Ghyll:Orthogonalities

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 previous century.

Ghyll proper is just one of the many orthogonalities that jointly make up MetaGhyll. In the first year of the Encyclopedia, we have written mostly about Ghyll proper, but the Xurient is a separate orthogonality of MetaGhyll. It's not clear whether Down There is another orthogonality or just mythical.

The theory is an application of four-dimensional geometry, and as such impossible to visualize and not easy to understand formally. What follows is an analogy that preserves the appearances, rather than a strictly correct model of Rancticirchiretic's theory.

We will completely ignore the third dimension, and visualize the surface of Ghyll 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 (which is marked by the Pretty Impressive Fence) is. It is believed, but not proved, that every orthogonality intersects with every other orthogonality. (Don't bother trying to visualize the shape of MetaGhyll.)

However, 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. The probabilities of passing into either alternative orthogonality or remaining in the one you are in is roughly equal, so it may take several tries to cross over. People tend to do so at in a running leap, to minimize the possibility that different body parts end up in different orthogonalities. The Pretty Impressive Fence has gates marking the safe transition points.

Turning points are rare in the central of Ghyll, 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 of Ghyll—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 transition 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 Folktown this is, or even if it's the same for every orthogonality. It is believed, however, that transition points increase uniformly in every direction, which is why Ghyll 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 Ghyll 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 that Rancticirchiretic has found, with obvious enormous ramifications.

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 entry points into orthogonalities as one approaches the border of Ghyll proper. In summary, he believes there to be a hundred and fifty, though of course only twenty are significantly populated.

Doc Rockett was charged with explaining the new theory to the public, which he did in his textbook Invariant Transforms, with Historio-Physical Applications.