Rancticirchiretic worked on the theory of orthogonalities from shortly after his investiture as president of the Bureau until well after his retirement. It is considered the greatest scientific discovery of the past century.
"Ghyll proper", used herein to refer to the greater area surrounding Folktown, is just one of many orthogonalities that jointly make up what he termed "MetaGhyll", though common usage applies the term "Ghyll" to both the primary orthogonality, or "Ghyll proper", and the entire collection of otherwise known orthogonalities. In the first year of this Encyclopedia, we mostly wrote about Ghyll proper, but the Xurient and Down There are separate orthogonalities of this "MetaGhyll".
Visualizing the Theory
Rancticirchiretic's theory is an application of multi-dimensional geometry (at least six, maybe more, dimensions) and, as such, impossible to visualize in three-dimensional space, nor is it easy to understand formally. What follows is an analogy that preserves the appearances of the theory rather than a strictly correct model.
Let us completely ignore the third dimension and visualize the surface of Ghyll proper as a pure two-dimensional disk with its center near Folktown. Each alternative orthogonality is another disk intersecting Ghyll along some line (or possibly circle, ellipse, parabola, or hyperbola or part thereof), known as an intersection line. Thus, it is not really true that the Xurient is 230 lele east of Egron; rather, the intersection line between the Xurient and Ghyll proper (which is marked by the Pretty Impressive Fence) is. (It is believed, but not proved, that every orthogonality intersects every other orthogonality. Don't even bother trying to visualize a shape for MetaGhyll as a whole.)
You cannot cross from one orthogonality to another just anywhere on an intersection line. Rather, you must go to a turning point, which is the intersection of two intersection lines. At these points, it is possible to transition into either of two orthogonalities. There is only one turning point for each possible combination of three orthogonalities, which creates a directional triple such as "Ghyll proper–Xurient–Down There" (a hypothetical example). The probabilities of passing into either alternative orthogonality, or remaining in the one you are in, are roughly equal, so it may take several tries to cross over. People tend to do so at a running leap so as to minimize the possibility that different body parts end up in different orthogonalities. The Xurient's Pretty Impressive Fence has gates clearly marking reasonably safe turning points to other orthogonalities.
Turning points are rare in the central region of an orthogonality, but become more common the further one goes from the center. Rancticirchiretic measured the distance between known turning points and found that they increase exponentially as one travels towards the borders - which is why, of course, exploration of these areas becomes increasingly difficult as turning off onto another orthogonality becomes ever more difficult to avoid. The outer edges of an orthogonality are very dangerous: if you cross over, there may be another turning point just a few inanits away, or even inside your body!
There is, technically speaking, no final outer edge to any orthogonality, but there is an effective edge based on the distance from the center which is incompatible with life. We don't know how far away from the center this is, or even if it's the same for every orthogonality. It is believed, however, that turning points increase uniformly in every direction, which is why Ghyll proper and the other orthogonalities can be modeled as having circular surfaces.
Rancticirchiretic also devised the name "orthogonality". A name had never been needed before, since (as the Cartographer's Nerves principle states) measurement of an orthogonality is so difficult that "approximations like 'Near' or 'To the West' are so common not only in informal communications but also official literature, legal documents, and scholarly work." This is very, very slowly beginning to change, thanks to the cracks, and their enormous ramifications, that Rancticirchiretic has found.
Though Rancticirchiretic has been unable to explain why Pinky and Perky look exactly the same from every orthogonality, he has been able to provide the best available approximation of the number of safely transitionable orthogonalities, based on some complex mathematics involving the increase of repetition of turning points into orthogonalities as one approaches the border of Ghyll proper. In summary, he believes there to be a hundred and fifty orthogonalities though, of course, only twenty are significantly populated.