Cataloging these variations is an important first step, said Dr. Niedenthal, but it can’t deliver an answer to the enigma of smiles. “People like to make dictionaries of the facial muscles to make a particular gesture, but there’s no depth to that approach,” she said.

Some researchers have tried to move deeper, to understand the states of mind that produce smiles. We think of them as signifying happiness, and indeed, researchers do find that the more intensely people contract their zygomaticus major muscles, the happier they say they feel. But this is far from an iron law. The same muscles sometimes contract when people are feeling sadness or disgust, for example.

The link between feelings and faces is even more mysterious. Why should any feeling cause us to curl up our mouths, after all? This is a question that Darwin pondered for years. An important clue, he said, is found in the faces of apes, which draw up their mouths as well. These expressions, Darwin argued, were also smiles. In other words, Mona Lisa inherited her endlessly intriguing smile from the grinning common ancestor she shared with chimpanzees.

Primatologists have been able to sort smiles into a few categories, and Dr. Niedenthal thinks that human smiles should be classified in the same way. Chimpanzees sometimes smile from pleasure, as when baby chimps play with each other. But chimpanzees also smile when they’re trying to strengthen a social bond with another chimpanzee.

Dr. Niedenthal thinks that some human smiles fall into these categories as well. What’s more, they may be distinguished by certain expressions. An embarrassed smile is often accompanied by a lowered chin, for example, while a smile of greeting often comes with raised eyebrows.