The Fractal Murders

by Mark Cohen

  “What can I do for you gentlemen?” he asked. He sat in one of the chairs. I went to the sofa. Scott walked back into the kitchen to make sure nobody else was in the house, then claimed a chair across the room to make sure Gauss/Tobias was between us.

  “We know you’re Thomas Tobias,” I said. He looked down for several seconds.

  “I can’t imagine who might have hired you to find me,” he finally said.

  “We weren’t hired to find you,” I said. “We’re looking into the deaths of some mathematicians, and your name came up.”

  “My name?” He seemed genuinely surprised.

  “Did you know Paul Fontaine?” I asked.

  “Of course,” he said.

  “Carolyn Chang?”

  “Sure,” he said, “she was at Nebraska.”

  “Donald Underwood?”

  “What’s this about?”

  “They’re dead,” I said.

  “All of them?”

  “Yes.”

  “Murdered?”

  “Two were murdered,” I said. “Underwood’s death might have been a suicide. We’re not sure.” I gave him time to process it.

  “They each specialized in fractal geometry, didn’t they?”

  “Yes.”

  “And you’re searching for people who worked with all three of them?”

  “Worked or studied with them,” I said.

  “How did you find me?” he asked.

  “It wasn’t that hard,” Scott said. A little hostile. I don’t think he’d figured it out yet.

  “You think I killed these people?” he said, incredulous.

  “We have to check every lead,” I said. “Just give us a rational explanation for the disappearing act. If it’s not related to our case, we’re out of here.”

  “I have AIDS,” he said. He crossed his left leg over his right.

  “I’m sorry,” I said. There were purple lesions on his skin. I saw Scott’s hostility fade.

  “I’d been HIV positive for some time, but when the symptoms began to appear, I decided to live my dream. Buy a place in the Southwest, plant a garden.”

  “You couldn’t do that as Thomas Tobias?”

  “I’m gay,” he said. “Mother’s a devout Christian. She never accepted my lifestyle. I don’t have any other relatives. I didn’t want to spend my last years listening to her sell me religion. I just wanted to start over. It’s better this way.” I didn’t know what to say. “Don’t feel sorry for me,” he said. “My medications are working. I have friends here. It’s a good life.”

  I told him about our case and how we’d tracked him down. “You worked hard to find me,” he said. “I’m sorry it was all for naught.”

  “It wasn’t,” I said. “We’ve eliminated you as a suspect.” I stood and Scott followed suit.

  “Would you like to stay for lunch?” he asked. “I’m preparing vegetable soup and salad.”

  We’d already eaten, but he seemed to want our company, so we accepted his offer. Though New Mexico’s growing season is long, he’d constructed a small greenhouse behind his home to enable him to garden all year long. The soup and salad, he explained, were made with vegetables he’d started inside and transplanted a month ago. His pride in his vegetables was obvious.

  When we had finished, I asked him to keep what we’d told him to himself. He said he would and asked us not to reveal his whereabouts to his mother or anyone else he’d known. We gave our word and climbed into the truck for the long trip home.

  “What’s Heidegger got to say about that?” Scott asked.

  “I haven’t gotten that far,” I said.

  20

  THURSDAY MORNING IN Nederland. With Tobias no longer a suspect, I felt dejected. I’d been searching for a common thread and Tobias had been it.

  A few tufts of white cloud punctuated an otherwise blue sky. Perfect running weather. I took Wheat for a morning run, hoping the exercise would clear my mind. Just to be different, I headed up Caribou Road. It will take you to the Continental Divide, if that’s what you want, but the first few miles are easy. The dirt road rises gradually and follows a meandering creek.

  Two miles up the road is a pasture where donkeys graze. That’s usually where I turn around. I picked up my pace as we approached it. One of the reasons I seldom run on a track is because it’s too easy to quit. If you start at home and run three miles out a country road, quitting isn’t an option. You still have to get home. It was like that with my case. Despite feeling the investigation had stalled, there were at least three reasons not to quit. First, I felt in my gut that the deaths were related. Second, I had nothing else to do. Third, I wanted to know Jayne Smyers better and quitting didn’t seem likely to help my cause. I had to finish what I’d started.

  Wheat reached the pasture before me. He ran under the barbed-wire fence, but stopped short of the big animals. “Hello, donkeys,” I said as I did a wide turn and began running back toward Colorado 119. I’ve always liked donkeys and mules. They’re reputed to be stubborn. Maybe I see a bit of myself in them.

  Running downhill required little effort, and I found myself thinking about something I’d been toying with since reading the article sent by Monica. Fontaine had enjoyed following the stock market. Carolyn Chang had been keeping company with an economist. Underwood had written about neural networks—computer programs apparently useful in predicting market behavior. It wasn’t much, but it was a thread. And it was all I had.

  So I spent the rest of the morning doing some on-line research. By eleven-thirty I knew Fontaine’s portfolio of nineteen stocks had beaten the S&P 500 every quarter for the past three years. Sometimes by as much as 7 percent. His portfolio had outperformed some of the hottest funds in the world.

  I made some phone calls unrelated to the case, then closed my eyes to consider my next move. I needed more information about the use of fractal geometry in economic forecasting. I started my truck, slipped an old Bob Dylan tape into the cassette deck, and headed to Boulder.

  I parked at a meter by the University Memorial Center and walked north to the Norlin Library. Norlin is the main library at the University of Colorado. Joy and I used to seek refuge there when we wanted to escape the law school crowd. It had been years since I’d been inside, but the only difference was that computers had replaced the card catalog. They were everywhere.

  The semester had ended, but there were a half dozen students working quietly in the reference section. I claimed the nearest vacant terminal and began reading the instructions. Articles in magazines or academic journals were indexed in sixteen different databases. I could search by subject, title, or author. I was familiar with MathSciNet, but there was also a business and economics index, and that’s the one I chose.

  I first searched for articles written by Fontaine, Chang, or Underwood. Nothing.

  Then I typed “fractals.” The monitor indicated my search had found thirty-seven articles. I scrolled through the abstracts just to get a sense of the literature. Many sounded highly technical, but I had expected that. What I hadn’t expected was an article by Dale Hawkins. His contribution, “Weather and the Fractal Structure of Crop Markets,” had been published shortly after he would have met Carolyn Chang.

  Sadly, technology had not yet advanced to the point where I could press a button and print all thirty-seven articles. I could print the list, but I’d have to find the articles in the stacks and copy them individually. In for a dime, in for a dollar. I hit Print.

  A dot-matrix printer two tables to my left suddenly came alive. It was noisy and painfully slow, a fact that caused everyone present to look up to determine who had printed thirty-seven abstracts. I retrieved the printout and was about to head upstairs to the periodicals when I decided to conduct one more search. Might as well see what else Hawkins had contributed to the literature.

  It took only a minute to learn that he had authored six other articles, all prior to meeting Carolyn Chang. From the abstracts, it didn’t appear they had anything
to do with fractals. I hit Print once more, took the printout, purchased a twenty-dollar copy card, and began the tedious task of finding and copying the articles. I eliminated some that appeared unlikely to further my education. “The Fractal Nature of Urban Japanese Population Patterns” seemed like one I could safely skip.

  It was past four when I finished copying. I left the library and headed in the direction of the math building. Clouds were building over the mountains and afternoon thunderstorms seemed a distinct possibility.

  I hadn’t read the articles—I would do that at home—but I found it curious that Hawkins had written an article incorporating ideas from fractal geometry. Perhaps I hadn’t asked the right questions, but he had conveyed the impression he knew little about the subject: “I knew what her field of expertise was, and that she loved teaching, but beyond that I can’t tell you much.” He’d also insisted he and Carolyn had seldom talked about their work. Yet the only article he’d ever authored having anything to do with fractal geometry had been written after they’d started seeing each other.

  The door to Jayne’s office was closed and nobody responded when I knocked. I scribbled a note on my legal pad and slid it under the door.

  My truck was four miles up the mountain when the rain began. By the time I reached Boulder Falls, it was coming down so hard I had to pull off because I couldn’t see. I leaned against the driver’s door, swung my feet up, and treated myself to a brief nap.

  The rain subsided within a half hour. I got home, let the dogs out, and ordered a pizza from Backcountry. There are three pizza places in Nederland, but Backcountry is the only one that delivers. I could drive to any of them within three minutes, but sometimes it’s nice to have them come to you.

  I clicked on the TV just in time to catch the start of the CBS Evening News. Some say Dan Rather has a liberal bias, but I don’t see it. Besides, he’s an ex-marine, and that’s more than you can say for Tom Brokaw or Peter Jennings.

  A dreadlocked kid delivered the pizza just after six. I tipped him two dollars, ate three slices, and put the rest in the refrigerator. The prospect of reading several dozen articles on economics didn’t excite me, so I channel-surfed till seven, then tuned to the Thursday Night Fights.

  The first bout was a four-round match between two lightweights of moderate talent. The second fight, another four-rounder, was somewhat more interesting, if only because the fighters were women. The main event started around eight-fifteen. It featured an up-and-coming heavyweight from Montreal against a journeyman from New York. Both black. The younger one was taller and in better condition, but kept his fists too close to his head in a peek-a-boo style, like Floyd Patterson used to do. It’s good to keep your hands up for protection, but you have to keep them out far enough that they offer some value as shock absorbers. They were in the fourth round when my phone rang. I hit Mute and picked up the cordless.

  “Pepper Keane,” I said.

  “Hello, Pepper Keane,” she said, “this is Jayne Smyers.”

  “Hi, Jayne.”

  “What are you doing?” she asked.

  “Just reading,” I said. It was a little early in the relationship to spring my passion for boxing on her. “How about you?”

  “Laundry.” I was absorbed in the fight and didn’t respond. “Are you still there?” she asked.

  “Yes,” I said. “I’m sorry.” I clicked off the TV.

  “I can call back.”

  “No, that’s fine.”

  “Anyhow, I got your note, so I thought I’d call. Do you have some news on the case?”

  “I do,” I said, “but that’s not why I asked you to call.”

  “Really?”

  “I wondered if you’d like to do something this weekend?”

  “What did you have in mind?” she asked.

  “I don’t know,” I said. “Maybe go hiking.”

  “Sounds like fun,” she said.

  After firming up our plans, I told her the disappointing news about Thomas Tobias and summarized the angle I was now pursuing.

  “That’s interesting,” she said. “I don’t know much about fractal geometry in the business world, but if you need help sorting through the literature, let me know.”

  “I will,” I promised. We talked for a few more minutes, then said good night, but when I clicked on the TV, the cable network had switched to stock-car racing—a sport in which men compete to determine who can execute a specified number of left turns in the least amount of time. I still don’t know who won the fight.

  21

  WHERE’S AURORA?” GILBERT asked.

  “It’s a suburb of Denver,” I said. “Why?” I had just finished lifting weights in my basement.

  “The gun we found traces to a pawnshop there.”

  “I thought the serial number had been filed off.”

  “It was,” he said, “but whoever did it should’ve done a better job. The forensic people at the state patrol did some work on it, used solvent to clean it, then ran it through some kind of high-tech X-ray machine. They’re certain they’ve got the last four digits.”

  “Fantastic,” I said.

  “Ever hear of Saul’s Pawnshop?”

  “No, but I’m guessing it’s on Colfax Avenue.”

  “Gold star,” he said.

  “There’s a dozen of them on that stretch,” I explained. “It’s not a good neighborhood.”

  “Old Saul did his best to keep it that way,” he said. “He sold a five-shot Taurus to a career hoodlum named Delbert Gaffney a few years ago.”

  “The last four digits match?”

  “Yup.”

  “Any leads on Gaffney?”

  “He ain’t the guy. The feds caught him selling crack last July, and he’s been in custody ever since.” Fontaine had been murdered in September.

  “Did you get the FBI reports?”

  “Yeah, but he was unarmed when they busted him, so nobody asked him any questions about guns.”

  “They search his house?”

  “No weapons there either.”

  “That’s a first,” I said. I’d prosecuted dozens of drug cases, and I’d almost always been able to add two or three firearms charges. “Where is he now?” I asked.

  “He’s in the federal correctional institution in Florence, Colorado, wherever that is.”

  “It’s west of Pueblo,” I said.

  “I figure he sold the gun on the street or loaned it to one of his gangsta friends before he got busted.”

  “Maybe some crackhead stole it from him,” I said.

  “Maybe,” he admitted. “Why don’t you take a drive down there and ask him?”

  “The city won’t pay for you to fly down and question him?”

  “Kid’s twenty-three and doing a mandatory twenty. Why’s he gonna talk to me?”

  “Why’s he gonna talk to me?”

  “You’re a lawyer,” he said. “They love talking to lawyers. It’ll take him a full day just to tell you all the mistakes his lawyer made.” I laughed.

  “That’s a maximum-security facility,” I said, “but I’ll give it a try.” It was eleven-thirty on a Friday morning.

  I spent the afternoon eating almonds and reading the articles I’d copied on Thursday. The crossing of economic and mathematical theory had yielded a bumper crop of jargon, but the upshot of everything I’d read was that the random walk theory popular in the sixties and seventies had been discredited. Whereas many economists had once considered market trends no more predictable than the outcome of a coin toss, the development of fractal mathematics had convinced most that market behavior is not random in a mathematical sense.

  This was good news for Wall Street. Taken to its logical extreme, the random walk theory suggested that a blindfolded lemur throwing darts at the financial pages could select a portfolio that would perform as well as one selected by experts. Consequently, Wall Street had never accepted it. Dismissing the random walk theory as nonsense spouted only by out-of-touch academics, Wall Stree
t had long been divided into two camps: the fundamental analysts and the technical analysts.

  Fundamental analysts seek to determine the true value of a stock by examining earnings per share, book value, price/earnings ratio, and so forth. Technical analysts, on the other hand, focus on the psychology of the market. They assume the price of a stock is determined by the expectations of all buyers and sellers. Relying on past price movements, they try to predict what a given stock (or the market as a whole) will do in the future. Because they chart the past in order to predict the future, technical analysts are also called chartists.

  To fundamental analysts, the belief that you can predict stock trends or market behavior by studying the past is absurd. For this reason, they had long viewed chartists with disdain. To them, the difference between the two schools was like the difference between astronomy and astrology.

  But the discovery and growth of fractal geometry had bestowed new respectability on chartists. Using theories borrowed from fractal mathematics, empirical studies had confirmed that economic markets possess a type of memory. When volatility is low, the market follows trends: It rises or falls for longer than random. When volatility is high, trends persist for shorter than random. In either situation, the market is partly predictable. Economic markets do not conform to the Gaussian distribution.

  This came as no surprise to chartists. A market consists of people second-guessing one another. The more people share a belief, the more the belief is likely to come true. Suppose a stock has traded at $50.00 to $60.00 a share for several months. A chartist would call these the stock’s resistance levels. If the price suddenly rises above $60.00, the chartist concludes something has happened to change investors’ perceptions. The breakthrough convinces investors that other investors believe the stock is undervalued. As a result, investors buy the stock and drive up the price. It’s a self-fulfilling prophecy.

  It was interesting reading, but for my purposes the most interesting aspect of it was that fractal mathematics had resulted in the computerization of Wall Street. Today’s brokerage houses and fund managers are constantly processing vast amounts of data and searching for recurring patterns. And that led me back to the article Monica had sent. Neural networks. Underwood.