To a square we assign the name Zero.
To its left is a square of equal size, and the right edge of that square is the left edge of the square name Zero. Our purpose now is to name that second square.
In order to illuminate the problem somewhat, first we name the sides of square Zero: left, top, right, and bottom. We also name the corners of the square top left, top right, bottom left, and bottom right. Now starting from top left we measure one third of the distance to top right along top, and call that point top of first dividing line. Then we measure one third of the distance along bottom from bottom left to bottom right and call that point bottom of first division. Connecting top of fist division and bottom of first division to form a line segment, we call that line segment first dividing line.
Second dividing line extends from a point two thirds of the way from top left to top right to a point two thirds of the way from bottom left to bottom right. Third dividing line extends from a point one third of the distance from top left to bottom left on left to a point one third of the way from top right to bottom right on right, and fourth dividing line extends from a point two thirds of the way from top left to bottom left on left to a point two thirds of the way from top right to bottom right on right.
The four dividing lines divide Zero into nine squares in three horizontal rows. The three squares in the top horizontal row are named, from left to right, Zero:One, Zero:Two, Zero:Three. The three squares of the center horizontal row are named, from left to right, Zero:Four, Zero:Five, and Zero:Six. The three squares of the bottom horizontal row are named, from left to right, Zero:Seven, Zero:Eight, and Zero:Nine.
Next, we describe a square with a side length three times the side length of Zero. We name it One. Zero and One share the same center. Applying the method we used to divide Zero into nine squares and name each of them, we divide One into nine squares and name them. By this means we can derive the name of the square to the left of Zero and equal to it in size: One:Four.
Square Zero defines an infinity of two dimensional space, divided into an infinity number of squares, ranging from infinitely large squares to infinitely small squares, and a unique name can be derived for any one of these squares, using the system described here.
For the sake of completeness in describing the system, we can call square Zero a native square, and define native squares as squares with side lengths some positive integer multiple of three times the side length of square Zero, and sharing the same center with square Zero. Every native square can be divide into nine squares as described, and these are called divisions. Divisions of divisions are also called divisions. The name of a native square is a non-negative integer number and the name of a native square whose side length is three times that of another native square is the name of that other native square plus one. Additionally, any square can be divided into nine divisions, and the name of a division is derived by appending to the name of the square being divided the name of the location of the division in the square being divided.
This nomenclature is designed to serve a purpose. That purpose has to do with a division of the field of computing called the browser, which is a division of the larger field of computing called windows. I can describe the browser in terms of our experience with the browser, which is that pages are displayed in the browser window, and we are familiar with the fact that pages often extend beyond the browser window, so that often only part of that page is displayed in the window at any one time. Page elements are geometries which are described, in the inner workings of browser math, by identifying points with reference to the window. These points, as a rule, are indirectly listed in a file the browser opens. The browser, however, constructs a list of specific points by interpreting the contents of that file, and displays geometries defined by those points. More generally, though, the browser constructs, based on that list of points, a map of colored pixels. Each pixel is in some location relative to the browser window, with, in most cases, some of the pixels being in the window, and some of them being outside of the window space. A more correct statement would be that the pixels are located with reference to the page, and some part of the page is generally in the window space, and other parts of it are generally outside of the window space. This adds a layer to browser math. First browser math constructs a list of points representing the geometries to be displayed, based on information in the file being opened, and then it constructs an array of pixels representing the page, and then it colors the pixels of the window based on the array of pixels representing the page and information about what part of the page is in the window and what part of it is outside of the window.
This procedure is fairly effective, but it has limits. Beginning with the first layer, the number of geometries which can be described in a file that the browser can open is theoretically very large, and the sizes of those geometries can range from extremely small to extremely large without requiring that the file itself be inordinately large or cumbersome. If, however, the described geometries are very large, or if they describe a very large expanse of space by being very distant from each other, the array of pixels they describe becomes very large, quite possibly unmanageably so. The problem becomes even more acute when we move to the third level of browser math, if the array of pixels representing the page is extremely large. At every stage, it is a problem of deciding which of the geometries described in the previous stage belong in the window, and which do not. The stages of browser math are levels of memory. At the file level, very large geometries can be described in great detail, without exceeding the memory capacity of the computer. At the page level, a still spacious but nevertheless more limited amount of space can be described, and at the window level, only a very limited amount of space and detail can be described.
The upshot is, page size ultimately cannot exceed some fairly limited bounds, as things stand today. The mathematics described here is designed to move the limit on page size far beyond its present level, and very near to infinity, and it does so by modifying the function of the first level of browser math, the file.
As implemented today, the file the browser accesses for information about the geometries to be displayed contains very little information about the location of page elements. The information it contains is sufficient to construct a map of element locations, but this map must be constructed through a process, and the process is problematic in this sense: essentially, the entire page must be described, as an array of pixels, in order to ascertain the location on the page of any given element.
In other words, today's Web file organizes information about a page by describing in some manner the identity of each of various page elements, without in any direct way describing the location of each element on a page. The characteristics of each element are described, but largely with reference to itself, and the page must be constructed essentially by comparing the described elements with each other.
What is described here, then, is a Web file which is organized according to the locations of page elements. One way to think about this is to propose that the very first item of information in a Web file be a geometric description of the entire page, abstracted from any description of page elements. If we begin by assuming that the page is rectangular, or by defining it as such, then the entirety of that information can be contained in a single number which represents the proportions of the page, that is, a ratio of its with and its height.
This, however, is still slightly indirect. A completely direct approach, if we accept the system I've described for naming regions of space, would be to simply locate page elements relative to square Zero. Let us say that a particular page element is listed as being in square Zero, in the file, and then let us say that the browser has been instructed to display square zero at a certain size and location relative to the window. The browser would request information about elements in square zero, and in any adjacent squares that might be in the window, and there would be no need for it to retrieve information about squares which are not in the window.
The reader will note, however, that assignment of a page element to square zero, or to some other square, remains, as I have described it thus far very briefly, quite arbitrary. I will leave a fuller discussion of that for another time.
To summarize, what I am describing is a Web file, which the browser can access to render a page in the window, which can describe an essentially infinitely large page. And I am aware that readers may wonder why such a thing is necessary, or what advantage there might be to it. I, of course, have my own reasons for wanting such a facility, and for wanting to develop such a facility, but I find that a discussion of reasons presents a somewhat difficult challenge. If I attempt to analyze the nature of that challenge, I find myself concluding that it has to do with the way we generally think about the Web, and Web pages, and their purpose in computing, or in media. I believe I am proposing a reassessment of this question of purpose, and that people are generally reluctant to consider that possibility. I am attempting to move our thinking about the Web into slightly unfamiliar territory, and, in my experience, it tends to do little to argue my case for entering that territory. People want to know why they should move into new territory at all. But I am interested to see whether, having described that new territory in purely technical terms, now, however incompletely, readers might surprise me by seeing some point in it, without my needing to insist that it does have a point, or purpose, or needing to elaborate, for the moment, on what that point or purpose might be. I suspect that my anxiety regarding this is in part a kind of bias of my own, so, for the purpose of this essay, I will rest my hope on that or on the possible value of testing that notion on the reader.
