Soon innovations in textile design moved from the artisan’s studio to the industrial factory. Designers like Harry Bertoia and Pierre Paulin relied on new elastic fabrics, stretched across metal frames, to create chairs. At the same time, fiber artists (mostly women) began to explore the sculptural possibilities of weaving, creating gorgeous but long misunderstood works that dissolved borders of art, craft and design. The Polish artist Magdalena Abakanowicz commands an entire wall with “Yellow Abakan” (1967-68) , a weaving of coarse-grained, fraying yellow sisal suspended like a jacket on a hook.

Sheila Hicks , who learned weaving techniques from Anni Albers at the Yale School of Art, is here with two glorious, recently acquired sculptures of beige and coral linen, bundled like ponytails and heaped like doubloons. (Ms. Hicks is also one of the artists in “Surrounds,” where her flowing column of synthetic colored fibers stands at the entrance of the sixth-floor galleries.)

Like all the opening exhibitions at the new MoMA, “Taking a Thread for a Walk” draws almost entirely on the museum’s deep holdings. Its curators have clearly taken some pleasure in exhuming the outliers of a collection that has been assembled less deliberately that some suppose. Who knew that Lillie P. Bliss, one of the three founders of MoMA, donated a Coptic tapestry of the enthroned Christ from around 800 AD?