Part 1 (2/2)
”No.”
”A fractal,” she said, ”is a type of geometric shape.” She paused. ”I don't know quite how to explain it to you.” She tilted her head slightly, paused again, then said, ”Picture a coastline.”
”Okay.” I didn't know much about geometry, but I'd been a marine officer for three years and I knew about coastlines.
”If we take a small section of that coastline, we can use a straight line to represent it on a map. But if we look closely at that section, we will see that it is made up of many small inlets and peninsulas, right?”
”Sure, and each inlet and peninsula has its own smaller bays and headlands.”
”Yes,” she said, ”that's exactly right.” She sipped her coffee. ”And if we continue to look at smaller and smaller sections of the coastline, we'll find that this pattern is always present.”
”Right down to the last grain of sand.”
”Yes. That's the interesting thing about fractal objects: Their pattern remains more or less the same no matter how closely you examine them.”
”So a fractal is just a shape with a random pattern?” I took the white handkerchief from my pants pocket, blew my nose, folded it gently, and placed it back in my trousers.
”Not a random pattern,” she said, ”an irregular pattern. Strictly speaking, there's no such thing as a random pattern. The two words are inconsistent. It's an oxymoron, like military intelligence.” I let that pa.s.s without comment, though my high and tight haircut should've suggested I had once served in uniform.
”You're saying the shape of a coastline is not random?”
”Not in a mathematical sense,” she said. ”Each point on a coastline is linked with the points next door. If it were truly random, one point would have no relations.h.i.+p to the next. Instead of gradual curves, you'd see lines going all over the place. One point might be up here, the next might be way down there.”
”Okay,” I said, ”I'll buy that.” I waited for her to continue, confident that sooner or later the reason for my presence would become apparent.
”Did you study geometry in high school?”
”Tenth grade.” I wondered what Mrs. Clagett was doing these days. Probably in the Aspen Siesta nursing home suffering recurring nightmares about McCutcheon and me.
”The problem with traditional geometry,” she continued, ”is that triangles, squares, and circles are abstract concepts. You can't use them to describe the shape of things like mountains, clouds, or trees.”
”Or a coastline.”
”Or a coastline,” she agreed. She was becoming more animated; she clearly enjoyed the subject. ”Traditional geometry-what we call Euclidean geometry-has to ignore the crinkles and swirls of the real world because they are irregular and can't be described by standard mathematical formulas. Then, about twenty years ago, a man named Mandelbrot invented something we call fractal geometry.”
”Fractal geometry,” I repeated. I sensed the lesson was nearing its conclusion.
”Mandelbrot realized that although many natural phenomena appear to be chaotic, there is frequently a hidden order in them. In fact, he called fractal geometry the geometry of nature.” Another sip of coffee. ”No two coastlines are identical, yet they all possess the same general shape, so there is a certain order there. Do you follow me?”
”I think so.”
”Fractal geometry provides a way to identify patterns where there appears to be disorder. It allows us to model and predict the behavior of complex systems. It's a language,” she said. ”Once you speak it, you can describe the shape of a coastline as precisely as an architect can describe a house.”
I doubted that. ”Give me an example,” I said.
”Certainly,” she replied, eager for the invitation. ”One of the tools we use to compare fractal objects is the concept of fractal dimension. For example, the coastline of Great Britain has a fractal dimension of approximately one point two-five, but the more rugged coastline of Norway has a fractal dimension of better than one point five-six.”
”I'll take your word for it.”
”I'm sorry.” She sighed. ”I've probably told you more than you need to know. I hope I haven't bored you.”
”No, it's interesting.” Not as interesting as the way her delicate bra straps traversed her bony shoulders, but interesting nonetheless.
”This will all make sense in a minute. I promise.” She sipped her coffee, and I noticed a silver Navajo bracelet on her right arm. No wedding ring on either hand.
”Take your time,” I urged. Despite my strong preference that people get right to the point, experience had taught me that the best way to conduct an interview was to shut up and listen.
”As I said,” she continued, ”my specialty is fractal geometry.” I noted the Ph.D. from Harvard on the wall to my right. ”Last year I began working on a paper I intended to present at a conference this fall. It's publish or perish, you know.”
”So I've heard.”
”When I completed my draft, I wanted someone else to critique it.” She finished her coffee and set the mug to one side. ”The last thing you want to do is publish a paper that contains a flaw.”
”So you have your colleagues read it in advance to see if they can poke holes in it?”
”Yes, but my colleagues here wouldn't be much help. Fractal geometry is a rather narrow specialty, so I compiled a list of five of the most respected people in the field and attempted to contact them to see if they would be willing to critique it.” Her slender neck became visibly tense and I thought she might be having trouble breathing.
”Are you all right?” I asked. She took a deep breath and nodded affirmatively.
”Mr. Keane,” she continued, ”when I attempted to contact these people, I learned that two had been murdered and a third had committed suicide.”
”Over what span of time?”
”All within six months of each other,” she said. ”Do you know the odds against that?” It was a rhetorical question, but I had a hunch she could tell me the odds right down to the decimal point if she wanted to.
”And you want me to find out if these deaths were related?”
”Yes.”
”Did you report this to anyone?” I asked.
”I called the police.”
”And they said it wasn't their problem?”
”Yes, because none of the deaths had taken place in Boulder. They suggested I call the FBI.”
”Did you?”
”Yes.”
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