In “Tim’s Vermeer,” Tim Jenison dives into the theory that Johannes Vermeer used a camera obscura to create such paintings as “The Music Lesson” in the 17th century. (Tim Jenison)

What if Johannes Vermeer were a machine?

That’s the question posed by Tim Jenison in “Tim’s Vermeer,” a highly watchable documentary about Jenison’s obsessive investigation into how the 17th-century Dutch painter was able to paint with such verisimilitude and luminosity. In the course of Jenison’s five-year odyssey, the inventor, video engineer and unapologetic non-painter attempts to create, by hand, a painted replica of Vermeer’s “The Music Lesson.

He’s not interested in copying the painting itself. Jenison saw the original once, when he was granted a 30-minute viewing by Queen Elizabeth II, who owns it. Nor is he interested in making a copy from a printed reproduction. Instead, Jenison actually builds a scale model of Vermeer’s studio and fills it with life-size replicas of every stick of furniture, as well as human models, in the painting. He’s interested in painting exactly what — and possibly how — Vermeer painted.

Jenison’s theory is that the artist used readily available 17th-century technology to help him create his art: specifically, a room-size projector, known as a camera obscura, to translate the three-dimensional scene in his studio into a two-dimensional image on a flat surface.

This is not the first time someone has alleged that Vermeer may have used such a device. Most notably and controversially, painter David Hockney and author Philip Steadman have both put forth this argument.

But Jenison takes their speculation a couple of steps further, building a souped-up camera obscura that eliminates some of that antique technology’s inherent drawbacks — reduction in image brightness, for one thing — and introducing a second tool: a small yet ingenious mirror that allows Jenison to compare the projected image with the picture he’s painting of it. (And yes, mirrors were available in the 1600s.)

Directed by Teller of the magic duo Penn and Teller, and written and narrated by his partner, Penn Jillette, the movie puts forth an utterly fascinating and fairly compelling argument, not to mention the question: Was Vermeer less genius than geek?

Put another way, if an unartistic tinkerer like Jenison can produce a reasonable facsimile of “The Music Lesson” using so-called trickery, is it not also reasonable to ask whether Vermeer himself might have availed himself of those same tricks? The film can’t offer proof that he did, but it’s intriguing enough to sway even skeptics.

The one drawback of “Tim’s Vermeer” is that we can’t see the finished product of Jenison’s labors in the flesh, though on camera his “Vermeer” comes across as remarkably close to the original. Hockney, who is shown inspecting it with Steadman, pronounces it “better” than the original, which sounds slightly fishy. What he means, of course, is that it’s too good a copy, too precisely rendered.

At another point, someone refers to Jenison’s painting as a “dead ringer” for Vermeer, which may inadvertently be closer to the truth, at least judging by the glimpses we’re shown of details of Jenison’s painting. In those close-ups, Jenison’s paint handling looks tight and somewhat lifeless: soul sacrificed for precision.

Without seeing the two pictures side by side and in person, it’s hard to judge. “Tim’s Vermeer” makes a convincing case that Vermeer could have painted the way Jenison says he did. It also makes a pretty powerful ancillary point: that some people are both geniuses and geeks.

★ ★ ★

PG-13. At Landmark’s E Street Cinema. Contains strong language. 80 minutes.