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At the top, we waited. And waited. Machu Picchu is cloudy in the morning, and we could only catch glimpses of the city below between the mist. John pointed to the far end of the site, where groups of Inca Trail hikers who’d awakened at four to meet the sunrise at the Sun Gate—virtually every tour company uses this supposedly magical moment as a selling point—were waiting impatiently for their first glimpse of the city.

“The Sun Gate at sunrise is a complete waste of time,” John said as we looked south. “The sun doesn’t actually arrive at Machu Picchu at the sunrise. And when it does rise everything’s covered in mist. All my clients used to insist on doing it because they’d been brainwashed. And then everyone always used to say, ‘That’s it? I wish I’d slept in.’”

Directly above us, at the very top of Huayna Picchu, a gaggle of college kids sprawled across a pile of enormous boulders, arguing about who had made it up the fastest and napping away hangovers. Most of the bars in Aguas Calientes, I’d noticed, offered a four-for-one pisco sour special. John prowled the peak, taking GPS measurements as I chatted with a mother and son from Düsseldorf. They thought it was absolutely vital that I know that the Hellmann’s mayonnaise in Europe is inferior to the stuff we can buy in the States. “Ours is more bitter,” the mother told me, then lowered her head as if she’d unburdened a dark secret.

I should have asked them if there was a twelve-syllable German word meaning “even more sublime.” When the fog finally lifted and we could see Machu Picchu stretching out below, the chatter atop Huayna Picchu stopped. From this vantage point it seemed entirely credible that the city had been lost to the outside world for centuries, and that more wonders might be hiding in the cloud forest. Evidently the Incas were fond of the view from Huayna Picchu, too, because several temples and a large usnu had been built near the top.

“You’ll notice that the temple here looks straight onto Llactapata,” John said, pointing west with his right hand. His left was pointed south, so that his arms formed an L shape. “The Intihuatana is due south of here, and Salcantay”—a twenty-thousand-foot peak and one of the Incas’ two most sacred apus—“is due south of the Intihuatana.” I had to take his word for it. Due to low clouds and smog from the seasonal fires, I hadn’t caught a single glimpse of Salcantay. On a map, a line connecting the two peaks and the carved rock could be drawn with a ruler. The Incas, leaving nothing to chance, had carved a wedge of rock on the peak of Huayna Picchu that aimed like a compass point due south directly at a wedge-shaped hollow carved into the base of the Intihuatana, which in turn pointed at Salcantay.

The anthropologist Johan Reinhard has observed that the Intihuatana stone may actually be an abstract rendering of Huayna Picchu mountain. If viewed from the correct angle, he says, the shadows cast by the sharp edges of the Intihuatana mirror those that move across the face of Huayna Picchu. Thus the Intihuatana may have been several things at once: a link between two sacred peaks; a sculptural homage to one of those peaks; and a sundial, though not one that marked the hour according to the angle of the shadow it cast. Rather, in the age-old Andean tradition, the time of day could be assessed by reading the shadows on a mountain’s face (or a smaller, carved model of the mountain’s face)—just as my friend Nati had learned to do in her village as a girl. There were any number of buildings at Machu Picchu that were designed with such multiple uses in mind. And John and I hadn’t even discussed how certain buildings were also astronomical observatories by night.

John kept pointing out alignments, but when I tried to imagine all the solstices and trails intersecting with Machu Picchu, the only picture I could conjure in my head was a web of lines intersecting like a gigantic cat’s cradle. I felt an overwhelming desire to sit.

By early afternoon, John and I were back down at the main site, resting