So I’ve been thinking about the biangles in Cloudward Ho, and I have some theories and thoughts. Honestly, this is more notes than anything else, but I figured I’d share. Long post under the cut.
First, a biangle is an actual shape with a mathematical definition. Biangles are two-dimensional: they occur on a surface, and they don’t have a thickness. A triangle has 3 sides and 3 points. A biangle has 2 sides and 2 points—so it only works in geometry based on shapes where two straight lines can intersect twice, like spherical geometry:
That crescent shape is a biangle. If you want a physical example, you could make a biangle out of an orange peel.
The Effulgent Biangle occurs on Gath. Presumably, because it emits light, we can expect that it only occurs in the atmosphere surrounding Gath, which explains why it takes the shape of a biangle: it likely triggers as an electromagnetic phenomenon between the two poles of the planet.
Problem: Zood and Zern are tubes. Here’s the map:
Biangles can’t occur on the surface of a tube. You can’t make a closed shape on the surface of a tube with only 2 points and 2 straight lines. Further, Brennan suggested that there is atmosphere dividing Zood and Zern, so that would also be in the shapes of tubes. Thus, a biangle cannot occur on the surface of Zood or Zern in the same way that the Effulgent Biangle occurs on Gath. At least, the Effulgent Biangle might not occur naturally in Zood or Zern—which is probably why Torse doesn’t believe there is a way to return to Gath.
But wait. There is a sphere in this world that might have the conditions necessary to experience the Effulgent Biangle based on electromagnetic events, such as solar light and wind. Brennan also described the moon as a "perfect" circle... which is a little ominous when combined with the fact that this is depicted as pure black, and there is no white moon on the map.
Regardless, there might be a way to use the Effulgent Biangle on that land mass to get back to Gath. We need more information to be sure.
However, the Colefactory Biangle seems to generate heat. Unlike light, heat can permeate solid matter, so the sphere where the biangle manifests doesn’t necessarily need to be limited to the atmosphere around a landmass. It might even be smaller. Maybe like one of the phenomenon we saw in the opening cinematic:
That opening may be a biangle with a very wide angle, and the color seems to match the description of the Straka energy. That might be the Colefactory Biangle at the Ectic Research Station. If they can be that small, stable,and contained (using seawater as a coolant), then it’s not a stretch to think there are Colefactory Biangles in both Zood and Zern that Comfrey has been using to go back and forth.
We need to know more about how heat and cold are distributed in Zood to be sure. Brennan has given pockets of exposition that the climate is very strange. it's possible that there sections of Zood that never get warm/cold, and perhaps they host biangles relating to that circumstance. We'll see.
That leaves the third: the Obtenebrate Biangle, which has been associated with dark, water, and mirrors. The Eyeless Hand’s materials located among Comfrey’s research at the Ramansu power station were the only thing that mentioned this one. Other factions might have info on it, but the Eyeless Hand is particularly interested in it.
One hiccup has been that Mordecestershire is a member of the Eyeless Hand, and there have been members in Zood for centuries. I think that perhaps Comfrey got The Eyeless Hand's assistance in Gath (specifically, from Mordecestershire) by offering to locate the Obtenebrate Biangle, not the Effulgent Biangle. In episode 2, he didn’t specify:
Mordecestershire: Did she find it?
Max: Find what?
Mordecestershire: The Biangle!
If Mordecestershire was primarily interested in the Obtenebrate Biangle rather than the Effulgent Biangle, and Comfrey had either worked with the Eyeless Hand or actually joined the organization, it would explain numerous loose ends and line up with what we know about Comfrey:
Comfrey repeatedly picked up financiers to support her research and drained them for all she could until they turned hostile.
Comfrey had "captured intelligence" and correspondence from Mordecestershire in the Ramansu power station--perhaps because it was her correspondence and they gave her these materials to study.
If Mordecestershire already knew how to intercept the Effulgent Biangle, that explains how he caught up to the Zephyr quickly as it fled the South Pole.
Mordecestershire hit the ground running in Zood and murdered Haunch Saxon with incredible efficiency; if he had previously been receiving reports via Comfrey's interplanar radio (whether from her or the Eyeless Hand) about what to expect before he made the trip (and before he lost contact), he could have been ready to do that for a while.
Mordecestershire's lack of knowledge about Ramansu doesn't undermine the previous point; if Comfrey was taking him for a ride, it makes sense that she'd have kept the Ramansu base (financed by the House of Fehujar) secret from him and the Eyeless Hand.
Comfrey didn't seek free assistance from the people of Zood and instead got additional financing from the Bank of Fehujar; perhaps she was jealously guarding her own research and didn't want to do favors, and perhaps she didn't want to leave a trail that the Eyeless Hand could easily follow when they realized she wasn't devoted to their cause.
Even if only some of those are true, it seems as though the Obtenebrate Biangle is what Comfrey is focusing on currently, though we don't know to what end.
In episode 3, at the South Pole station, Marya and Max explored some architecture that seemed to be from an old castle. Brennan gave the following description:
The center of a circular, stone chamber, lichen covering the walls inside. You see that there is a faint outline—in what be ash or soot or pure shadow—of a hand on the back side of the door, as though it was reaching for this latch: a circular iron ring latch on the back of the door. In the center of the chamber, the floor is ice, pure ice, and in the center, a massive, almost naturally formed, craggy hole that drops into darkness.
The cast hypothesized that this castle room had been airlifted and brought to Gath from Zood. At the very least, the ceramic and crystal device that was hidden in the well was made to detect light and an opposing force (presumably darkness), which is more relevant to Zood technology than Gath technology so far. Whether it was the materials or only information came from Zood, there has to be some level of travel or communication occurring in that direction. It also seems that an early hint of the Eyeless Hand was left inside the room. (This also bolsters the theory that interplanar radios had been used casually by at least Comfrey’s teams.)
In episode 8, the crew found another stone room in Ramansu power station with another well in the center with another ceramic and crystal device. Brennan did not specify if it was circular, and the players didn't ask.
However, the structure of the circular room at the South Pole is interesting with respect to the Biangles. If the Colefactory Biangle can be as small as the image from the opening cinematic, then who's to say that an artificial Effulgent (or Obtenebrate?) Biangle couldn't be created in a spherical room? What if Comfrey had been going back and forth from Ramansu in Zood and the South Pole in Gath through those rooms? We can't be sure, but it's a possibility.
However, the most tantalizing loose end about the Obtenebrate Biangle is that Comfrey's notes at Ramansu theorized that it's not "a true biangle." That's a strange way to phrase it. Currently, we do not know why these phenomena need to be in the shape of a biangle. If they could be in other shapes, then spheres would not be necessary, and perhaps it would be easier to create an artificial method of traveling between Gath, Zood, and Zern. However, presuming that something very close to the shape of a biangle is necessary, why not refer to the precise shape (triangle, square, etc), angles, points, etc?
I think the takeaway is that an imperfect biangle requires an imperfect sphere.
(We could get really pedantic that Gath is not a perfect sphere and therefore the Effulgent Biangle is not a true biangle, but the atmosphere around Gath is probably the actual sphere where the Effulgent Biangle spawns, and so the geography of the surface wouldn't impede that.)
If not for the phrase that the Obtenebrate Biangle might not be a "true" biangle, I would have expected that the shadow on the moon of Gath could be it. However, because Brennan described the moon as a "perfect circle," I have some doubts about that. Maybe it could be the location of an Effulgent Biangle, but probably not the Obtenebrate Biangle.
Which brings me to Van Chapman.
In episode 3, while Siobhan is working on the encoded message after narrowly escaping at the South Pole (when her arm went to the red for the first time), Brennan narrates, “I think you stay up working on it because to go to sleep, you’d have to close your eyes, and when you close your eyes, you see the dark.” Maybe Brennan is just getting in her head, but I don't think he would so casually connect Van to the Eyeless Hand's obsession with the Obtenebrate Biangle and also emphasize this specific fear to Van. He could have evoked any number of feelings of water, tides, etc. to instill fear of the ocean and the curse, and instead he chose darkness.
In episode 8, Monty reviewed Comfrey’s file on Van, where Comfrey had kept note of "a workroom in the inner tower," which is the stone room where all of the Eyeless Hand materials were, as well as the Eyeful Hand (which could be Van's severed hand). All that reinforces the crew's theory that the Eyeless Hand needs Van for something. The crew even hypothesized in episode 9 that maybe the patron of the trade that caused the curse is this Nameless God.
Assuming that the Colefactory Biangle is indeed the strange construction from the opening cinematic, then we can conclude that the Biangles aren't necessarily required to be as massive as the Effulgent Biangle that streaks across the sky of Gath. They can be smaller, about the size of a submersible or Straka. If the rooms at the South Pole and Ramansu are methods of creating artificial biangles, then even the size of a room.
There is a very common imperfect sphere that features what looks like a biangle on its surface, and it is primarily used to process light.