<Yes, let's.> He will keep Mhalir at this all day if no one interferes.

Mhalir interrupts to get Leareth to send half a dozen messages for him, including one to Amanda telling her not to worry, he's fine. Eventually, midway through the afternoon, he extracts himself with apologies, he's due for the Yeerk pool first thing tomorrow and should catch up on some other things first. If Leareth is up for it, though, he can come back in a couple of days to cover more? 

Leareth hadn't known it was possible for his brain to get this tired, apparently Mhalir can manage longer periods of more intense focus than he can do alone, but this was fascinating and he's now starting to catch up a bit and understand more of it again, so sure. 

Cayaldwin is fine with continuing in a couple of days, this has suggested some promising avenues he can keep himself busy with. 

Perfect, Leareth will arrange that whenever it makes sense for Mhalir, in the meantime he's got Gate-targeting research to get through. 

He Gates them back. 

:I hope you enjoyed your day off, Amanda. Are you ready to have me back now?: 

"Yeah." She reaches for him. "Did Leareth take good care of you."

Having left Leareth's head, he can't answer until he's back in hers. <Yes! We did some fascinating research, which unfortunately I cannot tell you about for security reasons. I will probably go with him again in a couple of days.> 

She's glad. She feels vaguely protective of him. 

(She went out to a bar and met a guy and considered whether she ought to tell him she had a Yeerk because some people would want to know or not tell him because, like, maybe then she'd be targeted by the anti-Yeerk resistance or something, and decided that in a TV show this definitely gets her kidnapped or something, and didn't tell him, and they went home to his place and she's in a much better mood now.)

Aww. Knowing that, he can definitely arrange the next trip to involve staying overnight too. 

Leareth goes back to his hyperspace targeting and Cayaldwin's program for mapping planar interactions, and gets started on repeating the process he used to find Earth from Velgarth, this time to the Andalite home planet from Earth. Beforehand he's not sure whether it'll be much easier because he's done it before, around the same difficulty, or harder because of distance. 

Knowing what he's doing really helps but also the distance means that a lot of simple routing solutions just don't work; there's probably some way to find one that does that's cleverer than just brute trying-all-the-planes.

Okay, hmm, so the thing he's doing here is taking advantage of the fact that planes don't spatially map one-to-one against the Void (or z-space), and specifically, different planes have different Void-mappings. So he can cut through Void to the Elemental Plane of Water, 'search' a feasible distance there, and slip back to the Void very 'far away', in a position where his search can more feasibly reach the place that maps to the Andalite homeworld. 

The trouble is that really, he probably needs to do that more than once, and that makes it a stupidly complex optimization problem. But he's got a lot of planar maps from near Velgarth, and can get some more near Earth, and he knows the straight-line normal space direction and distance to the Andalite world (it's just not one that anyone can travel across, even in many many lifetimes), and he knows the z-space route too. There's still going to be trial and error involved, but he can make better guesses about what to try. 

...The Elemental Plane of Fire and the Abyssal Plane are both very 'dense' and thus can skip across large equivalent-distances in z-space, but also very hostile to magic, so he doesn't want a spell going there for long.

He works steadily on it, twelve hours a day, a lot of it is just staring at computer modeling programs rather than tiring magic. When his brain needs a break he morphs Andalite and practices tail-sparring with the others. 

He's getting good enough he won't kill anyone by accident and can try without a protective tail-cover, now. The protective tail-cover subtly affects trajectories so it's better to practice without it as long as you're not liable to kill people by accident. (Maiming them is fine; Andalites can fix that.)

Matirin's friend who morphed Yeerk on the other Dome ship is here now, tail-sparring with Matirin.

(Leareth is pleased about practicing without the tail-cover and having it be more accurate to real fighting; he'll try pretty hard not to maim anyone even if it's fixable.) 

He watches them while he looks around for someone who wants to spar with him. They're both very good and it's enjoyable to watch. 

They're both very good but Finleran's better at it. When he wins he holds his tail-blade to Matirin's throat for distinctly longer than traditional, until Matirin takes a risky-looking step forward to touch foreheads with him. Their tails twine. 

Then they trot off to work. Matirin wishes Leareth a good morning as he trots by.

Huh. Leareth tries to remember if he's seen any of the other Andalites doing that before. His best guess is that they're - romantic gestures? 

<Good morning> he says back to Matirin. Asking about it hardly seems like the highest priority here, though he is kind of curious. Instead he focuses on his sparring-partner, once he finds one - as usual he loses, but he's making them work harder for it now - and then he demorphs and returns to his Gate-work. 

Cayaldwin thinks that this is the sort of search computers are in principle very good at. You'd need data on the spatial correspondences between planes at many many points in a computer-readable format for the computer to get anywhere, but if you had that it could tell you precisely where to route through each plane.

Interesting! How many points and how far apart - if they need to do hyperspace travel to get additional points then that's costlier, although maybe still worth it if it's, like, a candlemark's trip there and back in each direction that could cut the research time on this by a week. 

Well, it kind of depends on whether there are mathematical regularities that describe how the spatial correspondences between the planes work. Probably there are; there are among the planes the Andalites know of. If there's not, then they'll just have to repeatedly go and measure. If there is, they probably want hyperspace travel, but a candlemark might be enough.

Leareth is fairly sure based on his past work that there are mathematical regularities, although complex ones, enough so that he can't map them out without computers to solve the Andalite-home-world routing problem. 

Well, computers are good at the kinds of mathematical regularities that are complex if you're just using, uh, an abacus? He doesn't know much about what people used before computing. He can show Leareth how to plug spatial correspondences into a twelve-dimensional model. (You can only see three at a time, but with practice you can get used to which ones various turns bring into view. Some people can actually hold eight in their head, that way. No one can do twelve, but, well, that's what the computers are for.)

Neat! Leareth's done some of this kind of depiction with illusions plus paper notes before, and he can arguably hold a four-dimensional model in his head although it's pretty lossy. Turning it to different views is really fun to play with and also kind of makes his head hurt. (Less so if he morphs Andalite for it.) 

He asks Matirin about borrowing one of the hyperspace-capable ships so he and Cayaldwin can go collect a lot of data points and get the computers tell him how to route his Gate. 

Matirin is delighted to assist with that. He can have a ship.

Does Cayaldwin actually want to come? Leareth doesn't technically need him now that he knows how to enter his data into the program, and he's not sure how well Cayaldwin can work on his other morph research if he's spending all day on a ship flying around in z-space gathering data.