“When you think you’re not thinking about something, you probably are,” says Dr. Jung-Beeman, who has a doctorate in experimental psychology.

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Another theory is that typical approaches to problem-solving may decay or weaken during sleep, enabling the brain to switch to more innovative alternatives. A classic switching story, recounted in “A Popular History of American Invention” in 1924, involves Elias Howe’s invention of the automated sewing machine: after much frustration with his original model, which used a needle with an eye in the middle, Howe dreamed that he was being attacked by painted warriors brandishing spears with holes in the sharp end. He patented a new design based on the dream spears; by the time the patent expired in 1867, he had earned more than $2 million in royalties.

Spear-wielding savages make for compelling stories, but creative insights directly induced by dreams are rare. In general, people are unaware of sleep’s effects on their performance.

Dr. Ellenbogen’s research at Harvard indicates that if an incubation period includes sleep, people are 33 percent more likely to infer connections among distantly related ideas, and yet, as he puts it, these performance enhancements exist “completely beneath the radar screen.”

In other words, people are more creative after sleep, but they don’t know it.

This lack of awareness makes it hard to identify specific aha! insights that have been prompted by sleep.

“It’s more that sleep brings a change of approach,” explains Mark Holmes, an art director at Pixar Animation Studios who worked on the film “Wall-E.” “You can get tunnel vision when you’re hammering away at a problem. You keep going down this same path, again and again, just tweaking, making incremental changes at best. ” He continues: “Sleep erases that. It resets you. You wake up and realize  wait a minute!  there is another way to do this.”

Business attitudes toward sleep may be starting to shift. Claire Stapleton, a spokeswoman for Google, says “grassroots” interest in sleep led to an on-campus talk by Sara C. Mednick a napping expert. Google also installed EnergyPods, leather recliners with egglike hoods that block noise and light, for employees to take naps at work.