Chapter 2 of F begins, if possible, even more obscurely than Chapter 1. The first line of F, Chapter 1, is ‘The Baroque refers not to an essence but rather to an operative function, to a trait’ (F:3). This might be a dense sentence in that it’s implications will need to be unpacked and explored but compared to the first sentence of Chapter 2 it seems relatively transparent.

‘Inflection is the ideal genetic element of the variable curve or fold.’ (F:14) So begins Chapter 2. It continues – ‘Inflection is the authentic atom, the elastic point. This is what Klee extracts as the genetic element of the active, spontaneous line’ (ibid).

One of my fellow readers at the group had done some useful background research and traced the diagram or illustration that occurs at the beginning of Chapter 2 (F:15), tracking it to Klee’s ‘Pedagogical Notebooks’ where I didn’t notice any immediate reference to inflection but where the curve is described as ‘an active line on a walk for a walk’s sake’, which a number of us commented on as it seemed close to the image of the schizophrenic on a walk that Deleuze and Guattari use at the beginning of Anti-Oedipus.

These ‘backgrounds’ that can be filled in by tracking down some of the more allusive and elusive sources that fill Deleuze’s work help in the activity of familiarising ourselves with the text. In particular the diagram, which stands in the text unsourced, becomes less random and seems located, allowing us to feel like there is a work of unpacking to be done in reading F that is not without some point or purpose – that we’re not, as it were, on a wild goose chase. Nothing in the Klee reference, however, immediately illuminates quite what this notion of ‘inflection’ is doing here.

Another reader had tracked down some background that more specifically focused on the meaning of inflection, tracking it to a a possible geometric source where we can find that there is a use within the realm of differential calculus, where an inflection (inflexion) point has a specific role to play. Now it is not the case that the geometric usage needs to tally with the claim Deleuze makes (‘Inflection is the ideal genetic element of the variable curve or fold.’) since it is not a *geometric* claim that is being presented, at least I am not taking it to be such. It is rather a *philosophical* claim. It is clear from the presentation that it is Klee, not geometry, which Deleuze is drawing on and moreover it is Klee’s ‘methodological’ or ‘philosophical’ comments. Quite what philosophical claim is it, however, that Deleuze is attempting to put forward?

Only a paragraph later we find that ‘inflection is the pure Event of the line or the point, the Virtual, ideality par excellence’ (F:15). Deleuze then goes on to pursue again this *productive* model of the Event which the inflection is being mobilised to articulate. Here, in this second paragraph, the presentation brings in not just Klee but the work of Bernard Cache, an ‘independent architect and furniture designer living in Paris’ according to the MIT website which publishes the translation of ‘L’ameublement du territoire’, a book that Deleuze considers ‘essential for any theory of the fold’ (M:15, fn3)^{1}. In the case of both Klee and Cache what it looks like is happening is that Deleuze is drawing on *something like* an account of essence but one which is not mathematically orientated but artistically and creatively focused. This is the point of the inflection notion, to begin or continue something like an account of the essence of a thing, or in this case, the essence of an event. It is, of course, only *something like* an account of essence because the very notion of an essence, as usually understood (or ‘as usually not understood’ we might say), is not to be presupposed, indeed might even be opposed. The task of the ‘inflection’ is to give us a new way of thinking *something like* that which we try to think with the concept of an essence.

- This may well be a kind of self-congratulation since Cache himself has been described as a ‘Deleuzian’ architect ↩

This sounds very intriguing. Especially the end regarding the notion of ‘essence’.

What would a new way of thinking through the concept of essence be? If it is not, by what you demonstrated, supposed to be the traditional mathematical view, or the view that holds it to be that without which a thing can’t be thought – the necessary property so to speak?

You seem to shift here away from a thing, to considering an event as the distinction from which to take our bearing. Then, likewise, to shift from mathematics to art (or creativity).

The inflection point in differential calculus has many approaches regarding its meaning, but it is almost always adhesive to the notion of the constant ‘c’ of a graph that resembles f(x)=ax^(z)+c. This is perhaps because the inflection point concerns a point in the graphs development where either x or y =0. The constant is very definitive of a function as it marks its transition from one element to the next, e.g. from the negative to the positive. The same however maybe said of all constants which are parts of lines, including tangents of curved functions. The constant seems to affect a function’s appearance, but it only shifts the function on the x-axis or the y-axis. In other words, it only moves it around the page, it does not alter its form. The constant of every function does not alter the form of a function, only its position on the grid, which may be something interesting to consider. The form of a graph seems to be altered more powerfully by the immediate effect on x (or y) (e.g. multiplication or raising to the power).

I wonder why Deleuze would liken an essence to an inflection point? It would seem very much counter-intuitive, then again we are asked to not view it mathematically, and to not view essence it traditionally… What would you expand on regarding this final move in your notes?

Well I’m going to have to expand on a lot, although your comments are fascinating and I would like to get to grips with them a little more. I think I understand but get the sense that some graphic exhibition of the point your making might help and wondered if there were any maths sites you know of where this is shown.

I’m settling back into the notes again and will post some more soon – at the moment the ‘shortest’ and not too useful response to the connection between an ‘essence’ (in the new form, ie: as an event) that Deleuze is developing in the Leibinz book and the inflection point would seem to be a kind of analogy between the monad within the best possible world as a point on a curve of variability. The inflection point would on this reading be an inflection point within a particular possible world, offering the option for the same curve to have another inflection point within another possible world. That would seem to be where the analogy might break down, hence why I would want to explore further the concept of inflection point.

For example, does each formula have one and only one inflection point? Would it be possible, for example, to generate a variation of inflection points on the same curve, each of which formed on the basis of the ‘grid-ing’ that enframed the curve?

I don’t know any websites because I’m using my old A-level textbooks. Although I found these on the internet, they seem to say the same thing:

http://www.absoluteastronomy.com/topics/Inflection_point

http://www.absoluteastronomy.com/topics/Inflection_point

http://mathsfirst.massey.ac.nz/Calculus/SignsOfDer/POI.htm (this one is good)

Please scrap parts of my above definition, I would like to elaborate and clarify. I made a mistake between the inflection point and the intersection point. They are very much different but more often than not they do relate, given the function.

Now I don’t know how higher-level Maths or other corners of Maths use the notion of the inflection point, but the one that I learned in college was best seen as this:

The point of inflection is the point where the direction of the line alters and seems to come back on itself such that if a tangent is drawn then it will cut through the line. It is the change in element (or sign, neg/pos) of a line. Think of a river that meanders such that it comes back to face itself but will not touch itself, instead it changes trajectory, it only views itself from a distance and continues on in another direction. In some definitions it is seen as the point where the curve alters its concavity from either upwards or downward (when viewed as a movement from left to right). This may be significant to the Leibnizian monad.

It is categorized as the point where there is a transition in the overall trajectory of the graph from one sign to the next. Usually these points are indicated by x or y being equal to zero, but it is not necessarily so. The way they find inflection points is through 1st or 2nd derivatives, depending on the function, but usually the first suffices. The maximum or minimum extremity of the first derivative is usually where the inflection point of the function is found.

These are just little particularities though, that are not so significant. The one thing to take is the relationship between the change of element and the inflection point.

In answer to your first question: yes, a function can have many inflection points e.g. a sin graph. However, it is only the coordinates of the inflection point in the sin graph that change, and usually they descend by a constant or rise by a constant e.g. if inflection in sin graph is at 0,0, next one will be ([-]6,0), the one after ([-]12,0)… ([-]18,0) etc. Other graphs may be different, but if the function is constant I am pretty sure the relationship between inflection points will likewise remain constant. Or some functions may have no inflection point, even though they change direction, but do not change sign, i.e. x^2.

I think to answer your final question is a little difficult, as I am not entirely clear myself regarding the necessity and relationship of either of these two definitions:

-a) The tangent that crosses the graph.

-b) The point where it begins to change sign.

I would take b) as more essential, because one can draw a tangent and cross the graph, but not be at an inflection point. As such, I think its best if we take definition b) as necessary and sufficient, but a) as only necessary.

The key component of the inflection point is that it changes a sign. Also if we draw a tangent somewhere on the graph and that tangent crosses the graph itself, then you have an imminent inflection point. I hope this helps a little more. Also, I found the little diagram on inflection point on wikipedia quite useful for visualization. Alternatively you can draw your own graphs by downloading this brilliant little program:

http://www.padowan.dk/graph/Download.php

I would recommend the first download due to derivatives.