Deleuze uses the upper level of his metaphorical Baroque house to reintroduce the ideal space of singularities which we described in detail in Chapter 4. It returns in The Fold couched in a new language of folds and points of inflection, but most of its elements remain familiar. We can introduce points of inflection by considering a simple curved line, like an ‘S’ shape. Just as in previous chapters, we treat this curve purely as a geometrical figure, regardless of what it represents. On this curve, as we saw in Chapter 4, there is a certain point that stands out from the others as remarkable or singular. There we called it a singularity; here in The Fold, Deleuze calls it a ‘point of inflection’. We can identify it precisely by calling it the point on the curve where the curve changes direction. Mathematically, we say that the curvature of the curve changes sign, from negative curvature to positive curvature. Imagine the curve is the path taken by a car: the point of inflection, halfway through the journey, is the moment when the steering wheel switches from left to right, or vice versa. Initially, however, Deleuze turns to art, rather than mathematics, to explain points of inflection, and in particular to the theory of Paul Klee.

Just like in chapter 1 of The Fold, Deleuze initially approaches the central issue of the chapter from an angle, through a more-or-less analogical discussion of art. Unlike chapter 1, however, we’re no longer dealing with Baroque architecture, but with an expressionist painter. This is important for two reasons. First, there is the obvious fact that Paul Klee is not a Baroque artist. Above, our comparisons between Baroque architecture and Leibniz's conception of inorganic matter were justified by appealing to universal traits, expressed by, but by no means limited to, Baroque architecture. Now, for the first time, we are confronted with the possibility that Baroque traits are not even limited to the Baroque period itself. Deleuze writes that Klee's emphasis on the point of inflection in any given line or curve demonstrates his ‘affinity with the Baroque and with Leibniz’ (LP 20). The third chapter of The Fold and the question it poses (‘What is Baroque?’) tries to show how Baroque traits are easily identifiable in artistic and philosophical works produced outside of the narrow period of history that the Baroque strictly refers to.