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. 

Probably it's more efficient for Leareth to take some other pilot, but sometimes their conversations end up being startlingly productive and he could use a break.

He nods to Matirin and ignores Finleran (who everyone else seems vaguely deferential towards) and heads off with Leareth.

Leareth is playing with the twelve-dimensional modelling program, and attempting to test whether morphing Andalite gives him an advantage to holding more than four in his head at once. (<A leg up, one might say> he says to Cayaldwin with amusement.) 

<I wonder if it is possible to morph anything that is natively multiplanar, I bet they have good intuitions for it.>

<Like a - Velgarth god?> Uneasy tail-lash. <What an uncomfortable concept. I am not sure how one would acquire them, though, they do not have a physical presence at all.> 

<There are probably things that are less natively multiplanar than that - in a sense Gifted people are, right - but yes, something like that except ideally also having a physical presence.>

<Right. There are other creatures native to Velgarth that might be very interesting to morph, and - actually that would be an extremely useful way to research them, since many are incredibly dangerous. Colddrakes, for example, have a kind of native mind-magic that lets them paralyze their prey by looking at them, so they can eat you at their leisure.> 

<Huh. ...wonder if there's a way to composite-morph that.>

<...What a concept. Now I am trying to think of all the morphs one could acquire in Velgarth, some of them might be very useful to Andalites in combat. Gryphons are a race that Urtho engineered with magic before the Cataclysm, they are basically designed for combat - they are large and strong, they can fly and have very sharp claws and teeth, they are strongly magical. They also have many medical problems, their design was imperfect, but that would matter less if one were simply morphing them. Unfortunately I only know of gryphons still surviving on the very far western coast, where Nayoki hails from.> 

<Few Andalites acquire morphs for face-to-face combat, it's rare and we like our native bodies for it, but I could imagine flight being useful in some circumstances.>

<That makes sense. There are also a couple of Pelagirs creatures with native invisibility magic - illusion-magic, technically, but very well adapted for concealment in particular. And basilisks, which were created to eat toxic magical waste for some reason and can eat and metabolize nearly anything. They are also disgusting, though. Hmm, and then there are all the other sentient species, some of which have Gift variants that are rare in humans - dyheli have absurdly strong Mindspeech, for example, they are herd animals like Andalites, and the herd leader can control all of the others via Mindspeech in an emergency.> 

< - huh. Perhaps sometime we should go to Velgarth and collect morphs.>

<That might be very valuable! And fascinating, of course.> Leareth keeps poking at the program. <How is your other research going? I am not quite caught up enough on the nuances of morph theory to contribute directly, but it occurs to me that the spell-artifact I was working on for demorphing people trapped in morph might have some relevance for the demorphing-elsewhere challenge as well, since it also involves modifying the tether to the construct body.> 

Swish swish. <Setting up a second tether requires substantially modifying how they work at all, and probably requires disabling the nothlit-safeguards entirely, and I worry it makes the spatial instability problem worse...>

He can talk about this all the way out to where they can hyperspace jump.

Leareth keeps up reasonably well. He takes notes. When the timer running in his head approaches his morph limit, he demorphs and stays human, pacing in that form instead.

And then they can travel about an hour out in z-space, pop back to normal space, and Leareth can take Cayaldwin along in rapport to gather measurements on every single plane he can access. Including the Abyssal Plane, which is pretty dangerous to visit even mentally, but Leareth is skilled enough with magic to manage it. 

Improved hyperspace routing is very militarily and commercially valuable in its own right and he's so excited about it.

It seems so valuable! The peacetime implications for trade and people-transport are almost more mindboggling than the military applications. Especially now that Andalites can morph arbitrary Gifts - in fact, once any of them are skilled enough to learn the interplanar Gating technique at all, they can specifically composite-morph Vanyel's mage-gift, use the electricity-to-mage-energy setup as a 'node', and pull off far more impressive Gates. Vanyel specifically has a confusing Gate-disability which Leareth doesn't understand the cause of, but he expects it's not about the mage-channels and wouldn't carry across to a composite-morph. Leareth may want to do that for the Andalite homeworld Gate in particular, doing magic in morph with someone else's much stronger mage-gift throws off his control so he hasn't used it that much but it'd be worth it for this. 

He morphs Andalite again just because he can, and paces and speculates. His body language is a lot more expressive in Andalite form, possibly because he's absorbed much of it from Cayaldwin, who is very expressive when he's excited about some research problem. 

Leareth is a good Andalite and it's nice when he's in a form where his body language is legible. He starts looking through the data they collected to make sure it all makes sense and figure out what else the computers might need.

If they need data on additional planes that are known to Andalites but not studied in Velgarth, Leareth can try getting them there as well? He's unsure how to do this but he's one of the most skilled people in all of Velgarth at mental projection to other planes, so he can give it a go. 

They mostly have good data on the planes they use for hyperspace travel (less good out here around Earth, far from Andalite space, but tolerable). Would Leareth be able to route a Gate through there, if one of the computers thought it was a good idea?

Probably? He would have a harder time teaching another mage the technique, in that case, but he also isn't sure he can teach anyone else to route through the Abyssal Plane or the Elemental Plane of Fire, since they require a lot of care due to their hostility to human magic. It seems worth adding to their calculations anyway, and then if he can't, they can take it out again and get the next-best route.