The abstract painter Thornton Willis, who was born in 1936, is a rarity in today’s art world: He is a direct link between mid-20th century American painting and contemporary abstraction. Both old-world and new-, Mr. Willis has the painterly swagger of an abstract expressionist, the charm of a Southern gentleman, and the rugged good looks of an aging leading man — albeit one who spent countless hours smoking, drinking, and talking into the wee hours at the White Horse and Cedar taverns. Listening to him wax on excitedly about the New York School, which he does without nostalgia, is to be brought face-to-face with the continuing struggle of being an artist in the city. And looking at his 20 paintings in the knockout retrospective, “Thornton Willis : Painting: 40 Years,” at Sideshow Gallery in Williamsburg, Brooklyn, is to be reassured that talented, committed artists can continually find ways to challenge themselves and to further the language of painting.

Mr. Willis studied at the University of Alabama in Tuscaloosa, with the Abstract Expressionist painter Melville Price. Price, a close friend of Franz Kline and Willem de Kooning, encouraged Mr. Willis to move to New York, where he has lived since 1963. Mr. Willis’s work has been exhibited at the Museum of Modern Art, and at galleries including Paley and Lowe, Sidney Janis, Andre Zarre, and, most recently, Elizabeth Harris. But this is the first time we have been offered four decades of his work at one venue. Sideshow’s retrospective, which has been mounted in cooperation with the Elizabeth Harris Gallery, includes recent works on paper and paintings from the 1960s “Slat” series — luminous, layered, horizontal, and striped pictures made with 3-inch-wide rollers; the basically two-colored, figure/ ground 1970s and 1980s “Fin” and “Wedge” series, as well as the interlocking, faceted, grid, and triangular paintings of the last 15 years.

The show opens with two large acrylic “Slat” paintings, “Red Wall” and “Black Wall” (both 1969), each of which is roughly 10 feet wide. Mr. Willis is reported to have made these “Slat” pictures in 14-hour stretches, and with their countless overlapping and jostling stripes rushing across the rectangle, they suggest a quick, painterly attack. But their experience is anything but fast: The stripes, intermixed on the canvases, breathe slowly, waver, and rock in the plane. At times they flutter like ribbons. They sprint midway across the rectangle, then they pressure upward, outward, and downward against the horizontal forces, vibrating like plucked stings. Speed gives way to abrupt temperature changes. “Red Wall,” anchored by black, is hot pinks, reds, oranges, and yellows — burning like embers. The hardboiled “Black Wall,” composed of blacks, pinks, and yellows on a dirty white ground, is dark and sanguine, translucent yet full-bodied. Its sides, like latticework (as in “Red Wall”), create checkerboard edges that lock the ever-advancing stripes to the rectangle.

Next, we encounter the large, vertical, two-colored, figure/ ground paintings in which a single vertical wedge or makeshift letterform moves from top to bottom or from edge to edge within the canvas. “Bisby” (1977) is an elongated gray shark fin on a raspberry ground. “Big N” (1984) is a lime green “N” on salmon-orange. “Big Twister” (1985), a cobalt blue “Z,” is racing stripes zipping through lemon yellow. And “Evelyn’s House” (1986), roughened and cool, is a green wedge on muted mauve. In all of the paintings in these series, the central form is firm but not completely hardedged. The paintings are made up of generous underpainting, which bleeds through the top colors, especially where form meets ground. Dripping and wavering edges rock the intersections, causing the ground at times to pulse forward and in front of the wedge or letterform. This causes the forms to soften and sway, melding figure and ground into an interactive dance.

The most recent paintings are the most complex, engaging, and dynamic. Based in cubist or constructivist spatial interactions, they are made up of elongated, interlocking rectangles and triangles that fold and buckle the rectangle into origami or crystalline compositions. “Gray Harmony” (1993) is based on the grid. Its warm, creamy yellows, whites, blacks, and grays shimmer like heat rising in shadow. “Gray Harmony,” slowly prying itself open, is a painting about stasis and transition: Its frame timidly, rhythmically steps in and out of the picture; and its rectangles are just beginning to move into diagonals and triangles — to rupture the grid and to open the plane.

After “Gray Harmony,” the newest pictures’ interactions between figure and ground, shape and contour, become increasingly daring. “Coyote” (2000), like all of the recent paintings, suggests stained glass. Cool in places, warm in others, its lines and forms are seen as if through a veiled light or icy mist. Black lines evolve into triangles, and forms and contours become translucent, as if, faceting and breathing in the plane, they are changing from edge to volume and back to contour right before our eyes.

“Space Out” (2006) and “Study For Entanglement #2” (2007) are both icy-cool and as fluid as molten glass. They are broken and folded box kites that advance like arrows or the prows of ocean liners; they shoot upward like rockets, and they turn themselves inside out, as if they were Klein bottles. Their triangles recede and then step forward; interlock and release; become opaque and then transparent. The forms, shifting from side to side like sliding puzzles, are as loose and free, as brushy and drippy, and as playful as if they had been let out for recess. Yet they move with mathematical precision, and they hold together as if locked by the tracery of stained glass.

The only problems with this retrospective are that it is much too small and that it is not at a major New York museum, where the large canvases could be given the room they deserve. Like the recent retrospectives of the abstract painters Pat Adams and Joan Snyder, at Zabriskie and the Jewish Museum, respectively, Mr. Willis’s retrospective has been truncated. Still, even as a sampling, the show demonstrates that Mr. Willis is a major talent who is, and always has been, going strong.

Until November 15 (319 Bedford Ave., between South 2nd and South 3rd streets, Williamsburg, Brooklyn, 718-486-8180).