"Of course, kid.

Just one turbine today! It's for pumping and heating water, once the new aqueduct is done."

The foreman takes the team in through the main entrance and down several flights of stairs to a mezzanine. Though an arch, branching left, down some more stairs and into a long straight corridor. If Calsa weren't there it would be pretty dark. "This is actually the new aqueduct we're in now!"

After a lot of walking in the dark, they reach an active team. Werewolves roll table-sized disks of stone into a slot in the wall.

     "Is Breath early or am I just that energetic today?"

"Hey Flesh, Breath passing you on the side, look out. We're early to try an experiment on the turbine chamber! If it works there'll be a party at my place when you wake up, featuring our guest of honor, this kid Nico!"

        "Aw, we don't get to watch?"

            "If it works we'll all be doing it until we're sick of it."

   "You don't have to shout, we can all see the cat with you!"

        "Oh hi Calsa! I know you! You're Merta's kitty!"

     "Yeah, Nico! Right on!" says someone who has never met him.

They follow the empty-handed werewolves, staying away from the ones rolling the disks. The tunnel narrows up ahead, where werewolves are slowly peeling disks off the walls, and continues beyond that, single-file. After more walking, the tunnel widens to a chamber up ahead, but before reaching it they turn to the right into a nook, up stairs, and onto an elevator hanging in an open shaft, suspended by metal chains. "Stand clear, hold firm! Unlocked!" The elevator lurches, and then smoothly descends, picking up speed as they go. "Stand clear, hold firm, wait!" It plops to a halt surprisingly gently. "Locked!"

"Okay, kid, this is going to be the turbine chamber! See this cone above us? It needs to be exactly the right shape. Normally we'd wait until the shaft is in the middle, then set up a whole contraption with a bunch of strings and rods, and then shape it. And then remove the whole mess.

But with you here, we can get all the measurements now. It will take a few minutes for us to get ready, but let's start thinking about the first measurement now." Which is a horizontal circle 15 feet above the ground, 10 feet in diameter.

This is sooooo cooooool. Aqueduct! Giant turbines for the whole town! Amazing!!!

"I need someone to point exactly where the center goes and where the edge should go..."

"...Waaaaait. The way I was thinking of doing it would make a ball, not a circle."

He holds his hands up. Two sparks appear, then a bunch more, flickering around as if on a sphere. He drops it quickly, to conserve his white.

"Yeah. Ball. I can't just tell them to stay this far apart from the first one..."

The werewolves climb up the wall to the needed height, hanging from inset rungs, with the exception of the foreman and one carrying a folding tripod draped with equipment.

"Daneia, would you drop a plumb line down the center?"

    "On it!" She lays her tripod on the ground, runs to the elevator, pulls some levers causing thuds from above, and ascends.

  "If we extend to the ball across a band a hands-breadth wide, I think that should work? Even if we can't tell where the exact farthest edge is, the wall doesn't change that fast and we can smooth out the narrow parts later."

"Good thinking, and if we had a couple hundred balls, yes, but we don't. If we're going to hold up bits of the template against only a few measurements, they have to be precise. It's not just a smooth curve, it also marks where to put the stators."

        "Say, why haven't we been getting a whisshopper for this all along?"

          "Ooh good point! I guess it's not really a whisshopper's style so no one thought of it?"

      "Can a whisshopper guarantee the measurements?"

          "I don't know!"

    "If the floor was flat and level we could measure the heights later with rods?"

  "That sounds like more work than hanging the template."

    "Maybe we could measure from the bottom of the shaft, or the center of the socket?"

  "So then we don't have to level the whole floor!"

"Nico, could you do a couple hundred balls if you only had to hold each one for a second? If not, I think we could make a few balls work, if you hold them for several minutes each? It's okay if you can't."

    "We could make the rods now, and a thing to hold their base to the center of the socket, and use that to find the right height to mark on the edge of the ball!"

  "Anyone good at square roots?"

       "Or put Daneia on the socket watching the bottoms of the vertical measuring cords we already have to make sure they're level with where the base of the shaft will be."

  "Oh, nice and simple! I love it!"

Lin can calculate square roots pretty quick, but she's relieved not to have to choose between straining her cover and letting the project fail!

"I don't think I can hold a huge ball for minutes. That's a lot of sparks compared to a circle. There has to be a way to make it a circle instead of a ball." He is peering at the tripod. Just asking for a circle probably wouldn't work. You have to be sneaky. "I just need to... Tell it where the flat is with more dots maybe? I can do betweens and closer-tos and far-away-froms..."

She can't resist. "If you have two distances to two points, can you put the sparkles everywhere that's the correct distances to both points? The second point doesn't have to be at the center of the socket. It would actually be easier to have the second point 3/4 of the radius away from the first point, wherever that falls along the plumb line, and then the second distance is 5/4 of the radius."

"Umm... Tell it to be five-fourths away from this one and four-fourths away from that one, you mean?"

Sparkles appear between his hands. Two, one above the other, and then more that settle into a wobbly circle. "...Wow. Why does that work?"

That's awesome. She wishes her first daddy male parental figure could see this.

"Well, the whole thing is defined by the two points and the three distances, with nothing else going in any other particular direction, so the result has to be round. A circle or a ball or a bunch of circles or something.

If you think of each point-and-distance pair as defining a hollow ball, the result is that the sparkles follow the rules for both balls. Depending on the sizes and positions of the balls, that might be nothing, a point, a circle, or a hollow ball.

This circle is special: it's the biggest circle you can have using the first ball. It cuts the first ball in half at its widest point. That's special because it makes the top slice of the ball the same as the bottom slice. If you put the second point above the first point and kept the rest the same, you'd get the same circle.

Let's imagine that you move your illusion so the two points are on the wall - a smooth vertical part of the wall. You can actually do it if you want. Remember, the shape has to be round, so we don't need to worry about all of it. Whatever points are exactly on the wall already tell us everything about the shape.

So, here's the first point, the second point, a point on the circle to their left, a point on the circle to their right, and we could have gotten this by putting the second point above the first, so let's imagine another 'second point' too. Now, there are four triangles which are all identical in shape although some of them are flipped around. See how they all have a squarish angle, where they meet at the first point? Since they're all the same shape, that angle has to be the same in all of them. Four of them make a complete turn around the first point, so the angle is exactly a fourth of a turn. That's what makes this work.

Any triangle with a square angle at the first point would have worked. The fun part is that a triangle with lengths of 3, 4, and 5 of whatever - fourths of the radius - has a square angle and has lengths that are easy to measure out.

To prove that, we're going to need several triangles like this of different sizes. The shape of all of them is the same, just the size is different. The first one is five times bigger: its sides are 15-20-25 instead of 3-4-5, but it's still a 3-4-5 triangle. The 'something' in 'three somethings' is five. Five something-smallers.

Now, we draw a line from the square point of the big triangle to the other side, so that it makes two square angles against it. That gives us two smaller triangles.

This one has the narrower angle of the big triangle and it has a square angle, so it has the same shape over all.

This one has the wider angle of the big triangle and it has a square angle, so it also has the same shape.

How big are each of these smaller triangles? This one has its 'five side' pressed against the big one's 'four side' which was 20. Twenty is five fours, so it must be measuring in fours.

The other one has its 'five side' pressed against the big one's 'three side', which is 15, five threes, so it's measuring in threes.

The question is, does this diagram actually work? There's two lines which we can measure two ways. Do we get the same length both ways?

The inner line is four threes, which is twelve, and also three fours, which is twelve again.

The 'five line' of the big triangle is five fives, which is twenty-five, and also four fours plus three threes, sixteen plus nine, which add to twenty-five."

If Nico points out the circular argument, he can probably follow a more rigorous proof, and if not, that's also fine.

He lets the circle fade again; White's burning away. He listens raptly. And bounces slightly. He doesn't think to question the proof. Some things just are. And he's not quite picturing it right, maybe? There's no sand table to draw with here, and Lin can't do sparkles to help.

"Wow. You learned this when you were little, right? This is the kind of thing I wanted to go to the [temple school] for, but they wouldn't let me! Yes math is useful, Dad, you can do shape magic with it! I want to hear more later. I think I'm confused by some of what you just said. Like if it wasn't quite three fourths up, the bigger invisible ball would... Push the circle where the two invisible balls meet down and smaller? ...But right now I just need to make a circle."

He turns to Quiet Pavel and the foreman.

"Can you measure something down here to make sure this works right? Or should I just try when the plumb line is ready? Umm, fifteen feet off the ground, and then also fifteen plus three quarters of ten, so... Twenty two and a half feet up, too. Right?"

"Fifteen, and fifteen plus three quarters of five, which is eighteen and three quarters. Hey Daneia, you hearing this? Can you mark the points for the first five circles?"

    "No problem!"

And soon they're all ready. The werewolves can shape a smooth circle all the way around in about a minute, and will happily continue until Nico is out of white if not stopped.

Once, a faint rumbling thud trembles through the floor. The mining crew ignores it.

He holds out a hand towards Lin for a fistbump!

Well, if they're not worried, he's not worried... He has QUESTIONS about turbines and digging, if there's any major lulls, but they seem really efficient, so there probably won't be.

The last few circles are thin and faint, tiny sparkles spread almost foot apart from each other and moving quickly clockwise. And eventually he does run out entirely. He doesn't seem tired at all but insists there's no more illusions to be had until tomorrow. The magic's just not there.

Um okay, fistbump.

There's a pause of a minute between each circle, and more time ask questions as they go back to the end of the aqueduct that's still being dug out, where the Shift of Breath team takes over for the Shift of Flesh team. They can talk while they work though, and they do, bantering with the foreman about the party he has to throw. The visitors are welcome to stay as long as they like.

If that's as much as Nico can do without rest, they probably won't need him again until the next turbine. "I know I promised you another hundred to do it again later, but the circles would have to be really bright, six of them, and you'd need to do it without looking at all, while the turbine is spinning and has some fire in it. Sorry kid. Did you make any plans counting on having that hundred?"

Yeah, fistbump!

"That makes sense- What's it for, checking everything over?- I'm supposed to get better at it and stronger, but maybe not in just a week. Without looking at all... Maybe something weird with triangles could get all the right dots in the right places, but I don't know. And no. I just want to earn money for later, mostly. It's never bad to have some. Though Lin and Quiet Pavel deserve some of mine for figuring out this is possible! Even if you didn't pay, it would have been worth it to see all this amazing stuff and learn about circles!" Bounce! 

(Is Calsa still around?)

Calsa is right behind him, keeping a watchful eye.

"In this turbine, there are several layers of blades, alternating with 'stators', which look just like the turbine blades but they stick out from the wall instead of the shaft, and they're made of stone instead of steel. We pull the mass of them out of the walls as we put the rotating parts in, and do the final shaping hot -

Huh. It just occurred to me that we could separate those steps if we could make the stators precise from the beginning. The normal way is to send catfolk up the inside of the shaft to look at how close the stators are to the blades, instead of hanging a whole new template for each layer. But if you can guide us with circles, just regular circles like today, while we put the stators in, we can skip most of the visual inspection.

We'll be wanting you again after all!"

    "Aw kid I didn't do anything, just admired your show."

"You had the idea! That's important. At least want a hamburger or something! Maybe I can make... Uh."

He turns and frowns at Calsa. Then back forward. "...Some kind of tool with string and angles to help with doing circles and things."

...He impulsively hugs Calsa around the legs. Briefly.

And then has more questions about why turbines work!

Hm? What was he just thinking...

Hug.

    "If you squish the air before heating it, it expands with more force..."

        "But takes force to squish. The point is that the upper half works better when the pressure is a lot higher in the middle, and that's worth making it complicated."

    "Works 'better'?"

        "Like, more of the energy goes into the spinning."

  "But we're not trying to spin anything, we're trying to move the air?"

        "Right! Flame by itself would make the air really hot, but only move it a little. The turbine sucks in a lot of air and blows it out under pressure, and raises the temperature somewhat."

          "So if you only wanted the heat, you wouldn't need a turbine?"

        "Right! The turbine can't get you more heat out of the same amount of fire. "

  "Oh! I've seen catfolk making glass jewelry that way! Just a long tube of fire."

Quiet Pavel will accept a burger. "You're going to be at the party, right? End of Breath shift."

Nico is happy to hang out in the area for the next little while, cheerful and glad to have 'been blue', which he clarifies means 'working smarter, not harder' and plays off as just a saying. He doesn't wander, and treats the workplace with seriousness, staying out of the way and going patiently quiet when asked. (He has IDEAS: Turbine on a wagon? Pump water up high and let it flow down again for people to use or through another turbine when catfolk aren't around, at his old house they had a water tank on the roof- Or is that what they're going to do anyway? Do they know about balloons?), but then wants to return to the sand table to talk about MATH with Lin. Math is cool and Lin is so amazing for knowing so much of it!! And he will also buy something small for Lin if she wants it!!

He doesn't stop talking about the mines and turbines and circles all day, even when he's taking care of the mice and practicing the language with Calsa. At the party later, he talks more than he roughhouses, about the power of circles and how awesome the tunnels are, and can't show off his illusions any more because he's been playing with circles all day whenever he got any capacity at all back. It's nice, though. Everyone being loud and having fun and eating together.

Days pass.

He hangs out in the square doing illusions, and helps with any more turbine work he can, and spends the money-of-Kef the foreman gives him on little bits of werewolf fur that he washes and then dries to try and make them less smelly, for the mice. And diligently grinds down some of the sand into something softer and less abrasive. He tries, tentatively, taking a point of Green on one that has the worst sore paws. This is classic Green work that everyone can do, right? Specialized healers exist, if you're sick, if you're poisoned, you can hurt yourself worse doing it if you're very thirsty or hungry, but anyone can just do a little- Get-well-now.

It feels weird, but seems to work. Its paws don't ache anymore, He reserves one green point for Diagnose and one for Heal on the worst-off mouse every day. He makes Siki's fireplate, and then tries to experiment with proper [enchantment] from what he can remember from the book, but nothing works. There were all sorts of complicated shapes and runes... He talks a lot with Calsa, practicing his words and grammar and telling her everything he's trying with magic, and almost as much with Lin, and some with Merta and Siki. He tries to learn to write, too.

He doesn't get any more points in the next week or so. He tries not to feel impatient about it. You don't get points every moon.

Promising ideas! There's not really anything to see from a balloon out here, is there?, but hmm... A turbine the size of a water pump on a wagon sounds like a disaster waiting to happen, but somecat tries a smaller one and soon it's a popular game to putter around on it.

The big pumps are for sending water to Argolake, where it joins an older aqueduct that goes all the way to the drybright.

Siki takes him to his workshop and shows him the watches and clocks and gyroscopes and the mechanism for an orrery. "It will get decorated in the city." The planets are, from the sun outward, Equim, Vulpim, Kappim, Ansaf (where they are now), Viridios, Lacrimos, Astros, and Dallekos.

His illusion practice in the square is popular, with many making a point of showing up every cycle.

Lin doesn't need anything small. Can he make a 'potion' or something to give other people the ability to make sparkles?

The mice love the bits of fur. The ground sand is kind of dusty though.

And then, he comes home from practicing on the square, and there's a nervous crowd around the house. Inside, Merta is lying in the huddlebed, her legs encased in rock, with Calsa holding her.

The workshop is amazing and he us exquisitely careful not to disturb anything, in case he gets in trouble. He's more fascinated by the gears in the orrery than outer space.

.........He can, after a couple of experiments, make a potion that sparkles for about an hour when exposed to air, out of a teeny pinch of charcoal and water. It makes pretty paint? And Lin's ability to make airless bottles came in handy to stop it from going off right away!

His first impulse when people are hanging out near the house is to hide and listen for a bit. Only then does he slink inside and go quietly up to Calsa. He doesn't say anything, just looks worried.

"We should give them space."

    "This is the obvious place to go to talk about it. Where would you prefer, the courthouse?"

        "Not the courthouse!"

"You're worried about getting in trouble?"

        "It looks bad. I don't want to make the Magistrate twitchy."

            "It's her fault. Using a wolf to do an elph's job."

  "So you'd have Kef violate the ratio?"

            "The Allheart Alliance does it lots of ways. The ratio is just Sota."

  "You want to rebel against Sota?!"

            "Why not?"

  "And to violate the ratio too! Might as well join the Lei empire!"

"I'd hate to be on that fire crew. They must feel terrible."

      "They didn't hear the screams because they were too engrossed in the story one of them was telling. I think they should feel terrible."

         "Come on, that doesn't make any sense."

  "I wouldn't be surprised if a link in the sluice gate chain turns up missing if you know what I mean."

    "A wolfy cat like Calsa, I bet she wouldn't even notice if that frogold snuck out."

         "What is up with you two and your conspiracy theories."

"Hey kiddo. It's okay. There was an accident at the waterworks, but she'll be fine. The rock is just to keep her clean while she heals."