Durer's grid, Alberti's voile...they both amount to the same mechanism. Da Vinci also had occasion to use something similar, as did Mr. Neville, the protagonist of Peter Greenaway's 1982 film The Draughtsman's Contract. These were all ways to achieve a similar goal: maximum verisimilitude while maintaining accurate proportionality, taking utmost care, to paraphrase the fictional Mr. Neville, to never never distort nor dissemble.
Put another way, it is really no more than a simple matter of mapping. Focusing on one quadrant at a time, one maps one small square of Nature onto one corresponding small square of Representation. It should come as no surprise that once the Grid Man had seen Mr. Greenaway's break-through film, the personage of Mr. Neville who, you will recall, is almost never seen without his grid tripod, quickly became our hero's new hero. So much so, that the Grid Man took to watching the film once every day at precisely four o'clock in the afternoon, thanks to a compact DVD player to which the authorities granted the Grid Man personal access. This arrangement involved some initial hesitation on the part of the doctors, but all apprehensions were quickly relieved when one enterprising young Hungarian nurse had the brilliant idea to cover any sharp edges the DVD player possessed with 2 layers of thick, industrial-strength duct tape.
The Grid Man was clearly fond of this concept of mapping. Although, as his condition worsened, rather than a calming, reassuring notion, mapping became a hideous nightmare whereby the requisite one-to-one correspondence increasingly deteriorated into a state of arbitrary non-correspondence. In this late phase, the mapping process lost all coherence and structure, hence, meaning. And mapping without meaning makes for chaos, madness even. Sometimes even death. But I am getting ahead of myself.
Let us return to mapping in its most ideal, Utopian form. Under these circumstances, a grid is a thing of beauty. It imposes order on an unruly universe -- metaphorically. Metaphorically, it makes reality not only manageable, it can even offer the possible transformation into the transcendent realm of Art.
Of course, one could also argue the grid is a double-edged sword. Metaphorically. In addition to these positive manifestations, one could argue, especially in connection with our hero the Grid Man, that a grid is also a crutch. It is a subtle, but unmistakable barrier. Think here of the French word voile, which means "veil." This barrier separates the subject from reality, mediates reality for him, as it were, allowing him to keep a discreet but definitive distance from the messiness of Life, even offering him the seductive possibility of never venturing forth from an alternative reality which is as safe and comfortable in its neat logic and lack of emotion as it is artificial and misleading.