If one’s sole companionship in the space of two months is the Chinese takeout guy and the treasurer of American Origami, a few changes may be in order.

The depressive master folder Ilana Andrews (Kellie Overbey) finds herself in this situation in Rajiv Joseph’s dog-eared but nonetheless ingratiating “Animals Out of Paper.” Mr. Joseph clearly has seen his share of romantic comedies as well as examples of the similarly oversaturated mutual-uplift-through-mentorship genre: His work, the second of his to be presented by Second Stage’s new works summer series, could be described as equal parts “Creaseless in Seattle” and “Good Will Folding.” Still, director Giovanna Sardelli and her nimble three-member cast manage to — sorry — paper over some of Mr. Joseph’s creakier passages and provide just enough surprises to keep his love story afloat.

Ilana’s off-the-grid existence has been interrupted by Andy Froling (Jeremy Shamos), an endearingly nebbishy high school math teacher who has long cherished an essayistic origami primer that she wrote years earlier. He’s the treasurer. The play’s third and final character is not the Chinese delivery guy, although Andy briefly mistakes the unkempt pile of takeout containers in Ilana’s living room for a conceptual origami project. (Beowulf Borritt’s inviting set is stuffed with actual works of origami, from a gorgeous swooping hawk to intersecting tetrahedra; a note in the program thanks no fewer than nine origamists for their onstage contributions.)

Rather, it is Suresh (Utkarsh Ambudkar), a prodigy from Andy’s school who’s far more interested in hip-hop than calculus. Suresh has displayed a phenomenal gift for origami, and Andy brokers an independent study project between his under-stimulated pupil and his idol. Ilana, still smarting from a recent divorce, soon finds herself in an unconventional but not too unconventional love triangle as she vacillates between Andy, who’s adorable in a maladroit, stalker-ish sort of way, and Suresh, who’s adorable in a sullen, poseur-ish sort of way, and learning from each along the way.

These lessons, alas, frequently come in predigested morsels. Mr. Joseph has a habit of stating and then restating metaphorical conceits that might have more fruitfully remained implicit. Ilana has reached a sort of folder’s block for the last two months, stymied by an assignment to create a mesh sleeve that can enfold — are you ready? — the human heart. Her inability to plumb the heart’s dimensions, a rather tired notion to begin with, comes in for several further descriptions.

Mr. Joseph introduces other ideas with comparable lead-footedness, including Suresh’s demonstration of intuitive thinking via the rap music that he loves. (The winsome Mr. Ambudkar is at his most interesting when his character retreats into a thuggish defensive crouch, offering only brief glimpses of the tender, dutiful teenager underneath.) As if this display wasn’t sufficient to differentiate the characters, Mr. Joseph has Ilana protest that “it’s not about improvisation, it’s about rigor.”

Ms. Sandelli deploys a few schemes to make the play’s schematic construction less apparent. She’s not afraid to spill her actors into the deepest corners of the set and have them face away from the audience, and she has clearly given thought to bridging the play’s scenes with a diverting fluidity. (Both Andy and Suresh chart passing time through extreme — and extremely graceful — makeovers on Ilana’s apartment, each in ways appropriate to the characters.) And while Ms. Sandelli’s conception of Andy hews a bit too closely to type, she and Ms. Overbey get the hang of Ilana’s recessive, even combative nature as the play leads her to an inevitable but nonetheless satisfying epiphany.

Andy’s life is literally an open book to Ilana: He inadvertently leaves in her apartment a painstakingly compiled list of the blessings he has counted over the years — he’s hovering near 7,900 when the play begins, and his encounter with Ilana quickly puts him above 8,000. She can’t help reading it from cover to cover, which puts Andy at a relative disadvantage as their wary friendship burbles into romance. “It’s weird to not be able to lie a little to someone,” he grumbles during one would-be romantic dinner.

Freudians would question just how accidental his forgetfulness was, of course, and his sudden transparency opens him to a new level of vulnerability. “Animals Out of Paper” probes earnestly and often engagingly at the risks as well as the rewards of learning more about one’s companions. Even when it sags under the weight of its labored metaphors, Mr. Joseph exploits this tension with a tantalizing willingness to let his characters make unlikely and unsympathetic choices, even as they head to a familiar conclusion. May he continue to learn from his talented collaborators and trust his viewers to apply their own rigor.

Until August 24 (Broadway and 76th Street, 212-246-4422).