Cardinality
Summary: A serial killer is stalking the citizens of Las Vegas. Started out as a case file but the G/S aspect demanded equal representation. This is the safe version for this site. The original version can be found at my web site.
Rating: R for subject matter
A/N: No real spoilers. Many thanks to Burked for teaching me how to be a serial killer, among many other things.
Disclaimer: Obviously, I don't own anything related to CSI. If I did, certain characters would be getting more screen time – together!

"I cannot help it; in spite of myself, infinity torments me." – Alfred de Musset

Chapter 2

"Well, if it isn't little Miss Sunshine," Catherine teased as she glided into the locker room that night, finding a clearly upset Sara stashing away her personal items.

"Don't start."

"Okay," she said, backing off at the harsh tone. Leaning against the far locker, she gave the younger woman a friendly smile. "You know, the mayor's office and the sheriff have been riding his ass about this case since it started. He's a little stressed. Don't take it personally."

Ignoring her colleague's comments, Sara went directly to gathering her evidence and headed to the Layout Room. Catherine followed cautiously, sensing there was more to this than Grissom's terse behavior.

"Did things go okay at the university?"

"Not really."

"Mathematician couldn't help?"

"Couldn't find one," Sara sighed, explaining the day's events.

"He didn't," the blonde exclaimed lightly when she got to the part about flirting, her lips twitching. "Sorry."

"He did," she muttered, failing to see the humor. She'd found the entire exchange insulting. Seeing the younger woman's discomfort, Catherine gave her a friendly smile and shrugged, as if to suggest what else would you expect from Grissom.

Sara glared briefly before pulling a note from her bag. "I spent all afternoon talking to his secretary. Not a nice woman. He was out of town today, but I finally arranged an appointment tomorrow morning to visit a retired, oversized flirt. One Professor Gustav-Maximilian Brandenburg."

"Ouch. Who would do that to their kid?"

"Why don't you ask him yourself?" Sara asked pleadingly. "You're … better … at this kinda thing. No offense."

"None taken," Catherine chuckled. "But the big guy's all yours. Consider it a training exercise."

"Thanks, Cath. Thanks a lot!"

Grissom walked down the hallway, his fingers playing nervously with his file. He needed to apologize, at least according to Catherine. She had entered his office, plopped down in a chair and asked him pointedly if pissing off Sara was a deliberate effort or a natural talent.

The statement caught him by surprise. After proclaiming his innocence, Catherine had proceeded to explain to him why Sara was upset. He apparently had compounded his error by pointing out she didn't need him to hold her hand when she was insulted.

"Get off it, Gil. Not even you are that dense," she fumed, ignoring his warning look. "You stood by while her work was dismissed because she was a woman, then you tell her to be a good little girl and go bat her pretty eyes at some big slob to get him to help us."

"I didn't," he insisted, but dropped his eyes to the assignments in his hand. Had he? It hadn't been an intentional slight. He'd been too distracted by the mayor's irritating page to pay attention to her mood.

Looking sheepishly at Catherine, he promised to apologize, but as he approached the Layout Room, he wasn't entirely sure how to proceed. People accused him of being an enigma, but she was at least as much as one. To him anyway.

While Sara could get emotionally involved in cases, and had a notorious temper, she could also be evasive about her feelings. People might know something had upset her, but she rarely volunteered the reasons why. Even when something was clearly bothering her, she would insist she was fine. Why?

When he'd returned from his surgery, he'd expected Sara to be angry with him. He knew she hadn't been pleased by his response to her earlier dinner invitation. Instead, she'd been polite and professional, if not exactly friendly.

The entire situation was as confusing now as it had been when she first asked him to dinner. In hindsight, Grissom realized he could have handled that encounter better, but at the time he had been too distracted by his hearing troubles.

He'd had plenty of time to reflect on that day while he recovered from surgery. She'd been acting unusual after the explosion, from denying her injuries to her pulling a gun on a suspect. Had Sara's invitation been another fluke? It seemed that way; she'd never brought it up again.

Considering all the emotional and professional complications a relationship with Sara would entail, it would be best if they didn't get involved. It would be easier and safer. It was better this way.

Definitely.

Entering the Layout Room, he found Sara intently going over the photos, pausing occasionally to jot down some notes. Calling out softly as he entered, Grissom walked to the opposite side of the table and leaned forward.

"How did it go?"

"Professor Brandenburg was out of town. I have an appointment with him in the morning. It was the earliest I could get in."

Grissom nodded, fiddling with the file in his hand. That last sentence had clearly been a pre-emptive strike; she actually expected to be criticized. He gave her a forced smile.

"Good. I'm sure you can handle it."

"You're not coming?" Sara asked pointedly, looking up from her notes.

"No." Even if he didn't have a follow-up appointment with Dr. Roth tomorrow, he would have declined. If he had any questions, he could contact the professor himself at a later date. Grissom didn't want her to think he doubted her abilities, and he knew he had given her a reason to think that way, even if it had been unintentional.

"Can I help you with something?" she asked when he remained silent.

He tried another forced smile. Grissom shifted, then pointed to the folder in his hand. "There's been a breaking-and-entering over on Oakwood. The only thing missing is a stuffed cat."

"A toy?"

"No. As in Whisker's remains," he said, passing her the folder.

"You're pulling me off this murder to investigate a stolen dead cat?" she asked, her disbelief barely masking her pain.

"No! I thought you'd like it. I figured you might want a short break. Get some fresh air. If you don't want it, I'll give it to Warrick when he gets back from his carjacking."

"If he's on another case, I'll take it."

"If you're busy with this ..."

"I'll take it," she stated calmly, holding out her hand for the file.

Grissom sighed as he passed the folder over, giving an apologetic shrug. He had thought Sara would appreciate the bizarre case. It wouldn't take long, and she usually liked the out of the ordinary ones. Instead, she thought he was punishing her. Pausing at the doorway, he winced slightly when he noticed her abrupt motions as she packed away the evidence. He wondered when their friendship had deteriorated to this point, sighing as he admitted the answer to himself.

"Sara … you're … an … asset to the lab."

"Thanks."

Sara pulled her Yukon into the driveway of a large Tudor-style house, knowing she was early for the appointment. She was anxious to get to work on the case, and hoped the mathematician would be able to shed some clues on it. After returning from the case of the stolen cat, she'd tackled the equations again, but had made little progress.

She told herself she was being paranoid as she followed the secretary's directions to head to a side door. It was probably the entrance to an office, and wasn't intended as a put down. With the way this week was turning out, she wouldn't discount it, though. After ringing the bell, she scanned the area. While the neighborhood wasn't on the same scale as Summerlin, this house wouldn't have come cheaply.

A dour-looking woman opened the door, interrupting her examination. After introducing herself, Sara was escorted through a small room – complete with file cabinets, a desk and couch – into a larger room. From the setup, she gathered this had originally been used as a guest or an in-law suite. The anteroom would have been a sitting area, this room would have been the bedroom, and one of the doors on the far wall would lead to a bathroom.

The unpleasant woman told her to wait, saying the professor's therapy session had run late. Her displeasure in Sara's arriving early wasn't lost on the CSI. Walking around the room, she eyed the electronics with envy. Walking over to the fireplace, she noted the various diplomas and certificates over the mantle. If nothing else, the guy had an impressive education.

Turning her attention to the rest of the room, she tried to see if there were any clues about her host. Everything was oversized; the room had a 12-foot high ceiling, the doorways were taller than normal, and the antique carved desk and leather chairs were massive. She wondered if the intention was to intimidate visitors.

"May, when that officer gets here, send her to the kitchen and tell her to help herself to whatever's in there."

"Actually, I'm early," Sara said, turning around. Coming out of the bathroom was a very startled and very nude man looking up from pulling a plastic bag off the cast on his left hand. She dropped her head quickly, fighting the urge to chuckle. Out of the corner of her eye, she saw him quickly hobble to the desk to grab a knee brace off of it.

"Nice to meet you. I'm embarrassed," he said, a slight Southern accent coloring his voice. "You'll excuse me if I don't shake your hand," he added, as he backed towards the bathroom, the brace held strategically in front of him.

Keeping her humor in check, Sara dropped down in one of the chairs, deciding the oversized furniture suited their owner. A few minutes later, the man returned, wearing casual attire along with a deep blush. If she hadn't of noticed his normal skin tone before, she would have thought he had a bad sunburn.

"Professor Brandenburg? Sara Sidle, Las Vegas Crime Lab. Thank you for seeing me," Sara said, trying to maintain a professional image. The mathematician approached sheepishly, an apologetic smile lighting up his face.

"Yes, well, I don't imagine you were expecting to see so much of me," he said, his eyes twinkling with amusement. They were such a deep green, she first suspected he was wearing contact lenses. "I do apologize for that. I hope I didn't offend."

Sara found a smile trying to creep up despite her attempts to keep it down. She recognized his type. The man practically dripped charisma. He probably was the captain of the football team in high school, able to get away with any transgression with a smile.

"In this job, I've seen worse," she said, realizing belatedly how insulting that could be construed.

"So I'm not as bad as a two-week-old corpse? That's something, I suppose." His smile was teasing, but Sara resisted returning it. The green eyes widened in amusement as he took a seat.

"What can I do to help the Crime Lab?" he asked kindly. "Computer troubles?"

"Math actually. We were hoping you could decipher these," she said, handing him a sample photograph. "I haven't been able to make any headway."

Taking the photo, he leaned back in his chair, rubbing the cast over his chin thoughtfully. "I'm not surprised."

"I beg your pardon?"

"These forms are rarely used. It's easier to work with cosines and sines," Brandenburg said. Looking up, he offered her a contrite smile. "I have four sisters. I would never imply anything that stupid."

Opening a drawer, he pulled out a notepad, and began making notes. He paused briefly, giving Sara another friendly smile. "What is your background? Criminal justice?"

"Theoretical physics. Harvard."

The mathematician stopped, looking over the top of the photograph in his hand. Leaning forward, he whispered conspiratorially, his eyes dancing with mirth.

"Knowing the city needs someone who can do Lorentz transforms and understands the tunneling effect doesn't really inspire confidence. I hate to see what type of crime scenes you work."

"There's more to physics than relativity and quantum mechanics," she said, raising an eyebrow in a silent challenge.

"I'm sure there is," he said, managing to make the statement sound suggestive, causing Sara to blush. "How did you get stuck in Vegas?"

"We actually have one of the best labs in the country."

"Really?" he asked, cocking his head as he examined the photo. "Are these written in blood? That's something you don't see every day. At least, I don't," he joked, noticing the stacks of folders she brought. "There's more? Let's move to the kitchen."

He politely helped her with the files, escorting her through the anteroom into the kitchen. She quickly understood why he wanted to move there: the room was huge, with long granite counters running along the walls and forming a large island. Besides a big bowl of fruit, they were mainly bare.

Directing her to a barstool, he moved the few containers off the island, clearing room for them to spread out the photographs. Sara arranged them in piles, according to which wall they came from, while her host moved to the cupboards.

"Help yourself to something to drink," he said, placing a box of cookies and a glass in front of her and pointing to the fridge. "I'd offer you coffee, but I can't stand the stuff. I'm afraid I don't have any in the house."

Drawing up another stool, he started looking through the photos as Sara went to get a drink. The fridge was equally bare. The beer was tempting, but she knew caffeine would be more effective, not to mention professional, than alcohol. Grabbing the pitcher of tea, she poured a glassful, and went back to join her host.

"Do you have overall shots? That show the entire wall?"

"Here," she said, reaching over to open a folder. "There was writing on four walls in each of three rooms. Each stack is from a different wall."

"Odd," he muttered, writing down some notes. "This is going to take some time."

"If you'll let us know your hourly rate, I can submit a bill to the city."

"You couldn't afford me," he said with a knowing smile. "Will the city cover meals? I don't have much in the way of food in the house, and you wouldn't want to eat anything I cooked."

"That's easy enough," she said lightly.

"You haven't seen how much I eat," he said softly. "I have very … healthy … appetites."

"The city will cover meals, Professor Brandenburg," Sara replied, once again fighting the urge to return his smile. It was infectious.

"Max is much shorter," he said with a wink.

"Yes it is. Doctor," she replied, her own smile finally making an appearance.

Brandenburg chuckled, raising his eyebrows playfully, then turning his attention back to the photos. After several minutes, he started flipping through the stacks in a noticeable pattern, a curious look forming.

"Now that's twisted," he said, giving Sara a deep smile. "Literally."

Walking through the hallways, Grissom ignored Catherine's 'I told you so' looks, as well as the laughter coming from the labs. It seemed as if everyone was joking about his sending Sara to flirt with an oversized, retired consultant. More accurately, the source of amusement was what her reaction to it would involve.

At least now he understood Catherine's concerns. Earlier, he'd only considered the professional aspect of the assignment. It might be an unpleasant job, and she might not like the experience, but he had no doubts about Sara's ability to carry it out.

He hadn't considered the fact she might find it personally offensive.

Heading into the break room, he gave the male members of the team a disapproving look as they smothered their laughter. Grissom chose to ignore Catherine's exaggerated glance at the clock. Shift would start in a few minutes, but Sara wasn't here.

Usually, she arrived well before the start of shift, not only to chat with co-workers on the swing shift, but also to leisurely review her notes from the previous day's work over a cup of coffee. Trying not to read any significance to her not being early, Grissom walked over to the coffee pot, sniffing the contents cautiously. Deciding the sludge hadn't reached toxic levels yet, he poured a cup, turning around in time to see Greg go bounding by the doorway.

Looking out the window, he saw the lab tech reaching out to help Sara. Together, the two carried in a large box, similar to the ones they used to store evidence. On top of it were stacked books and rolls of paper. Perched on the very top was a large brown bag.

"Thanks, Greg," she said as they set their burden down on the break room table.

"Present from Gustav-Maximilian?" Nick teased, snickering in his coffee until he caught Grissom's look.

"Oh, man, you know that guy got picked on when he was a kid," Warrick said.

"You'd think so," Sara said.

"Everything go okay?"

"Yeah, Cath. Max rocks."

Grissom turned to look at Catherine, his eyebrow going up slightly. He knew Sara could handle the project, and her perky mood wasn't lost on her friends. Catherine merely shrugged.

"That's quite a load ya got there, Sar. You must have had to do a lot of flirting."

"Hey, Nicky, you know how seriously I take my job. I threw myself into it," she said, enjoying the startled looks from around the table. She ignored Grissom's troubled stare.

"So he's a big guy?" Nick asked, unable to resist the urge to tease again.

Sara looked up, turning to stare at him. Instead of a rebuke, a sly smile broke out. Leaning back, she crossed her arms and gave her friend a wicked look.

"Yeah, actually he is. But size isn't everything," she said with a wink, waiting until her meaning registered. "You should know that."

"Hey!"

Catherine darted her eyes around the room. Everyone had turned to stare at Sara. Grissom in particular seemed shocked, his mug of coffee held halfway to his mouth. They had expected her to be upset, not making bawdy jokes about the guy. And how exactly did she know anything to make jokes about?

"I take it he was able to help?" she asked, trying to bring the conversation back to the case.

"Oh, yeah," Sara said in open admiration, causing Grissom to give her quizzical look. "Wait until you see this. It's wild."

"You've been busy," Catherine said, looking at the stack of materials the younger woman was taking out of the box.

"Yeah, I was there all day. Just left," she said. When she noticed the others looking at her damp hair, she shrugged. "I took a shower after I woke up."

The rest of the team exchanged another shocked look as the implications of her statement sank in. Seeing the open mouths around the room, she rolled her eyes and picked up a book. "Some of the reference materials Max lent me are in other languages. I took a nap while he translated some passages that might be relevant."

"So, what's in the bag?" Greg asked.

"Lunch. I think. Max packed it for me while I got dressed."

"Okay," Grissom said, holding up a hand. He didn't like the direction this conversation was heading, but knew he was in no position to comment on it.

"Sara, grab some coffee, then set that up in the Layout Room. Catherine, check with Al and Tox, see what they have. Warrick, go with Vega. He's getting a warrant for your shooting case. Nick, hit-and-run on the Strip. When you get done with those, get back here and get to work on this murder. Greg, get to work on those blood samples. I'll go check with Jacqui."

Catherine watched as Grissom left the room, then she turned to give Sara a pointed look, but the younger woman merely smiled innocently at her. Gathering up her supplies, Sara headed to the Layout Room, her upbeat mood evident in her stride.

"Tell me you have something."

"Sorry, Jim," Grissom said as he exited the Print Lab. "We lifted 17 sets of prints from the victim's apartment. One set was the victim's, one belonged to the apartment manager, and there haven't been any hits on the others yet."

"Probably clients. Guy ran his insurance business from the apartment. People in and out all the time, according to the neighbors. No one remembers any vampires, though."

"Our Count would have been taking a to-go container with him," Grissom said.

"Yeah. We're checking old cases. No hits on math equations, whether in blood or not. We're getting a list of bodies missing half their blood."

"But if the body had been exposed to the elements or to predators…"

"There'd be no way to know if they had been drained," Brass said.

"Hey, guys," Catherine called. "Tox narrowed it down, so to speak. The warfarin used is a typical prescription strength, not rat poison. The bad news is that every hospital, pharmacy and nursing home in the country would have it. Doc says it was one large ingestion, about 30 hours before he died."

"I'll go see what I can find out about any warfarin heists," Brass said, nodding goodbye to the others.

Catherine followed Grissom to the DNA lab, where Greg handed over a printout.

"Ran the immunoassays series. Confirmed the two blood samples in the turkey baster match the crime scene. One came from the victim, the other matched the blood on the walls. Once the DNA results are done, I'll check CODIS and our internal database. See if the blood on the walls matches any of our old cases."

"Thanks," Catherine called, trailing behind Grissom, who was scratching his beard absentmindedly.

"So, somewhere there's a body missing up to half of its blood, and as far as we know, we know nothing about it. And someone has half of our victim's blood, saving it for his next scene," he said as he entered the Layout Room.

Both of them stopped once they were inside the room. On the walls, Sara had pinned up a three-by-four grid of large multi-colored printouts, each page corresponding to a wall in the apartment. On each page, the equations were broken down into sections, each in a different color.

"What the hell?" Catherine muttered as she scanned the papers.

"There was a pattern after all," Grissom added.

"Oh, yeah. There was nothing random about this," Sara said walking back to the grid. "Spirals. Of spirals. Each wall of equations can be broken down into sections. Starting with the north wall in each of the rooms, the top section of equations describes a type of spiral. The bedroom has a logarithmic spiral, the bathroom a parabolic spiral, the living room is the spiral of Archimedes," she said, pointing out each section.

"Now, move to the second section of equations on the east wall in another room. The equations are continued. Move to the third section of the south wall in the next room, and you get the next piece. Head back to the west wall in the first room, and the pattern continues."

"A spiraling spiral," Grissom said, examining the printouts in detail. "But I don't follow how he's getting from one section to the next."

"The killer skipped steps. You can't get directly from where he left off on one wall to the next section. Max wrote out all the missing steps," she said, pulling out a stack of printouts, and handing him a copy.

"So, it's not likely he did this off the top of his head," he said, flipping through the pages, still somewhat lost.

"Exactly. What's even weirder is the killer changed the equations into more complicated forms. Normally, a spiral can be described really simply using polar coordinates. But our guy didn't stick with those easy forms," she said, pulling out another thick stack of printouts for each of her colleagues. "Max wrote out the steps necessary to transform the equations from their original form to the form our killer used. It makes no logical sense. They are far more complicated that way."

"Man, I thought I knew trig identities," Catherine said, raising her eyebrows dramatically as she flipped through the pages.

"Yeah. Those aren't well known. They're like a type of mathematical trivia. There's no reason to use them. There are a lot easier trigonometric equations that would serve the same purpose."

"Or the killer's hiding a message within the message," Grissom said, setting down the printouts.

"Well, he knows his math."

"Maybe not," Sara added, pointing to a highlighted section of equations. "He's made mistakes. Some of those trig equations are undefined at certain values. The original forms wouldn't be. And there are other mistakes. Here. This should be the inverse cosecant of X squared. Instead, he's squared the inverse cosecant of X. Big difference."

"So, what do we have? Transcription errors?" Grissom wondered.

"That's what Max and I thought. The killer figured out his pattern in advance, wrote it down and then painted them on the walls at the scene. In the process, he copied some of it incorrectly, But some of the mistakes are real," she said, pointing out an example on one of the printouts. "That's totally wrong."

"He'd have the time to copy this. It would take hours for the blood to collect in the chest cavity before he sucked it out," Catherine pointed out, noticing Grissom's odd looks whenever Sara called the mathematician by his given name. "He's using 'X' for polar equations? I thought you stuck with r and theta for those?"

"Normally you do. But they're just variables. You could use anything you want as long as you know what you mean; it doesn't change the equations."

"What about the rest of the formulas?" he asked, pointing to the remaining colored sections.

"Other polar forms. Cardioids, limaçons, cissoids, lemniscates," she said, pausing when she saw the confused looks on their faces. Opening another folder, she pulled out additional papers. "You probably covered this in your second semester of calculus in college."

"It's been a few years since I was a freshman," Grissom said evenly, looking at the printouts.

"Yeah, and you've probably never seen them since. The names are based on what the graphs of the functions look like. Cardioids look like hearts, if you kind of turn them around. Lemniscates look like bows, limaçons look like snails, and the ivy-looking things are cissoids. We made printouts of what these all look like when they're graphed. There are also graphs of the mistakes, in case they weren't really mistakes."

"So what does it all mean?" the blonde asked.

"A guy with a polar fixation?" Sara ventured.

"That's any guy," Catherine sighed, ignoring Grissom's stare.

"He's spiraling out of control?"

"Interlocking spirals. DNA?"

"He's plain crazy?"

"We don't have enough to go on," Grissom said, giving Sara a sharp look at her last reply. "Anything else?"

"Yes. Max provided some reference materials on the trig identities used," she said, pointing out some of the books. "He also provided backgrounds on the different spirals and shapes the killer used. There's translations for those books," she said indicating some thick folders. "He's checking with some friends who are math historians. Seeing if there's any significance to these equations."

"Okay. Let me know when you hear anything. I'm going to go check on the bugs I pulled from the victim," he said.

"You did good, kiddo," Catherine said with a friendly smile.

"You should be thanking Max," she said with a shrug. "I'll be meeting him later. You should come along; you'd like him."

"Yeah," she said in disbelief. "You're not getting out of it."

Sara gave her a puzzled look before she started chuckling softly.

TBC