The Fractal Murders Part 4

"That's rea.s.suring," she said. Her sarcasm wasn't directed at me; she was just too young and good-hearted to accept the fact that any person can kill any other person at just about any time.

"Look at it this way," I said. "Other than a brief friendship with Carolyn Chang a few years ago, nothing connects her to any of the three victims. And if we a.s.sume Underwood's death was a suicide, the last murder took place more than four months ago." She said nothing. "If I find something connecting one death to another, I'll be in a better position to know whether anyone else might be in danger. You have to find the connection to understand the motive. It's a process of gathering information."

"I have an idea about that," she said. She flung her head to one side to prevent her hair from encroaching on her face. It was a s.e.xy little move.

"Let's hear it."

"We use the Internet. We send e-mails to mathematicians at universities around the country outlining what we know. I'm sure we'd receive lots of useful information."

"We probably would," I said, "but we can't do it." I leaned forward and waited until we had good eye contact. "No one else is to know Professor Smyers reported this to the authorities or that she hired me to look into it."

"But," she insisted, "that would be the most efficient way for us to gather information." Her continued use of the plural indicated she considered herself part of my team. I laced my fingers together behind my head.

"Have you ever heard of Heisenberg's uncertainty principle?" I asked.

"Of course." She did a poor job of hiding her surprise that I'd heard of it, but I'd learned some physics in my study of philosophy.

"What is your understanding of it?" I asked. She played along.

"It's a principle of quantum mechanics that holds that it is impossible to measure the position and velocity of an object at exactly the same time because the very attempt to do so affects both the position and velocity of the object."

"That's exactly right," I said, "and that's the situation we've got here. If these deaths are related, someone took great pains to make them appear unrelated. That suggests what in my Marine Corps days we used to call a 'highly motivated individual'-someone who would not hesitate to kill again if he felt threatened. We don't know who the killer is or what his motive was, but the very attempt to find out might alert him to my investigation and put all of us in danger. It's vital that this remains a secret. I want your word on that." I was coming on strong, but I believed what I was saying and wanted her to.

"But," she pleaded, "we'd just be contacting other math people."

"Mary Pat," I said, "if these deaths are related, the killer almost certainly has some link to the mathematical community." She broke eye contact and gazed at the floor.

"I guess you're right," she admitted. "Sometimes I dig in before I've considered all the arguments."

"Me too," I said. "It's a hard habit to break." I unclasped my hands and switched legs so that my left crossed over my right. There was a brief pause, but I enjoyed talking with her and didn't want to leave. "So," I began, "you're working on your master's?"

"I just have to finish my thesis. I hope to earn my doctorate here. I'm from Milwaukee, but I love Boulder."

"Not as nice as Ennis, Montana," I said, "but better than East Saint Louis." That made her laugh.

"You were in the marines?" she asked.

"I was a lawyer," I said.

"Like the lady on JAG?"

"Something like that," I said.

"My dad was a marine. I was born at Camp-" She cut herself off when Finn appeared in the doorway. He had evidently just returned from a long run in the rain. He wore a white tank top, maroon nylon running shorts, and top-of-the-line running shoes, the kind with translucent green gel beneath the heel. He was soaking wet.

"Jayne around?" he asked as he polished his gla.s.ses with a small white cloth.

"No," said Mary Pat, "but she should be back any minute."

He looked at me, put his gla.s.ses in place, then stepped forward and said, "I don't believe we've been introduced. I'm Stephen Finn." I stood up and Mary Pat followed suit.

"Stephen competes in triathlons," Mary Pat explained.

"Pepper Keane," I said. We shook hands. He gave me his triathlete grip, so I gave him my I-can-dead-lift-535-pounds grip.

"What do you do, Mr. Keane?" He tried to ask it in an offhand manner, but seeing me in my client's office a second time had him burning with curiosity.

"Mr. Keane is a private investigator," Mary Pat said. I was ready to thrash her, but she added, "He's helping Professor Smyers design a workshop on personal safety for women at the shelter."

"A private investigator? Fascinating." He tried to sound interested, but his tone was slightly condescending. He felt it wasn't an occupation for the educated. "Anyhow," he continued as he turned to Mary Pat, "I just stopped by to chat with Jayne. Will you tell her I was looking for her?"

"I will," she promised.

"Nice to have met you, Mr. Keane."

"Same here," I said. He departed and we returned to our chairs. "What does he teach?" I asked.

"Freshman stuff, mostly. Baby calculus, things like that."

"Low man on the totem pole?"

"Yes."

"How long has he been here?"

"Two years, just like me."

"He looks so young."

"He graduated from Harvard at twenty."

"Big deal," I said, "I started kindergarten before I turned five." She laughed, then gave her a.s.sessment of Finn.

"He can be a bit pretentious, I'll admit, but he's not a bad guy, really. His students love him. His biggest fault is that he tries too hard. He's up for tenure next year and he's obsessed with it." I was about to ask the nature of his relationship with my client, but she walked in before I got the chance. She was still as tall as I was and she wasn't wearing heels. Pink lipstick, powder blue sundress, white sandals. And it wasn't even Flag Day.

"Did I miss a good joke?" she asked. I stood.

"No," said Mary Pat, still grinning as she came out of her chair, "Mr. Keane was just recounting his academic accomplishments."

"Well, Mary Pat," she said in mock seriousness, "it's not polite to laugh. It destroys the student's motivation to learn." We shared a smile as she walked around her desk and began making a pot of coffee. "Did Mary Pat give you everything you need?"

"She sure did," I said.

"If you need anything else, Mr. Keane," said Mary Pat as she neared the door, "just let me know. You can leave a message with the department secretary."

"I will. Thanks again."

"Where are you off to?" my client asked.

"I have a date with the supercomputer," Mary Pat replied. "Oh, I almost forgot, I'm supposed to tell you Stephen stopped by 'to chat.'" They exchanged a knowing look and Mary Pat departed. My client stood at the window behind her desk and stared out at the rain. She had a nice view of the mountains and I had a nice view of her.

"I like the rain," she said, "but if it doesn't let up, I'm going to have to make a mad dash for my car."

"I have an old poncho in my truck, but camouflage might clash with your outfit."

"Oh," she said, "that's sweet, but it's not raining that hard. That's what I get for relying on the weatherman." I had intended it as a joke, more or less, but that's what I get for trying to be funny. She sat down behind her desk and poured some coffee into her coyote mug. "So," she said, "what do you think of Mary Pat?"

"Bright young lady."

"She's one of the most talented students we've ever had. She scored a nine-eighty on the Graduate Record Exam. That's as close to perfect as you get."

"She told me she wants to earn her doctorate here."

"I've encouraged her to keep her options open. She could go anywhere, but she likes Boulder."

"Does she want to specialize in fractal geometry?"

"That's an interesting question. One minute it seems to appeal to her, the next minute she's off on something else."

"If she can't make up her mind," I said, "there's always law school. That's what it's there for." She smiled to acknowledge my keen wit. Or maybe she was acknowledging the element of truth in my remark. Either way, I'd take it. She had a nice smile.

"You don't sound much better," she said. "You should see a doctor."

"It's on my list of things to do," I said. I don't trust doctors. I knew plenty of morons who had made it through law school, and my years defending navy doctors against malpractice claims had convinced me a similar percentage survived medical school. "Anyhow," I said, "it looks like I've got some reading to do, so I'd better be going. Thanks for having Mary Pat get right on this."

"You're more than welcome. If you need anything else, just call."

"By the way," I said, "when Professor Finn stopped by, Mary Pat told him I'm helping you prepare a workshop on personal safety for some women in a shelter. Which one of you came up with that?"

She smiled. "She did, but you have to admit it's a good lie."

"It's perfect," I said. "I am an investigator, and you evidently work with one of the shelters." I pointed to the plaque honoring her service.

"I'm on the board of directors of the Boulder Women's Shelter."

"That's terrific. If you ever do want to put together a workshop, let me know. I've had some training in verbal de-escalation and personal safety techniques."

"Well," she said, "let's get through this first." I was almost out the door. "You know," she said, "when I first met with them, those FBI men asked me not to discuss this matter with anyone else. Even after the bureau closed the case, their boss encouraged me to be discreet-he wanted to protect the privacy of the families-so I haven't mentioned any of this to my colleagues, but at some point you may want to talk with Stephen."

"Why's that?"

"He taught at the University of Nebraska."

"When?"

"Just before joining the faculty here."

"Really? Did he know Carolyn Chang?"

"Yes, he learned of her death from a former colleague. He was quite upset."

"Small world," I said.

4.

I DROVE BACK UP THE MOUNTAIN with twenty-eight papers on various aspects of mathematics. I speculated as to the nature of the relationship between Finn and my client as Mick Jagger explained why he had been unable to obtain any satisfaction. It couldn't be too serious if she hadn't shared her suspicions about the three deaths with him.

Arriving home at three-thirty, I let the dogs out into the rain, then watched as they reentered and shook from side to side. Nothing like the smell of wet hounds to make a house a home. I headed to my office to study the collected works of the three victims.

My office is in my home. When I left the U.S. Attorneys' office to found Keane, Simms & Mercante, I purchased, among other things, a mahogany desk, a matching credenza, and an original Robert Batemen painting of an Alaskan brown bear. When I left the firm two years ago, I took it all with me. I had enjoyed the tremendous view of downtown Denver, the plush offices, and the other trappings of success, but I don't miss paying four thousand dollars a month in overhead. I have everything I need right here, including a great view of the Continental Divide. And you can bet I take the home office deduction.

Most of the articles had been published in professional journals, but a few had appeared in publications intended for a broader audience, magazines like Scientific American and Omni. Surprisingly, Fontaine, who had spent most of his career at a college I'd never heard of, had written fifteen of them. Of the remaining thirteen, Carolyn Chang had published seven and Underwood six.

Twenty of the articles were highly theoretical. Of those, Fontaine had written twelve, but several had been early in his career and had nothing to do with fractals. Five of Underwood's six fell into the theoretical category, and the remaining three had been auth.o.r.ed by Carolyn Chang. The terminology was foreign to me. I had never heard of Julia sets, parameter s.p.a.ces, or the escape time algorithm, and I guessed I'd need two years of college mathematics to have a prayer of understanding them. Except for my inability to comprehend them, I was unable to detect a common thread in the theoretical writings of the three victims.

I stood to stretch and became hypnotized by the rain. The clouds hung low, obscuring the tops of the snow-capped peaks in the distance. Frightened by the thunder, Buck trotted into my office and sought refuge beneath my desk. I gazed out the window until a pa.s.sing squirrel on an electrical line turned and gave me a quizzical look. We stared at each other for a good twenty seconds, then he scampered on down the line and I went back to work.

The remaining eight papers were more in the realm of applied mathematics; each had something to do with how fractal geometry was being used in the real world. Of those eight, Fontaine had written three, Carolyn Chang four, and Underwood one.

Fontaine's first paper was a dull narrative about the applications of fractal geometry in electrical engineering, but the other two fascinated me. They were nontechnical discussions of fractal image compression, a technique that makes it possible to store tremendous amounts of data in a small amount of s.p.a.ce. An English scientist named Barnsley had developed the concept while searching for more efficient ways to send satellite images back to earth.

Barnsley believed that images of natural objects could be broken down into a small number of component parts by finding similarities among shapes. By digitally coding these shapes instead of using the conventional bit-mapping technique, a color image that would normally consume 1.4 megabytes could be described using just 25 kilobytes. Fontaine claimed this was what had enabled Microsoft to put an encyclopedia with six thousand color photographs on a single CD-ROM.

The technique had also been used to create backgrounds for video games and movies. Suppose you wanted to create an image of the lunar surface. To reconstruct the moon as it actually is, wrote Fontaine, would take the combined memory of ten thousand home computers. But fractals offer another way. Though craters vary in size, their basic shape remains the same. All you need to do is describe that shape to a computer and provide a formula (an algorithm) that tells it to reproduce that shape on varying scales until it has generated something that resembles the moon. To make sure no two craters are identical, you throw a few random numbers into the formula. In effect, the computer creates a "fractal forgery" that mimics the lunar surface without worrying about precise details. I didn't fully understand the mechanics of it, but I got the idea.

I stretched again, caught a glimpse of two green hummingbirds as they darted from one of the feeders I had filled with red nectar, then turned my attention to the writings of Carolyn Chang. The four nontheoretical papers she'd auth.o.r.ed were exceptionally well written; she'd had a knack for presenting complex concepts in plain English. Three of them addressed the growing use of fractal geometry in medicine. The articles demonstrated how fractal patterns obtained from medical images were being used to identify and cla.s.sify many types of disease. As an example, she cited several studies that had found correlations between the fractal dimension of heart rhythms and the presence of disease. She predicted fractals would play an increasingly important role in automating the science of medical diagnosis.

Carolyn's other piece intrigued me because it seemed out of place in a professional journal. It was an essay in which she had expressed the view that fractals could change the way artists and musicians portrayed the world. "Fractal geometry," she wrote, "offers a new way of looking at s.p.a.ce and form. Just as the discovery of geometric perspective transformed the way Renaissance painters represented depth on a flat surface, fractals may free artists to portray natural objects as they truly are rather than within the confines of Euclidean concepts of dimension."

Underwood had made one attempt to write for a broader audience, and that had been one too many. He had not possessed Carolyn Chang's skill with the written word. His sentences were long and flowed in no logical sequence. His topic was neural networks, computer programs capable of recognizing fractal patterns. Because of their ability to identify patterns, such programs were valuable tools in efforts to predict the future. According to Underwood, they had been employed in fields ranging from meteorology to finance.

By now it was past seven o'clock. It had taken nearly four hours to review all twenty-eight papers, but I hadn't discerned any pattern or common theme in the writings of the three victims. I pushed the stack to one side, fed the dogs, and started to boil water for spaghetti. Then I placed another call to the cop in Walla Walla. It was an hour earlier on the West Coast. "I think the lieutenant is still at the hospital," said the female dispatcher, "but I can take a message." I left my name again and called it a day.

Surveying my stacks of CDs, I selected a collection of old Jack Guthrie tunes. The cousin of Woody and a distant relative of Arlo, Jack had died in 1948, but I just love those old western songs. Jack Guthrie, Patsy Montana, Jim Silvers, and anyone else who can yodel.

I opened a two-pound package of spinach spaghetti, broke the thin strands in half, and placed the entire contents in the rapidly warming water. I chopped up an onion, some mushrooms, and a garlic clove, then sauteed the mixture in a Teflon pan. No hamburger because I've been flirting with vegetarianism for a few years, though I'm not real consistent about it.

I opened a jar of marinara sauce, added some ingredients of my own, including red wine, cinnamon, and clove, and began to heat it. I wouldn't eat it all, but I could save what I didn't eat for another day. That done, I deposited myself in my recliner and clicked on CNN. Wheat jumped up and sat on my lap. Ten minutes later the phone rang. It was the cop from Walla Walla, Lieutenant Gilbert. I picked up the cordless. "Thanks for returning my call," I said as I lowered the volume on the TV.

"No problem, what can I do for you?" He sounded like a regular guy. Maybe a few years older than me.

"I'm a private investigator. I'd like to talk about Paul Fontaine."