space. More generally proved using four-dimensional hyperbolic geometry that no Brunnian link can be exactly circular. For three rings in their conventional Borromean arrangement, this can be seen from considering the link diagram. If one assumes that two of the circles touch at their two crossing points, then they lie in either a plane or a sphere. In either case, the third circle must pass through this plane or sphere four times, without lying