Manhattan ought to be one of the simplest places to navigate. Its grid, together with its vaguely oblong shape, give it a rational, calculable feel to newcomers, in which the rules are first learned and then applied. The formula is not quite as sophisticated as Washington DCs, in which building numbers on the avenues correspond to the nearest cross-street (so 1254 Avenue K will be near 12th Street), but it is good enough.
Then the rationalism begins to slowly unravel in a curious way, that leaves one with a feeling of something mischievous and disorientating, but without ever quite descending into the pure a-rationalism of a historically contingent lay-out such as London's. It begins with the more renowned pieces of rule-breaking, such as the West Village, where streets illegally cross each other, rather than avenues (as in this map). But it advances as one begins to realise that there is not just one rationalism at work, but several and they are at play with each other.
Henri Lefebvre's The Production of Space identifies three ways in which space is produced. There are 'spatial practices', namely the routines through which we act to produce spaces in a certain way (e.g. commuting). Then there are 'representations of space', such as maps of one sort or another or photographs. Then there are 'representational spaces', the spaces we deliberately create to carry meaning such as parks. I imagine Lefebvre has been endlessly used to assess Manhattan, and apologies if I'm going over old ground. But the 'representations of space' that exist in Manhattan are particularly unusual.
It strikes me that modern representations of space generally involve a reasonably neat split between euclidean geometric maps and user-centric maps. So in London, we have overground spatial practices supported by the 'accurate' representation to the left and underground spatial practices supported by the famous, 'inaccurate' representation below it. (One might be tempted to say that these two images are both 'of' roughly the same part of London, but, in a faintly Wittgensteinian sense, it would be unclear exactly what one meant by that.) Streets are represented in euclidean terms, that is, in terms of where they 'actually' are, the direction they 'actually' point in, and with a large variety of other representations alongside them, to allow for the user to determine their own uses of the city. It's not inconceivable that an international consultancy, say, may produce a 'business map' of London, in which space was re-represented in non-euclidean, purely use-oriented terms for a businessman who had specific needs and no interest in the city. But by and large, euclidean geometry is what is used to represent the various potential objects and uses of the overground city. The tube map, on the other hand, is entirely different, with the only euclidean limitation being that lines cannot be represented as being headed in a direction more than 45 degrees from their 'actual' direction.
Now take Manhattan. Here the two are blended into one another, the implication being that the city is so neatly rationally organised that there is no need to split a user-oriented map of the subway from a euclidean map of the streets. After all, isn't the whole city planned around ease of use? And this is where the multiple, playful and mischievous rationalisms start to pull apart from one another. Thanks partly to the woefully bad signage on the New York subway, there is little clear distinction between what is going on underground and overground. 'Second Avenue' station has an exit on 'First Avenue' avenue, and both are indicated in an identical font on the station platform itself. Without a clear split (as in London) between what Manuel Castells calls 'the space of flows' (such as a subway) and 'the space of places' (such as a physical street), the subway user experiences a disorientating middle ground in which space is neither user-friendly nor quite euclidean either.
What makes this all the more extraordinary is that the same is true overground in Manhattan as well! The vast majority of euclidean spatial representations of Manhattan are, in orthodox geographical terms, entirely false. The recognisable map of streets and subways above shows Avenues running, as common parlance has it, 'North/South' and the streets running 'East/West'. But if you look at the genuinely euclidean representation of this same space on the left, you'll see that the angle of the streets is so far out from this division, that things could almost be spoken of the other way around. 'North' is actually 'North-East', and 'East' is actually 'South-East'.
The representations of space are therefore rationalist, without comitting to being 'true'. As yet another act of collective solipsism and surrealism, Manhattan creates its own space that sits outside of the distinction between euclidean-geometric space (orthodox maps) and user-centric space (such as the London tube map). Had Manhattan been a great naval power rather than London, the 'North pole' would presumably sit somewhere in Germany, a projection of a grid that is simultaneously rational and yet arbitrary.
Finally, as three forms of rationalist-surrealist space sit inter-mingled with one another (the flows underground, the grid overground, orthodox geography), the obliviousness to any clear distinction between the three is played out in the failure to construct different symbolic codes for each one. For instance, should one be on the platform of 4th Street subway station (some of which is on 4th street), notions of 'north' and 'south' multiply, but never quite distinguish themselves from each other in terms of signs, linguistic or otherwise. It is all enfuriating and mesmerising.