David Letterman sports a popular new status symbol, the achievement beard. PHOTOGRAPH BY RON ASADORIAN / SPLASH NEWS / CORBIS

A photograph of the retired David Letterman has been making the rounds recently; perhaps you’ve seen it. The photo shows the former late-night host stalking the street in khakis and a floppy long-sleeved shirt. The shirt has one enormous pocket. Letterman displays a deep, soul-summoning gaze. His hair, now gray and cropped close to his skull, stands in a victorious V of a widow’s peak. His eyes seem scrutinous and feral, as if spiritually unnourished by the generous appeasements of a CBS pension. He looks like someone who wants to sell you yak-wool sweaters for extreme weather, or a chapbook of his own apocalyptic poems. Most of all, he looks unshaven. Letterman now wears a wide and woolly beard.

The achievement beard—a marker of triumphant lassitude, the victory lap after a long job well done—has been gaining currency in recent years among men who might like to move through the world noticeably unnoticed. It has become standard issue for an entertainer on the comedown from a high-intensity career: Stephen Colbert donned a seaworthy achievement beard during the nine-month hiatus between his first show and his new post in Letterman’s stead, and Jon Stewart has been growing one since stepping down from “The Daily Show,” in August. Yet it doesn’t take a lengthy television run to grow a beard, and similar adornments have been cropping up to mark a range of distinguished endeavors. The retired soccer star David Beckham was recently seen sporting an achievement beard. Al Gore famously grew one a while back, apparently to mark the achievement of no longer having to run for office. The beard is a self-presented lifetime-achievement award, a modest way of underscoring what few people would dare to miss.

Exactly how is this beard unlike all other beards? The achievement beard, in length and vigor, occupies a place above James Franco but below Chester Arthur. It is not to be confused with the recently popular baseball beard, or with the performative crazy beard, or with the hipster beard, which marks no achievement at all. It is not a bohemian power beard, as worn by ZZ Top or The New Yorker ’s own Richard Brody, and it also shouldn’t be confused with the beard of disaffection, à la Ted Kaczynski, or James Mason during his cross period. Rather than saying, “I have given up on the world,” the achievement beard declares, “I am away, but not gone.” (Hence its popularity among showmen at leisure.) It holds none of the freewheeling, Manson-like menace of the “Joy of Sex” beard, but it’s not too fussy, either. (Note Letterman’s trimmed edges and untrimmed everything else.) It must never be something that eight minutes of shaving can’t undo. If there’s wax involved, the beard has ceased to mark achievement; it is now a beard of solipsism. The achievement beard is definitionally low-maintenance.

The overtones of the achievement beard are plain. Most examples have a way of growing out to regal length: Colbert's beard most closely resembled that of King George V. Most, too, are gray; an achievement-style beard on someone of youth and of moot accomplishment looks strange. (Consider the disturbing beards of Brad Pitt.) Occasionally, a longstanding avocational beard may become an achievement beard, the way that lifelong writing hobbyists sometimes turn into acclaimed writers (the Frank McCourt of beards). Generally, though, the achievement beard is notable not because it projects its wearer’s public image but because it doesn’t. The achievement-bearded Letterman is unlike the Letterman we know.

So which Letterman is the “real” Letterman, bearded or unbearded? Would the bearded man even exist were it not for the ascent of his clean-shaven counterpart? Such ontological questions confound the mind, but it is possible to solve them using diagrams. Let us first draw a time x-axis and a spatial y-axis. Next, let us plot the path of Unbearded David Letterman zigzagging rightward from Indiana to the majestic Ed Sullivan Theatre. Our starting premise is that Letterman remained continually unbearded all this time, but do we know that to be true? Conceivably, Letterman might have grown a beard at some point along the way—a weekend, say—and shaved it before returning to the airwaves. (Maybe he even grew one beard, shaved it off due to familial teasing, and then, defiantly, grew another beard, a thing known to happen during months of underemployment.) To take probabilistic account of the various beard-growing possibilities, known and unknown, we must diagram all beard-and-shaving paths and subject them to complex arithmetic. In the course of drawing these diagrams, however, we will have a curious realization, which is that a bearded Letterman is the same as a shaving Letterman travelling backward in time. (Whoa.) This complicates the calculations, but it also introduces certain beautiful properties of beardedness. For one thing, we can conclude that the fact of Unbearded Letterman does indeed produce the possibility of Bearded Letterman. Also, if the Bearded and Unbearded Lettermans ever meet, they will annihilate each other and produce a photon. (This has never been experimentally confirmed.)

With such questions resolved, it is possible to make accurate predictions about the beard of Stephen Colbert and, in fact, all beards in the known universe. It also helps to explain certain paradoxes of achievement-beard causality, including Sean Connery. When Connery grew his achievement beard, in 1975, he was chiefly known for action films, including “Zardoz.” But then his beard appeared, and his career as a serious actor took off. Was this coincidence? Or was it fate? To paraphrase Matthew McConaughey—recently seen in facial hair of uncertain motives—your present achievement matches the future beard of your past self.

There is, of course, a big problem with the semiotics of the achievement beard, which is that women cannot grow it. Does this mean that it’s a redoubt of the patriarchy? Yes, clearly. Possibly in roundabout solidarity with the cause of equality, or maybe merely in defiance of such laurels, certain men throughout the ages have de-bearded at the moment of their greatest triumph. Consider Henry James, who went bald in all directions during his late years. Or take a look at the Microsoft co-founder Paul Allen, who, though sporting an achievement-like beard at the outset of his corporate ride, now proudly bares his sixty-two-year-old cheeks to the elements. Such examples are heroic. Aspiring high achievers might be wise to heed them. This summer, Jack Dorsey, the interim head of Twitter, made news when he appeared on television with a very strange, long beard. The circumstances were misaligned enough to make the beard’s intent mysterious—it was presented in the style of an achievement beard, but what was the achievement?—and the beard was misaligned enough to make Dorsey’s head look frighteningly crescentic. This week, at last, he was named permanent C.E.O., and the beard appears to be in remission. Good for both of them. The last fact of the achievement beard may be the most unfair: there’s always something more to do before it’s truly earned.