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", 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 have written mostly about Ghyll proper, but the Xurient is a separate orthogonality of this "MetaGhyll". It is not entirely clear yet whether Down There is another orthogonality or simply mythical.

Visualizing the Theory

Rancticirchiretic's theory is an application of four-dimensional geometry 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 (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.

File:Orthogonalities 01.png

An example of a turning point at the intersection of three orthogonalities. More examples below.

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, 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, is 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 the safe turning points to other orthogonalities.

Turning points are rare in the central 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.

Orthogonalities in Pictures

The following pictures and text are excerpted from Mother Mutton's Golden Book of Orthogonalities, Neither Orthogonal Nor Nervous, But Always Coloring Fun. Remember: these images purport to illustrate a fourth dimensional concept in a third dimensional image; as such, they are analogous approximations at best. Note: for the sake of clearer understanding, the Encyclopedants have partaken in the coloring fun and colored each orthogonality for you. If you'd like the joy of your your own coloring, we heartily advise you to seek out your nearest bookseller.

File:Orthogonalities 01.png

"The three orthogonalities in this picture, named A, C, and T, intersect at a single turning point. We'll name this directional triple A-C-T, but if you prefer the triple C-A-T, so do we! Hugs!"

File:Orthogonalities 02.png

"Another example of three orthogonalities, this time G, C, and A. That's right! Here, the turning point is G-C-A! Or A-C-G! oOOh, G-A-C! Your friends told you orthogonalities were hard, but this is simple! You're smarter than your friends! Hold back a superior chuckle!"

 

File:Orthogonalities 03.png

"Remember how we said that intersections could be a straight line, but they might be curved as well? Here's an example of a curved intersection, where the orthogonality C intersects orthogonality G in a circular path. We'll throw a third orthogonality in, T, so that we have a valid turning point and thus, directional triple: T-C-G!"

File:Orthogonalities 04.png

"Our final concept! Here, we add a fourth orthogonality, A, and suddenly, we have a total of three turning points: T-G-A (or G-A-T, etc.), C-A-G (G-A-C!), and our old friend T-C-G. There's also a fourth C-A-T turning point but, due to our petty third-dimensional limitations), we can't show it. So... which colors did you choose for your orthogonality coloring fun!? Not as nervous as you once was, are ya?"

--The Encyclopedants