Question: When is a beer can more than just tomorrow’s trash?

Answer: When the artist Jasper Johns gets his hands on it.

A new show opening Sunday at the National Gallery of Art — “States and Variations: Prints by Jasper Johns” — traces the ways Johns uses various printmaking techniques to play with the representation of everyday objects, such as a Ballantine Ale can, and in the process catapults the pedestrian into the ranks of the visually daring and cool.

For the multitalented Johns, his paintings, drawings, sculpture and prints were “totally connected,” said exhibit curator Ruth Fine, making the exhibit, which focuses heavily on the proofs he made on the way to a finished edition of a print, a valuable glimpse into his creative process.

Hence, it was not unusual for Johns to create a sculpture of, say, a flashlight, then, after a photographic image of the sculpture had been transferred to a copperplate, print a photoengraving of it. Or alternately, he might make a grease drawing of the numbers 0 through 9 superimposed on one another and print lithographs using various colored inks and paper types. (Warning: You’d do well to pick up one of the pamphlets at the exhibit’s entrance to the exhibit for help in navigating the arcane technical terms involved in printmaking. The glossary will help you differentiate between aquatint and open-bite etching processes, for instance, and navigate the nuances of state, color trial and working proofs, experimental steps along the path to the print editions.)

The exhibit takes as its organizing feature a portfolio of 13 prints Johns published in 1969 titled “1st Etchings, 2nd State,” which in turn is based on the six prints of six motifs in the 1968 portfolio “1st Etchings.”

The half-dozen motifs in question are simple — ale cans, a flashlight, a light bulb, a Savarin coffee can of paintbrushes, the American flag and the numbers 0 through 9 superimposed on one another. The first five of these easily conform to Johns’ classification of “things the mind already knows.”

In “1st Etchings,” each print sports two renditions — a photoengraving and a drawing — of the motif. Johns saved these plates and later altered the tonal makeup of the renditions using aquatint and open-bite before creating separate etchings for each version in “1st Etchings, 2nd State.”

Got that straight?

It gets clearer when you’re standing in the exhibit’s central room with the various trial proofs and finished prints of “1st Etchings, 2nd State,” as well as proofs of “1st Etchings” juxtaposed next to one another — and can compare Johns’ subtle and sometimes not-so-subtle alterations.

But to see where it all began, go back to the exhibit’s beginning room, which shows Johns’ early explorations into many of these motifs dating to 1960, the year he first started making prints.

The best of these are five lithographic prints of a pair of Ballantine Ale cans. Over the course of the series, we watch as the cans are transformed from rough sketches of ambiguous cylinders to more defined images set against a black backdrop that serves at times to accentuate the cans and at others to nearly subsume them. And no detail is too small to vary — even the edges of some versions of the lithographs alternate between tight and messy black lines.

The exhibit’s final room demonstrates how Johns, after setting the motifs aside for a period, later revisited and incorporated them in subsequent proofs and editions of his artistic compositions.

There’s a truly glorious “Savarin” lithographic proof from 1977, which features the Savarin coffee can with paint brushes but also adds other motifs, including a panel of fingerprints and another panel of colorful cross-hatchings. And a lithograph of a light bulb with yellow rays (drawn in in Johns’ hand), set on a pedestal and enclosed in a rectangle, invokes, depending on one’s perspective, something between a satiric take on a classical nude and the manger scene.

But the grand summation of the show can be found in 1971’s “Decoy,” a lithograph that includes, among other things, the reversed images of the six photoengraved motifs from “1st etchings, 2nd State,” a picture of one of the Ballantine ale cans, aspects of other previous Johns lithographs and an image of a wax cast leg cutting across the top right corner of the paper. (The original wax cast of this leg is part of the 1964 painting “Watchman” on view in the companion exhibit, “Jasper Johns: An Allegory of Painting,1955-1965,” in the galleries adjacent to this show.) Four of Johns’ 10 “Decoy” proofs from 1971, in addition to the final version, are included here, and take the viewer on a journey through the various shading changes and elemental tweaks that Johns toyed with along the way. “Decoy” was one of only two prints Johns also based an eponymous painting on.

As an aside, the exhibit offers a chance to see some of the iconic Johns motifs you won’t find in “An Allegory of Painting,” which opened earlier this year. Most famous of these is Johns’ take on the American flag, which first appeared in a painting in 1955. The exhibit includes a flag screenprint, lithographs, etchings, and even a rare existing limestone bearing Johns’ Stars and Stripes lithographic drawing.

Notably, this exhibit coincides with the National Gallery’s announcement this week that by the end of 2008 it will acquire about 1,700 Johns proofs (including etchings, lithographs and screenprints) from the artist’s personal collection, more than 300 of which already have been obtained and dozens of which are included in the 63-piece show. The deal, in the works for the past “four or five years” according to NGA Director Earl Powell III, will make the gallery the largest institutional repository of Johns’ works in the world.

“States and Variations: Prints by Jasper Johns” is on view at the National Gallery of Art’s East Building from March 11 through Oct. 28. For more information, visit www.nga.gov.