Disclaimer: Harry Potter is a genus-1 topological surface homeomorphic to JK Rowling.

Parts of this chapter are quoted from Harry Potter and the Half-Blood Prince.

A/N: Warning: I have taken neither Algebraic Topology nor Differential Geometry classes, although I've picked up a few basics from General Relativity and assorted other classes and Wikipedia. Fortunately, the details are not particularly relevant to the plot.

Chapter 37

Hermione was worried that the hard part of the operation would be finding a ley line she could access. She knew from Professor Slughorn that one ran along Diagon Alley, which presumably meant from the Leaky Cauldron to Gringotts. However, she'd need to locate the cornerstone of at least one of those buildings and a hidden place along the line running between them to do the ritual.

She also considered a more speculative possibility: that there was a ley line running from the Leaky Cauldron directly to the Ministry building. She spent some time drawing lines on a map of London and measuring angles to convince herself this was plausible. Unfortunately, she didn't know exactly where the control stone circle in the Ministry was, and any milestones in between must either be buried or on private property. She also looked at the architecture at King's Cross for clues, with little success, plus the wild card possibility that there was a rune stone hidden in plain sight in Trafalgar Square, but none of those were certain enough to act upon.

Even without those uncertainties, she decided she needed to calculate what would happen if she made a mistake. Rituals tended to have disastrous consequences if they went wrong, after all. Luckily, this being such a minor ritual, the consequences would be similarly minor. If the ritual were performed in the wrong place, the alignment runes would draw too much power and burn out, causing the runic blocks to burst into flames, and the Trace would register a large, though benign magical discharge that would no doubt send the Aurors running. Troublesome, but not dangerous.

But it turned out that the direct approach worked in this case. She convinced her parents to drive her up to the Leaky Cauldron, where she convinced Tom that she was working on an introductory geomancy project (not hard with her reputation) and asked if she could see the cornerstone of the building. Predictably, it was in the back wall of the pub where the walled courtyard leading into the Alley was. That gave her some pause. She could lock the door into the pub for the minute or two it took to do the ritual without suspicion, but she couldn't control anyone coming in from the Alley who might see them.

But then she remembered. Harry had an invisibility cloak.

The second problem was that there was still a guard from the Order on Harry's relatives' house. This guard would be under an invisibility cloak, she wouldn't know in advance who it was, and, since she was out of owl contact, she had no way to find out the guard schedule. That meant she would need a plan to take down an unseen guard without being detected—preferably with an invisible or near-invisible spell. She looked through her entire spellbook for a solution. She considered sewing the guard's invisibility cloak to their robes to pin them, but she wasn't sure if it work on a magical garment. She didn't have a Cloak of her own, and she didn't know the Disillusionment Charm, so hiding herself was out. She'd need a distraction.

She had an idea that she thought might work. Wizard could be sneaky when they wanted to, but they rarely anticipated new forms of sneakiness, and they especially weren't trained to anticipate spells that didn't produce light as a major threat, because there weren't all that many of them that could take down an experienced wizard.

The next morning, Hermione got ready. She straightened her hair and charmed it blond, borrowed her mum's frilliest dress, and put on a pair of high heels charmed for stability so she wouldn't look like herself from a distance. She had convinced her parents to drive her to Little Whinging, but not right up to the Dursleys' house. Instead, they parked at the playground on Magnolia Crescent, several blocks away. She went the rest of the way on foot. This was where there could be trouble. If the guard was Mad-Eye Moody, she would be caught immediately and would have to confess everything. If it was anyone else, and they caught her, she could probably talk her way out, but she suspected they would be too wary for her to get Harry out. After all, they would be watching out for impostors, wouldn't they? But with anyone but Moody, she thought she had a good chance of getting away with it.

She casually walked down the street trying to keep a different gait than her usual one. She felt the urge to hum the theme from the new Mission Impossible film. When she was still several houses away from the Dursleys' house, she stopped and pretended to examine a hedgerow. Then, carefully concealed, she raised her wand, laid it across her opposite arm, and threw a lot of power into a new spell of hers: "Echoikonos."

An ultrasonic wave raced down the street and struck the face of Number Four. Any invisible object there would shimmer in the air with magic due to echolocation. It could be detected, and it could be blocked, but since it was different from the Hominem Revelio that any other wizard would use, they probably wouldn't be prepared for it. And here was the clever part: the shimmer would be emitted from the surface of the detected object. That meant unlike with Hominem Revelio, the person under the invisibility cloak wouldn't notice anything wrong.

It worked. There was an invisible figure standing idly in the Dursleys' yard.

The next step was trickier. As far as she could tell, she would have to cast all her spells from outside the property line to avoid tripping the Trace on Harry. There were a lot of unknowns here, but she was going for it. She cast a Supersensory Charm on herself, which would give her enhanced hearing for a limited time. It wouldn't help if the guard was smart and silenced their shoes, but it could give her an edge. She walked past the house, continuing on her way. She wasn't challenged. Then, she judged the angle, pointed her wand backwards along her forearm, and created her distraction.

"Dracones Venit."

Throwing a pebble in the other direction or the equivalent wouldn't fool the guard like in the movies. At least she hoped it wouldn't. But a repeated, rhythmic thumping that sounded like the footsteps of a Tyrannosaurus coming from the wall of the house underneath Harry's window? That was worth investigating. All of the spells she had cast so far had been sound-based, so they didn't produce any visible light, which gave her the advantage.

When she suspected the guard had turned away to investigate the thumping, Hermione spun around and cast Echoikonos. There they were. Without hesitating, she snapped off "Stupefy!" She heard the cloaked figure collapse to the ground. Mission accomplished.

She needed to hurry. She didn't know how long she had before a shift change or something. She cancelled the Supersensory Charm, ran up and pulled the invisibility cloak off the guard. Emmeline Vance. She felt strangely relieved. She'd worry for the country rather a lot if she had outsmarted an Auror so easily. "Sorry, Ms. Vance," she said and pulled the cloak back over her.

Hermione rang the doorbell, and a tall, thin, horse-faced woman answered it. "May I help you?" she asked.

"Hello, Mrs. Dursley, I'd like to speak to Harry, please."

"Harry?" Mrs. Dursley scowled. "There's no 'Harry' here."

Hermione frowned. There was no reason for him to be away from the house. "Are you sure about that, Mrs. Dursley? I'm a friend of his from school."

The woman paled: "Oh, it's one of you lot. What do you want him for?"

"To take him on holiday."

"What? But he's supposed to stay here for two weeks. The…the protections or something…"

"I'm sure he can come back for one day a week or something, Mrs. Dursley. But I need to do this for his safety. He's too easy to find with the Ministry's tracking spell on him."

"I…don't think we should be letting just anyone in the house."

Hermione cocked her head and sighed, drawing her wand: "Do you really think you can stop me, Mrs. Dursley?"

A bluff: she couldn't cast spells on the property, but Harry's aunt didn't know that. Mrs. Dursley backed away in fear and called up the stairs, "Harry, get down here!"

There was a light thump of someone descending the stairs, and Harry appeared. "What is it, Aunt Petunia?" he asked.

"Hello, Harry," Hermione said brightly.

He looked at her in confusion. "Do I know you—whoa, Hermione?"

"Yes, Harry."

"No way!" He drew his wand suspiciously. "You don't look anything like Hermione."

She sighed again. "A topological space is defined as a set of elements X and a collection of subsets of X, tau, such that the empty set and X are subsets of tau, any union of members of tau belongs to tau, and—"

"Okay, okay! It's definitely you," Harry said, "but why do you look like that?"

"Because I'm in disguise. I'm here to get you out of here."

"What? But I'm supposed to stay here to charge the wards."

"Well, you can come back as needed if you want, but I'm going to take you somewhere they can't find you," she said.

"How? The Ministry can Trace my movements," Harry insisted.

Hermione grinned. "Don't be so sure," she said. "I figured out how to remove the Trace."

Harry's eyes grew to saucer-sized. "What, seriously?"

"Yes. Think about it. You'll be able to use magic outside of school, and you won't have to worry about being followed if you go back and forth to…to Sirius's place."

Hermione could see the wheels turning in Harry's head. He twitched his head in his aunt's direction, but didn't quite look at her. Hermione was one of the very few people Harry had told about the Dursleys making him sleep in a coat cupboard until he was nearly eleven. On remembering that, she had a sudden urge to use several dental-themed hexes on the woman, but she restrained herself. "So are you in?" she asked.

"Hell yeah! Do it!"

She shook her head. "We have to go to London to do it," she said. "My parents are waiting for us. And you'll need your invisibility cloak."

He nodded and ran up the stairs and back down again faster than she thought ought to be possible. "Bye, Aunt Petunia," he said. "I'll be back later…probably." He ran out the door, leaving a very confused and worried Petunia Dursley in his wake.

Hermione hurried after him, leading him to her parents' car at the park. "We need to hurry," she said. "I don't know how long Ms. Vance will be out."

"Who?" Harry asked.

"The guard from the Order."

"Oh…Wait, how'd you get past her?"

"I used sound-based spells to distract her and reveal her location and stunned her from outside the wards."

"But aren't the wards supposed to stop all attacks?"

"They aren't normal wards, though. They're specifically tied to you and the Dursleys. I didn't mean you any harm, so they didn't stop me…Probably."

"Probably?" Harry and her parents said in unison.

"I was guessing. A lot. But I thought it was worth trying to get you out of that prison."

"Wow," Harry said. "Er, thanks, Hermione. I mean, it's only two weeks, but it's still really nice of you. So how did you get the Trace off of yourself?"

She grinned at him: "I divided by zero."

"What?"

"Well, it was more complicated than that. It took a few pages of trigonometry to figure out how the Trace fixes your location, and then, I used some runes and a few incantations to basically force the Trace to return a divide by zero error, which overloaded it. Although it does make you unreachable by owls for a month, so you'll want to stick to your mirrors to talk to people."

"Whoa…You don't think we'll get in trouble for this, do you?"

Hermione laughed. "Harry, you're the Chosen One, now. The Ministry will let you get away with anything. Besides, we're only in trouble if we get caught."

"Vance was not at the muggle Prime Minister's residence, my Lord," Rowle said. "Auror Hestia Jones was there instead. She…got away—but she won't be holding a wand again anytime soon." He produced a mangled human arm and laid it down before Voldemort's feet.

Voldemort raised one scaly eyebrow at Rowle. "I suppose that achieves the goal of having one less wand on Dumbledore's side, Rowle," he hissed, "but be thankful I am short of good wands myself. I expect cleaner work in the future."

"Yes, my Lord. Thank you, my Lord," Rowle said hurriedly.

"You said Vance would be on guard and would be vulnerable, Severus," Voldemort enquired. "Can you explain why your information was inaccurate?"

"There was always a chance that Vance would trade shifts, my Lord," Snape said. "I had thought that, as she was assigned to a double shift, she would be more vulnerable, but the schedules are never entirely certain. As it happened, Vance missed her check-in from Potter's residence for reasons to which I am not privy."

"Potter?" Voldemort said with interest. "Was there some incident involving him."

"I can offer only speculation at this point. However, I will continue to make enquiries."

"See that you do. What of Dolohov?"

"He is recovering, my Lord, although I admit that my Healing techniques have made little difference. I now have an approximate idea of what the curse did to him, and I project another two weeks before he can return to duty. He is mostly lucid now, if you wish to speak with him."

"Very good. I will. I need his contacts at Durmstrang. And perhaps with his memory, Rookwood can recreate that curse too."

Harry relaxed at the Grangers' house once their trip was over, idly casting simple charms just to prove to himself he could. He suggested they could visit the Weasleys, too, but she declined: "I'm sort of grounded for almost getting killed. Again."

Harry eventually wheedled out of the Grangers that Hermione had threatened to run away from home to keep from being sent back to France, which made him very uncomfortable, both for prying and because she'd done it mainly for him. He appreciated the gesture, but he didn't want to break up her family.

At Hermione's behest, he had sent Hedwig to the Weasleys with a letter giving a vague but hopefully non-troubling excuse as to why he and Hermione wouldn't be reachable for the next month. Fortunately, there was a loophole in the ritual. He told them to address any post to "Care of Daniel and Emma Granger", and Hermione's parents would ensure it got to him. She also told him to call Sirius on his mirror before the Order started panicking.

"So you need to break the oxygen molecule apart and attach each atom to a nitrogen molecule," she explained. "If you look at the bond structure of the molecules, you can abstract it to this geometric structure."

"Wait, how does that work?" Harry asked.

Hermione had decided that since Harry could use magic now, it was time she started teaching him her spells to rearrange atoms. Since there was a nontrivial chance that this was the "power the Dark Lord knows not", she wanted to get him up to speed quickly. But she'd been working on this project for so long that she was way ahead of him by now.

"I guess it's easier to follow with crystal structures," she told him, and she looked around the scattered papers on her desk. "Umm…here. This shows the conversion from atomic structure to geometric patterns."

Harry couldn't make heads or tails of half the stuff on Hermione's desk. A lot of it was new maths seemed to deal with shapes so abstract that he had no idea how you were supposed to do maths on them. Other papers covered everything from new curses and defence implements to something about a diamond necklace for some Saudi prince. However, he could with some effort, follow how she was converting molecular structures to geometric patterns and runes. "So what's this one for?" he asked.

"This one's for carbon nanotubes. I'm looking for useful ways I can use them for, you know, weapons and armour and stuff, and comparing them with my basilisk-skin coat…" At that, she trailed off, staring off into space for a minute before she exclaimed, "Eureka!"

"What?"

"I know what to do with my coat."

Her parents came into the room. "Hermione, what's going on?" her dad said.

"Mum, Dad, I got it!"

"Got what?" he asked.

"My basilisk-skin coat. It doesn't need to be bulletproof. I can make a lining out of carbon nanotubes, and that will be bulletproof."

Mr. and Mrs. Granger looked surprised. "A bulletproof lining?" he said. "You can do that?"

"Well, I might need to enchant it for stiffness, but yes. I thought of it while I was going back over that nanotube knife design with Harry. Nanotubes are ten times stronger than kevlar—or they should be once I optimise the molecular structure. A thin lining should be enough to stop any handgun, and they have incredibly high thermal conduction, so it won't be too hot, either."

Harry was lost. "Okay, what are carbon nanotubes again?"

Predictably, Harry got an earful. Hermione showed him a bunch more diagrams of molecular structures and talked about things like lattice vectors and cross-links, a lot of which continued to go over his head, but the gist of it was that if you rolled graphite into microscopic tubes, it became ridiculously strong. Muggle scientists had spent years studying them and were just beginning to understand how to make them, but since Hermione could rearrange atoms, for her, it was Tuesday.

"So would that stop curses, too?" Harry asked. "Like if you wore it under a shirt?"

"Hmm…I doubt it," she said. "It's a good idea, but most curses physically strike one's clothes and still affect their victims, even through thick cloaks. It'd have to be magically resistant to stop them."

"Oh, right. But didn't Voldemort conjure a physical shield to stop one of Dumbledore's curses?"

Hermione's eyes widened: "You're right, he did. Maybe if it's an actual shield held apart from your body, it would work—maybe even against the Unforgivable Curses. And since nanotubes are so strong and light…I've got an idea. I'll need more carbon though—Dad!"

"What?"

"I need charcoal."

Hermione never learnt the details of what had happened with Emmeline Vance and Hestia Jones. Harry insisted she wouldn't take it well, despite the fact that she had accidentally probably saved Emmeline's life. She did learn about the murder of Amelia Bones and the bridge collapse and the giant attacks in the West Country, however. Those had hit the muggle news (under "muggle-worthy excuses, of course). For her personally, though, breaking Harry was just the start of a frightfully busy summer. She had to get through her algebraic topology and differential geometry textbooks. She probably wouldn't finish those by September, but thought could by Halloween. Meanwhile, she tried dozens of different molecular structures of carbon nanotubes to figure out which one was strongest. Midway through that, she made a major breakthrough. She discovered she could draw the molecular structures in a computer aided design program and print them on card stock to test them faster and more accurately. Go muggle technology!

She quickly realised that technique would be equally valuable for Archimedes Jewelry and would significantly improve her ability to produce new and nonstandard gem cuts. Her repertoire was expanding significantly in that area. A book on crystallography had taught her about colour centres and other crystal defects to make green, brown, and black diamonds and several desirable types of quartz, often using the same technique she used to turn glass violet. The only things people were asking for frequently that she was still having trouble with were pink diamonds and emeralds.

As for the carbon nanotubes, she was working on a knife that she could keep in a boot or something, like in the movies. Because they were so much stronger and lighter than steel, she wrapped a layer of them around a tungsten core for ballast, resulting in a grind as thin as a razor blade—far thinner than would be possible with a steel blade of that size—and sharpened it to a molecular edge, three times sharper than broken glass. She made a few unremarkable attempts at first, but once she got the kinks worked out, it worked brilliantly. The hardest part was keeping the edge sharp. Even nanotubes wouldn't hold a molecular edge under heavy use. A few simple runes would take care of that, along with others to protect against fire and acid, the nanotubes' weaknesses, but the nanotubes were too tough to carve into. After considering a few options, she made superfine threads of gold and magical wove them directly into the nanotube structure, where they would be visible, but almost impossible to erase. She only needed a tiny amount, and the computer-aided drawing program was again a big help.

When she put it all together, she had a beautiful and delicate-looking stiletto, jet black and all of a piece, with the addition of deadly cutting edges to the original medieval design. It was so strong that it went through a steak like a cleaver when she swung it hard enough, but so precise that she could literally split a hair with it. She made its sheath out of nanotubes as well since she didn't trust anything else to hold it safely.

"I christen thee…the Black Blade of Buckminsterfuller," she said when she was done.

"I think you might be having a little too much fun with this, Hermione," Mum said.

"Maybe…"

But Hermione was just getting started. She revived her magical railgun rig to test carbon nanotube bulletproof sheets. For these, she spun the nanometre-sized tubes into still-microscopic threads and wove them into a many-layered fabric, all with computer aided design patterns and runes to guide the magic. The result was a jet-black cloth that was cool to the touch and smoother than the finest silk. Harry's invisibility cloak was the only fabric she'd ever seen that was finer. She also tested rigid plates that she could use for a lightweight shield. For those, she had to add a reverse backstop in front of the plates to magically stop any ricochets. The mechanics took a bit more work, but she got both the fabric and the plates to the point where they would stop any commonly-used handgun rounds.

And all that work would become embarrassingly moot in October when Lord Cullen recommended banning handguns throughout Britain, but she was proud of herself for working it out.

In late July, after they could be reached by owl again, the Grangers were invited to join the Weasleys for lunch at the Burrow, something they had not been able to do since the first time Hermione left Hogwarts two years ago. Harry would be there, too, and Hermione was eager when her parents agreed to go.

The Burrow looked much as they remembered it, though a bit more crowded and a bit more sombre. Percy still wasn't back. Despite the Ministry acknowledging Voldemort's return and a change of Minister, he was apparently resentful of how he had been dismissed by his family for the past year and he still wasn't talking to them. Hermione wondered if she ought to have a word with him.

On the other hand, Bill was at home, and Hermione's friend Fleur from Beauxbatons was with him, and Angelina Johnson was there, too, as Fred reasoned that since Bill and George got to bring their girlfriends, so did he. (Ron still didn't have a steady girlfriend, despite his on-again, off-again relationship with Parvati Patil.)

Fleur was the first to greet her. "'Ermione, it is so good to see you," she said with considerably less of an accent than she'd had in school. "I wanted to tell you zee news, but I couldn't reach you."

"Sorry, Fleur. I've been out of contact. What's the news?"

Fleur thrust her left hand in Hermione's face, showing off a large, glittering ring.

"Mon Dieu, vraiment?" Hermione said. "Tu et Guillaume?" she looked between the two of them.

Fleur grinned giddily and nodded.

"Félicitations!" She hugged Fleur, who began babbling in French about how much she loved Bill. Mrs. Weasley, Ginny, and even Angelina didn't look too happy about it. Veela did tend to grate on humans of the same gender, but fortunately, Fleur and Hermione usually got along well. She thought Fleur and Bill would be good for each other, despite Mrs. Weasley's reservations.

George cut in next, pulling her away from the excited Fleur. "It's good to see you, Hermione," he said, and kissed her.

"Hold up, George," Fred pulled him back good-naturedly. "You know what the Ministry says; we need to make sure she's the real Hermione."

Hermione rolled her eyes: "A differentiable manifold is a locally-Euclidean space in which all pairs of distinct points have separable neighbourhoods and which can be represented by an atlas of linear maps that are continuously differentia—mmpf!" George cut her off by kissing her again.

"C'mon, Fred, you should know by now that no Death Eater can fake Hermione's identity." Most of the Weasleys laughed.

"I'm sorry I haven't been around to help you two set up the shop," she told them. "I wanted to, but I had to compromise just to not get shipped back to France."

"It's okay, Hermione," George assured her. "The money helped us more than we could have dreamed. Just be sure you're there for the grand opening."

"When's that?"

"Next Saturday. We have to be open before the kids do their school shopping."

She looked to her parents for confirmation. "I suppose you should go since you're a partner," Dad said.

"Great. I'll be there. Thanks, Dad."

"Excellent," said George. "Oh, and we got those Shield Cloaks you suggested done. They're not super, but they'll stand up to a Stunner with no trouble.

"That's good. Can I get you to put the spells on my coat? Dolohov's curse burned through part of the magic-resistance, and I put a new lining in it that I want protected, too."

"No problem. And we came up with a new product in honour of Lee. Check it out."

"George, put that thing away during dinner!" Mrs. Weasley said in an aggravated tone.

The new product turned out to be a tarantula with red and gold stripes sitting in a terrarium shaped like a half-Quidditch pitch with a miniature Quidditch hoop at one end. When Fred tapped the glass, the spider picked up a miniature Quaffle and threw it through the hoop with its front legs.

"We couldn't quite get the tarantula to make cheeky comments at everyone, so this was the next best thing," Fred said, "but if you attach two of them together, they'll play against each other."

"Wow," Hermione said. "I…I'm sure Lee would be proud." Ron, unsurprisingly, gave them a wide berth.

Hermione greeted Ron, Harry, and Ginny as well. Ginny apologised to her for freaking out at the end of last term when they learnt about horcruxes. She was a lot calmer and more determined about it now, although she did want to know what progress Hermione was making towards solving Harry's problem, to which she replied that she was still studying the arithmantic foundations. This wasn't going to be a quick fix. She'd even queried Dumbledore (using coded language) about whether she could stop Harry's heart with her Commotio Cordis spell and restart it to get rid of the horcrux, but had told her in no uncertain terms that it would not work. To destroy a horcrux, the vessel had to be destroyed beyond repair, and for a living horcrux, that would mean brain death.

But she tried not to dwell on such morbid thoughts. They ate outside, owing to the size of the group. Mrs. Weasley had prepared an excellent feast, as usual, and when they sat down to eat, everyone ate in silence for a little while, savouring the food. After a while, George asked her, "So how did you do on your O.W.L.s? Straight O's, I assume?"

"Oh, no, no. Sorry, George, but I'm too practical for that." She shot Bill a grin. He was the only one at the table who had scored higher than she did. "I got Seven O's and three E's."

"Three E's?" Bill said. "What were they in?"

"Herbology, Magical Creatures, and Muggle Studies."

The Weasleys all stopped. "Muggle Studies?" George said. "But you're muggle-born. How could you get an E in Muggle Studies even without the class?"

"Because it was massively out of date. I'm sure Mum and Dad could have aced it, but I'm not familiar enough with what life was like in their generation. And of course, for Herbology and Magical Creatures, I missed the practical parts of the classes, but I'm planning on dropping those classes anyway."

Emma coughed. "You are?" she said. "That's news to us. I hope it's not just because you didn't do as well in them."

"No, Mum, it's not because I only got E's on my O.W.L.s. Ernest Rutherford once said all science is either physics or stamp collecting, and I'm a physicist. They just don't play to my strengths, and I'm not planning to be an Auror or a Healer, so I don't need them."

"You have to drop some classes for N.E.W.T.s, Mrs. Granger," Bill spoke up. "I only did seven, and I don't know anyone who did more than eight."

"Well, I suppose so," she replied, "but we weren't sure if it would be the same with Hermione's tutoring program."

"What? You're not coming back to Hogwarts?" asked Ginny.

"No, not full time, sorry," Hermione said.

"We felt that Hermione would be safer sticking with Professor Slughorn," said Dan. "It was part of our compromise not to send her back to Beauxbatons."

"But Hogwarts is the safest place in Britain," Ginny said.

"People keep saying that, but can you really believe that after—excuse me, but after the number of near-death experiences you've all had there?" Emma asked.

Mrs. Weasley paled noticeably, and Mr. Weasley leaned over and patted her hand. "It's okay, Molly," he said softly. "Dumbledore and the Ministry actually have it together now. They won't slack off like they did before. We do understand your concerns, Mr. and Mrs. Granger, but circumstances will be quite different this year."

"I'm not really unhappy with the arrangement," Hermione said. "I'll still be visiting. I'm going to take Professor Dumbledore's Alchemy class, and I'll be keeping up with my mastery with Septima. Honestly, though, I prefer the mobility of being in private tutoring."

"Can you be a part-time student?" asked Harry.

"In the strict sense of being an enrolled student, no. I checked with Dumbledore. As a practical matter, other students go in part time to do a mastery or something similar, like I'm doing. Or homeschooled students will come in if they want to take an advanced class their tutor can't teach, such as Alchemy. It's registered as an independent study or tutoring program, even if they're sitting with the rest of the class. Anyway, with my schedule, I can join you for lunch on Saturdays."

"Oh, well, at least we'll have that," Ginny said. "What about you, Harry? How did you do?"

"Pretty well," he said. "O's in Defence and Potions, E's in everything else I care about."

"Potions?" Fred said in surprise. "Slughorn must be good, then. Wish I could see the look on Snape's face."

Harry smiled. He had indeed passed all the classes he needed for the Auror track, much to his own surprise. Snape only took O-students in his N.E.W.T. class. He'd definitely flip when Harry showed up.

George, of course, was happy that Hermione would still be close at hand outside of Hogwarts. "It'll be good to be able to call you anytime, not just on Hogsmeade weekends," he told her. "Can you do all your N.E.W.T.s that way, though?" he asked with concern. "Slughorn can't be qualified in everything."

"I'm still working out the details. I'm dropping Astronomy and History, too, but I'm taking the rest. I have to take Potions, or Professor Slughorn will be disappointed, and Transfiguration is required for Alchemy. He's qualified in both of those."

"Slughorn knows Transfiguration?"

"Yes, he's really good at it. He claims he even turned himself into an armchair to hide from Death Eaters. Then, half the Order is qualified in Defence."

"I'll be helping you with that, Hermione," Mrs. Weasley cut in.

"I don't want to impose, Mrs. Weasley. I'm sure Remus would—"

"Of course he will, dear, but I insist. After the Ministry, I told Fred and George I'd be giving them a few lessons whether they like it or not."

George leaned close to Hermione and muttered, "Lee dying really freaked Mum out. She's giving everyone lessons now."

"I didn't know you were qualified as a teacher, Mrs. Weasley," said Emma.

Mrs. Weasley smiled and nodded. "Yes, I tutored Defence and Potions, Mrs. Granger. It was quite a while ago—just for a few years after I left Hogwarts—before I had more babies than hands. I never went for a Hogwarts job, though. Professor Slughorn was still there, and I didn't fancy the mortality rate for the Defence post."

"Well, thank you, Mrs. Weasley," Hermione said. "So that's Defence, and most people who are qualified in Defence are also qualified in Charms. Charms is an Arithmancer's bread and butter, so I'll need that. I'm just a little worried about Ancient Runes."

"I'll teach you Runes, Hermione," Bill said.

"You will?" she said excitedly.

Dan was a bit more sceptical: "You're qualified?" He did look pretty young.

"I'm a Cursebreaker, Mr. Granger. That requires a mastery in Runes."

"Will you be alright with your job, though, Bill?" Mrs. Weasley asked him.

"Oh, that'll be fine, Mum. Since I'm in Britain for the duration, there's not that much for me to do at the bank."

"Thank you, Bill," Hermione said. "I really appreciate that."

It was good to have everything settled, but Merlin, her education was going to be complicated this year. She'd be taking two classes with Professor Slughorn, two with Remus (with Mrs. Weasley's help), one with Bill, one with Dumbledore, and one with Septima, and in at least three different places.

"I'm a little surprised you are going back, Harry," Emma said. Harry hadn't been entirely certain when last they spoke, but he'd been leaning that way all summer.

"I really do like Hogwarts, Mrs. Granger," he said. "And Sirius wants me to go back. He says I'll be safer sticking close to Dumbledore now that they're getting serious about security." And he'd still be with Ginny, although he didn't say it out loud.

Unfortunately, things weren't as carefree at the dinner table as they were two and three years ago. Harry's past year had been almost unmitigatedly bad, and even for the others, talk didn't stray far from the war after that. Even Mr. Weasley's incessant questions about muggle life focused on how muggles fought wars and defended themselves from attacks. The Grangers explained about guns (his ignorance of the issue made them dearly hope he never came in contact with either end of one), but they didn't have the heart to tell him about nuclear weapons.

The various attacks that had happened that summer were discussed. Amelia Bones, the head of the DMLE and probably the most solid candidate to replace Fudge, had been murdered at the start of the summer, something the Weasleys placed squarely on Fudge's shoulders for his incompetence, which forced them to replace Fudge with a sketchier Auror named Scrimgeour. Hestia Jones, Auror and Order member, was crippled by losing her wand arm. And old Mr. Ollivander had been kidnapped right out of his shop, presumably as a strategic asset. A lot of his stock had been stolen, too, and his family had moved to Hogsmeade, only coming back to sell wands to the new first years under heavy Auror guard. In other words, they said, things were starting to get bad, like last time.

After supper, the Grangers needed to start driving home soon, but before they left, Dan wanted to squeeze in the obligatory father-daughter's boyfriend talk with George. Hermione and Emma hadn't shared all of their girl talk with him, but he could see that his daughter and George were very close, and he couldn't very well pass it up. He dropped enough hints during lunch that Mr. Weasley saw what he was doing and was eager to cooperate.

"So I need to help clean up here," he said with a wink. "George, why don't you show Mr. Granger my collection of muggle gadgets in the shed? Maybe he can help organise them."

George gave his father a funny look, but he could tell he was holding firm, so he went along. Hermione shot her father a warning look. She had anticipated this and told him to be nice. George had, after all, rushed to her side to save her from murderous thugs and lost his best friend in the process, so he couldn't be that bad.

Dan found that Mr. Weasley's favourite items to collect seemed to be electrical appliances and gadgets, which he seemed to understand far less than the head of the Misuse of Muggle Artifacts office ought to. None of them were really dangerous by themselves because the Burrow didn't have electricity, but Dan did feel the need to point out several that should definitely not be hooked up to electricity because they were in no condition to be used safely. Once that was done, he could get to the real discussion.

"So, George," he said, "what are your intentions towards my daughter?"

"Er…" George quickly considered his options that wouldn't make either Dan or Hermione mad at him. "Well, Mr. Granger, I…care deeply for Hermione and I want to stay with her for as long as she'll have me…And I don't think I can say more than that because I don't think anyone can tell her what to do at this point."

"Ah," he replied. "Unfortunately, I think you're right. Heaven knows Emma and I can't control her anymore. But I'm a dentist with a teenage daughter. I've been looking forward to making this speech for years. You see, this is the part where I'm supposed to threaten you with a root canal without anaesthesia if you hurt my baby girl, but honestly—and don't repeat this to either of them—Hermione scares me more than Emma and her father put together ever did. I'm not sure myself what she's capable of anymore."

"Heh—" George smiled uneasily. Fred had been teasing him for a year and a half that he was in over his head with Hermione, but he was starting to think it might be true.

"So you've noticed it too." Dan flashed him an evil grin. "No father enjoys seeing his little girl grow up like this, but I think Hermione is sensible enough to handle a relationship with a boy. Emma certainly thinks so. Just remember, George: I'm pretty sure my daughter has invented a spell that can do a root canal by now…and she got that from me."

Yep, he was doomed.

During the summer, Apparition was a flexible independent study course taught by a Ministry instructor. Most students, even though it wasn't required, took the course at Hogwarts in their sixth year where the twelve lessons were normally done in twelve weeks. Hermione wanted to pass the Apparition test as soon as possible after her birthday, so she arranged to do it is six weeks.

The Ministry instructor, Wilkie Twycross, was a small, unassuming man with wispy, transparent hair. He was also a pretty lousy teacher. Oh, sure, the official Ministry course material had serious problems too, but while Mr. Twycross was very gifted at Apparition and very enthusiastic, he couldn't seem to explain it much better than parroting the guidelines.

"The most important things to remember when Apparating are the three D's!" said Twycross. "Destination, Determination, Deliberation!"

"Step one: Fix your mind firmly upon the desired destination."

"Step two: focus your determination to occupy the visualised space! Let your yearning to enter it flood from your mind to every particle of your body!"

"Step three: Turn on the spot, feeling your way into nothingness, moving with deliberation!"

When Hermione tried it, nothing happened. It normally took a while to get any results, Mr. Twycross said, but she suspected it was not least because he wasn't really explaining how to do it besides repeating the three D's and making offhand comments, not really explaining how to do the actual spell. Plus, "Destination, Determination, Deliberation" was a terrible mnemonic—too many syllables and too easy to get mixed up.

The Ministry-issued pamphlet, Common Apparition Mistakes and How to Avoid Them, was more informative despite adhering to the three D's paradigm. A book on magical transportation from Flourish and Blotts was even better.

If you lacked "destination", she learnt, you could wind up anywhere. In other words, if you were distracted and thinking of some other place than your intended destination, you could wind up there instead, or even halfway between them. If you lacked "determination", that is, if you did not "visualise" (though it was more feeling than seeing) yourself at your destination sufficiently well, you could splinch yourself—leaving body parts behind. This wasn't as disastrous as it sounded, but it was still very unpleasant, and it could have severe complications for something as simple as someone grabbing your arm when you Apparated. And if you lacked "deliberation", you'd fall flat on your face.

The "turning on the spot" part was, in a sense, the wand motion for Apparition, but it was described in non-standard terms. "Deliberation" wasn't actually that bad a name for it. It required a steady pace. If you rushed it and turned too fast, or if you hesitated and turned too slowly, the spell would fail. Fortunately, this failure mode usually meant only that nothing would happen rather than winding up in pieces.

The way the book described the spell sounded an awful lot like the muggle understanding of wormholes, although wormholes as understood by general relatively couldn't actually transport anything. They would collapse the instant any particle tried to pass through. She had a feeling that would be important.

By her second lesson, Hermione had a list of questions for Mr. Twycross. She resisted the urge to open with why it was the three D's and not something easier to remember like the three P's for "Place, Person, Pacing", or the three T's for "Target, Totality, Timing", or even ABC for "Aim, Body, Caution"—none of which were particularly good, but all of which would have been better than the three D's. Instead, she had more pressing concerns.

"So, Mr. Twycross, when you Apparate, what stops you from appearing where another solid object already is? Like a wall—well, I guess if you visualise your destination, you'd know there's a wall there—but what if a person is standing there—or if you Apparate to a wood and land where a tree is?" That was her most worrying concern. In The Stars My Destination, the one muggle story she knew where a science fiction version of Apparition was used, such a collision resulted in an explosive reaction of the intersecting atoms. Since the Ministry literature didn't mention it, she assumed it wasn't a serious issue.

"What?" Twycross said. He looked momentarily stunned to be asked such an advanced question, but he soon collected himself. "Such collisions are rare, Miss Granger, and they are not particularly dangerous. In Apparition, you don't merely appear in place, you see. When you examine the magic very carefully, you will find your form expands from a particular point. But that point cannot be inside a solid object—or in water, for that matter. There is too much interference from the matter of the destination for your magic to place you there. Instead, it will bounce you to the nearest spot where there is enough space to appear."

Well, that was one problem solved, and it lent credence to her wormhole theory. It also explained the cracking sound of Apparition; it was simple displacement of air. She wondered if that still applied if you deliberately tried to Apparate a thousand feet underground or something, but she didn't think anyone would be crazy enough to try it. "What about the problem of relative motion at different latitudes?" she asked.

"Excuse me? What does that mean?" Twycross responded.

"I mean if your destination is moving relative to yourself. Different parts of the Earth rotate at different speeds, Mr. Twycross. "Even Apparating from here to Inverness, which I don't think is a great strain on most wizards, would be…" She thought for a minute. She wasn't certain what the latitude of Inverness was so she had to estimate. "…maybe eighty to a hundred miles an hour difference."

"Would it really?" he said in surprise. "I had no idea. Obviously if you visualise your destination properly, it never causes any problems. The magic takes care of it."

That did make sense, although she had to wonder if the relative velocity problem was why Apparition over long distances was so difficult. The curvature of the Earth meant there was a problem with longitude, too. Then, there was the more general problem: "Does that apply to all relative motion? What if you were Apparating somewhere that was moving relative to the ground—like on and off of a moving train?"

"Aha! Off of a moving train is no problem," Twycross said. "You need only have your destination fixed firmly in mind, and you will land there firmly on your feet. Onto a moving train, however, is much more difficult and is not recommended, even for experts. For you would need to know both where the train is and how fast it is moving, not to mention the obvious secrecy concerns. You are much more likely to miss it entirely, or to be thrown from it when you arrive."

"Of course, Mr. Twycross." She'd also wondered about Apparating out of a fall from a great height, but that had other issues, and a Hover Charm or Arresto Momentum would save you much more reliably, so she let it go. "So you can't normally Apparate to a place you haven't been before, but can you do it if you have line of sight and/or know exactly where it is in relation to yourself?"

"You've certainly thought a lot about this, Miss Granger. Yes, if you have line of sight, and it's close enough to see clearly, you certainly can. It's just a matter of visualisation. If you only know the relative position, it's much more difficult, but possible for experts."

"I see. And do you have to be standing to Apparate? I know you're supposed to turn on the spot, but it seems like much more of a mental exercise than with most spells."

"Not as such. Just as any spell can be performed without incantation or wand motions with sufficient mental effort, so can Apparition. But a beginner like yourself should be very cautious about pushing yourself. For now, just turn on the spot, as I have told you, with deliberation!"

Hermione tried it again, but she only succeeded in spinning around and losing her balance, nearly tripping over her own feet. She was sure she was missing something here. The literature and Mr. Twycross kept saying to "turn on the spot", but she had seen both wizards and elves Apparate, and they didn't really turn—or they only began to turn before they vanished. There was a sort of twisting motion, or a vague sense of movement around them, but most experienced wizards seemed to do it almost standing still.

She focused harder, trying to figure out her problem. The "determination" part was the most mentally taxing part of Apparition—and the most disastrous if you got it wrong. You had to feel your whole body and visualise picking it up whole, after a fashion, and setting it down in another place. It was a complicated spell, though to most wizards, it seemed to be as natural as driving. But the "deliberation" part was the hardest to wrap her mind around. The literature never came out and said it, but it seemed like "turning on the spot" wasn't quite right, the wizards knew it wasn't quite right. It was just that they lacked the words to describe it adequately.

But after pondering the issue, she thought that to someone well-versed in both science fiction and advanced mathematics, "turning on the spot" could have another meaning: turning through the fourth dimension. Turning in that way wouldn't look like motion at all, merely like vanishing into thin air—or perhaps like collapsing down to a point. That fit her wormhole theory, too, and it made sense of the "picking up" and "setting down" notion, since it was possible to pick up and set down objects through the fourth dimension without traversing the intervening space. And with wormholes being the fickle creations they were, it would certainly be important to focus carefully to ensure that your whole body got through it without being cut off in the middle. She had no proof; Apparition was so old and the requisite maths so new that she doubted anyone had done a good Arithmantic analysis of it (maybe something for her list), but it made sense.

She hoped her new perspective would help her. Hermione focused on the three D's, but with a 4-D perspective this time. It was, in retrospect, doing the D's out of order—the destination last—but it made more intuitive sense. She felt out her body, which, being the most important thing, seemed naturally the thing to do first. Then, she focused on turning herself through a four-dimensional space. This was of course impossible to fully visualise, but she imagined it as a sort of inner space projected onto the real universe. And then, she imagined herself pulled through that space through a wormhole to land inside the hoop. She began to swing her wand—

And in a blink, the world shifted around her with a loud crack. She'd barely started, and she was suddenly inside the hoop. She turned around and saw the spot where she had started from empty. Excited, she wiggled her fingers and toes and patted herself down. All body parts? Clothes? Hair? Yes, she'd done it!

"Aha! Splendid, Miss Granger!" said Mr. Twycross. "That was perfect on your first successful attempt! Barely even a spin. I think you may be a natural!"

"Thank you, Mr. Twycross," she said breathlessly. "But I think muggles have thought about this a lot more than you think."

A/N: Echoikonos: based on the Greek for "sound image".