Who invented zero?

This concept, so vital to modern math, was understood by the Maya, one of the first civilizations to use it. But that wasn’t the only innovation of indigenous peoples. Consider snow goggles. Or chocolate. Or suspension bridges. Or, in a way, sneakers.

“Native people did not invent Chuck Taylors,” said Duane Blue Spruce, project manager for the Smithsonian’s National Museum of the American Indian, referring to the classic athletic shoe. “But,” Mr. Blue Spruce added, “we are responsible for the chemical process of creating rubber.”

These discoveries and many more form the heart of the imagiNATIONS Activity Center, a permanent bilingual (English and Spanish) installation for youth and families that opened on Thursday in the museum’s New York City branch. Part of an approximately $9 million renovation, whose cost includes vast new retail space as well as heating and ventilation, the 4,500-square-foot center is the Lower Manhattan museum’s largest design undertaking since its founding in 1994.

Although the museum already has an imagiNATIONS center, for 4- to 8-year-olds, in its Washington building, this one is for students in grades 4 through 12. It also has an entirely new direction.