Ms. Chen examined several hundred arrangements over the next few weeks, and, she said, “there happened to be several that stood out as very dense.” Her best packing easily eclipsed what Dr. Conway and Dr. Torquato had found, with a packing density of almost 78 percent, surpassing spheres.

Image PUZZLING Researchers have been using Dungeons & Dragons dice to learn how to pack tetrahedrons. The record density recently hit 85.63 percent. Credit... Glotzer Laboratory/University of Michigan

“In fact, my adviser totally did not believe me,” Ms. Chen recalled.

After making physical models of tetrahedrons and demonstrating the packing patterns, she convinced Dr. Lagarias that her packings were as dense as she had said they were, and finally published her findings a year ago.

Meanwhile, Sharon C. Glotzer, a professor of chemical engineering also at the University of Michigan, was interested to see whether the tetrahedrons might line up as liquid crystals do. “We got into it, because we are trying to design new materials for the Air Force that have interesting optical properties,” she said.

Dr. Glotzer and her colleagues wrote a computer program that simulated the jostling of tetrahedrons and how they arranged themselves when pushed together. They found not liquid crystals but complex quasicrystal structures with patterns almost repeated yet not quite. “That is the most astonishing crazy thing,” Dr. Glotzer said.

Examining the quasicrystals, they did find a periodic structure that represented another leap in packing density: over 85 percent. Just as that finding was prepared for publication last month in the journal Nature, a group at Cornell, using a different search method, found yet another packing that was just as dense.

But while Dr. Glotzer’s structure was surprisingly complex  the repeat pattern consists of 82 tetrahedrons  the Cornell crystal was surprisingly simple, with just four. It is also puzzling to researchers why the tetrahedrons in Dr. Glotzer’s simulations tend to the complex quasicrystal structures if the best packing is actually a much simpler structure.

“That’s part of what’s so surprising about this,” said Dr. Cohn, of Microsoft Research. “Each of these packings feels very different.”