“I was pointing to the words on the page and trying to explain what a word was, but I’d watch him and realize that he didn’t have any idea what I was talking about,” he says. “This black-ink thing here is called a letter  I realized this was all very abstract.”

Mr. Marggraff likes to go to bed with one or more problems on his mind. “Typically, I’ll fall asleep chewing on it and then I’ll wake up at 4 in the morning with some sort of solution,” he says.

That’s a common theme in innovation, according to Mihaly Csikszentmihalyi, a psychologist at the Claremont Graduate University in California. “Cognitive accounts of what happens during incubation assume that some kind of information processing keeps going on even when we are not aware of it, even while we are asleep,” he writes in “Creativity: Flow and the Psychology of Discovery and Invention.”

This time, Mr. Marggraff awoke at 4 in the morning determined to “flatten out” the globe so he could use the Atlasphere’s near-touch technology on a single page and, ultimately, within a specially designed book to help children learn how to read. Though some would call this an epiphany, it took years of trial and error to make the LeapPad a reality.

“There’s an aha moment followed by a ton of work to figure out what it is that’s actually going to work,” agrees Douglas K. van Duyne, co-founder of Naviscent, a Web usability consulting firm. “It goes back to that old saw that invention is 1 percent inspiration and 99 percent perspiration. The idea of epiphany is a dreamer’s paradise where people want to believe that things are easier than they are. It takes a huge amount of determination and effort to follow through.”

Businesses want to believe that a brilliant mind or a brilliant idea can make or break their innovation efforts, Mr. Berkun says. The myth of epiphany has a long history because it’s appealing to believe that there is a short, simple reason that things happen. The myth has staying power because there is a tiny core of truth within it.

“But as soon as you dig into what happened five minutes before that magic moment, or a day, or a week, or a month,” he says, “you realize that there is a much more complicated story in the background."