Pablo Picasso, Chair Cannes, 1961. Musée National Picasso–Paris. Photo: 2015 Estate of Pablo Picasso/Artists Rights Society (ARS), New York

On October 9, 1912, Pablo Picasso wrote a letter to Georges Braque, his confrere in Cubism, whom he later derided as “my wife” and who, for his part, described himself and Picasso as “two mountaineers roped together.” “I am in the process of imagining a guitar,” Picasso wrote — a line that still sends shivers. He went on to declare that, after their summer working together in the South of France, he was hijacking two of the biggest artistic ideas of the century: collage and three-dimensional assemblage. Both would change the landscape of art: Via collage, painting took on a more physical body, changing its spatial presence forever; assemblage did all this, too, but in even more varied materials and space, a technique so radical that it essentially remapped the boundaries of painting, bas-relief, molding, and sculpture. Both ideas were Braques’s! As Picasso himself said, “Great artists steal.” Or as Matisse put it after a studio visit from Picasso, “He will put it all to good use in time.”

We don’t think of Picasso as a sculptor, but we should. He was a great one. In the years after that summer with Braque, Picasso performed a vivisection of 500 years of Western spatial perspective. For much of the 19th century, artists like Constable, Corot, Courbet, Manet, and many others tried to break the rigid illusionistic strictures and the structure of vaunted Renaissance perspective. Yet no matter what artists did, including Monet — breaking down every brushstroke into a physical thing that functioned at once as a mark and a picture, each one being absolutely equal to every other stroke, all but doing away with illusionistic space altogether — still, the borders and surfaces of the object reasserted themselves. With collage and assemblage, Picasso finally jarred space from a kind of 500-year sleeping sickness, a system that had silted up, impeded, and confined vision. With these works, Picasso broke forever from Renaissance tradition into modernist, Einsteinian relativity, the paradoxical space where things exist in different dimensions at once. It’s important to remember, of course, that Renaissance perspective was maniacally practiced only in the West. In Asia, Africa, and most of the rest of the world, systematic illusionistic space never caught on. In the West, however, Picasso (and the others) set space free.

Now, for the first time since the Museum of Modern Art’s epic 1967 “The Sculpture of Picasso,” MoMA returns to this fertile delta with more than 150 of his slabbed, shattering, avian-shaped, hallucinogenically assembled sculptures. This is a fantastic show. The work in this exhibition represents a macro evolution in the history of art, and at MoMA, you will see objects that look like loofahs shellacked with tough nuggets and nettles; vertebrae forms; craniums that look like crustaceans; dolls and Roman soldiers that might be from the Crab Nebula; iron-cage works that go off like slow fuses, morphing into movies, maps, and sliding diagrams. I found myself constantly having to catch my breath, reassessing what I thought about Picasso’s sculptures, even seeing a lot of what I thought were clunkers as packing information that scores of artists are still putting to good use. (One abstract cast of a crumpled paper bag basically gives rise to artists like Jean Debuffet and movements like Art Informel and Tachisme.) Whatever you do, don’t miss this exhibition; this is exactly the kind of show MoMA is made for. Museum co-curators Ann Temkin and Anne Umland (with Virginie Perdrisot) make MoMA’s fourth-floor galleries look more beautiful and useful than they have ever looked. Linger in these spaces; the chance won’t come again for a long time. “Picasso Sculpture” could be one of the great learning experiences of your seeing life.

Above all, this show lets you know in your bones that beneath it all, Picasso’s art was always sculptural — always structural, architectural, and tectonically imagined first. Picture the shapes and planes of his Cubist paintings as cut up and pieced back together as sculptures. Or traditional sculpture and objects crushed into bas reliefs, then given more dimension. That is virtually what he did. “It would have sufficed,” he said, “to cut up [the Cubist paintings] — the colors being no more than indications of difference in perspective, of planes … and then assemble them … [as] sculpture.” At another time, he said he analyzed “form into separate geometric components … presenting simultaneously several views of the same object as though the spectator were walking around it or turning it over in his own hands.” It is so wildly revolutionary that it makes you wish that artists like Tintoretto and El Greco had tried this centuries ago! Or reversed the process, and we would have seen Bernini tectonically flattened.

At MoMA, you will see all these separate components — faceted shapes, tectonic plains, abstract forms, axiomatic lines, and more — in sculptures that all point to undiscovered dimensions. But not in the first gallery. The figurative works here, from 1902 through 1909, are really just Picasso turning the key in the sculptural ignition; assimilating Degas, Maillol, and Matisse; African, Oceanic, and early Iberian sculpture, seeing what he can put “to good use.” The primitive-looking wooden totemlike objects have nowhere near the force of the Kongo power figures with hundreds of nails hammered into the surface that predate them; they look as if they might have been made by Gauguin, but lack his sense of decorative flatness. Or elsewhere, since Rodin did smushed female forms, so does Picasso. Still, two pieces from 1909 vibrate with a more pressing tension. The first, a small plaster Apple, which looks like a Chinese cube puzzle carved by Cézanne. It clues you in — despite not being an especially convincing sculpture — to the simultaneity that Picasso is after, as the apple looks like it is changing shapes and being seen from several perspectives at once. The other is the solid bronze Head of a Woman, whose angled, sluicing protrusions radiate the drama of two-dimensional Cubism that is already happening in his paintings and is about to detonate into sculptural space. But the bust is still leaden, bound to a space and governed by a gravity that are about to give way.

The rupture comes in the next galleries. Categories prolapse in constructed, cut, painted, assembled, and cast works that might be bas-relief, paintings, sculptures, all of the above, or none at all. You’re seeing something never seen before, yet Picasso does it with such utter clarity that you can feel the scales fall from your eyes. And even think, I could do this. Ponder the incomprehensibly influential guitars. For the first time in the history of Western sculpture, the inside, or the hollowness, of form is depicted — here, the hole of the guitar becomes an object with space and illusion all at once. Imagine Michelangelo’s David somehow also revealing chest cavity and organs flip-flopping and interfacing with the outer surfaces, seeing this beautiful boy from behind and in front at once. The guitars represent a new kind of illusion and a new realness in sculpture simultaneously, where the object comes to self-explanatory life and negative space isn’t a void anymore. In the same astounding gallery are the famous absinthe glasses, which do the same things but in more compact scale, using curves, torque, and twists, exploring different material states and gravity in ways that push dimensional limits into something like the uncanny. Are we really seeing these things from all sides — including the liquid — at the same time? The same thing is happening in the guitars.

Opportunistic superpredator that he was, when Picasso took, he took fast. Between that October 1912 letter to Braque and December 3 that year, he made paperboard, string, and painted-wire guitars, each of which exists in a spectrum of different non-narrative spaces at once. By 1915, in addition to groundbreaking collages, Picasso fashioned more guitars out of sheet metal, paper, wood, nails, string, canvas, and painted wire paper. Three of them are here. And still radioactive. These works change the sculptural game. The guitars are what Donald Judd later called “specific objects”: neither conventional sculpture nor musical instruments, but something else. (In fact, let’s use “something else” as one of the definitions of art.) At the same time, the guitars also seem to contain interior dimensions never before probed in sculpture. By 1915, there’s the splayed-open violin that lets you see the inside, outside, and left and right sides of the object at once. You think, Yes! He’s doing with sculpture what he does in paintings: finding a way to see breasts, anus, vulva, eyes, buttocks, belly button, and mouth at the same time! Picasso’s sculpture so encompasses this strange all-at-once dimension that today, flatfish with eyes on the same side of their heads are called “Picasso fish.”

In the middle of this gallery are the six painted bronze absinthe glasses, each with a real sugar spoon, each cast from an identical wax mold. Every one of them looks different; I have no idea how. They’re almost too much to deal with. Each comes to cartoonish life, at once slabbed and angular, Neolithic and modern. Here you witness the interior interweaving and untwining of form, temporal shifts, visual glitches, the force of an imagination dancing. Picasso decants the absinthe, lets you see the glass and through it. The liquid substance transforms into a decorated vessel, then slush, solid, translucent, polka-dotted plasma, all while appearing malleable, in changing states, or freezing. The sugar cube on the spoon turns into a boulder, a minimalist sculpture, and part of a pyramid about to be dropped into this vast alchemical caldron. Fins on the glass might be vestigial body parts, cooling devices, or mating plumage; spirals are really flanges; balance is impossible. Is the cup tilting? Twisting? Stable? Serving itself? We simultaneously behold this twisty thing from above, below, left, and right. In this gallery, Picasso’s works create the sculptural equivalent of what is known in geometry as a tesseract, ­a four-dimensional analogue of a cube, or a hypercube, which some speculate would, if entered in one place — say, MoMA — deposit you under the ocean or in another dimension. Whatever’s happening here, that’s what happened to art in 1912: It started in one place and ended in another.

After this outburst, there was sculptural silence from Picasso for almost a decade. The show picks up again in 1928, with two small, curvaceous bronze shapes that could be a woman running on a beach as you see the thunder of the Cubist guitars made more pliable, organic, sensual. Make a mental note; they foreshadow something spectacular to come. In fact, almost everything in this gallery is stuff that Picasso will put to real use further along.

The next gallery gave me whiplash. And got me hot. This space is a bacchanalian garden of voluptuous, abstract female forms made between 1930 and 1937. Here are new hieroglyphic shapes becoming faces, heads, breasts, mouths, vulvae — pendulous forms that are so unphilosophical and distorted that they become concretions of garrulous, graceful, free-flowing, libidinous meaning. This work doesn’t fit into any other concurrent movement of the time; Picasso is off the sculptural reservation here. Biographer John Richardson tells us that the woman Picasso was picturing was his “greatest sexual passion.” Her name is Marie-Thérèse Walter. Picasso spotted her when she emerged from the Paris Metro on January 8, 1927. Instantly smitten, he approached, proclaiming, “I’m Picasso! You and I are going to do great things together.” She’d never heard of him and was 17; he was 45 and married. So began a ten-year frenzied secret affair that cast Picasso as consuming monster, inflictor of abject cruelty (she was “a slice of melon” which he “denied … orgasm”). It is in these sculptures (and paintings of this period) that Picasso’s art gets fleshier and more organic, as he claims a larger part of himself and catapults himself from Cubism forever.

Many academics pooh-pooh this and much that follows in Picasso’s work. It’s true: At this point, he does break with modernist teleological progress, and certainly with Cubism; most of what comes next doesn’t fit neatly anywhere art-historically. Only in Picasso-ism. And he produced epic amounts of art, much of it not that good or even very avant-garde-looking. Yet it’s squeamish and persnickety to limit the greatness of Picasso to just Cubism and a few other high points. In this gallery, you see art that is blissed-out, brutal, Greek, goofy, a genus of modern Venuses rendered in new formal topographies. With Matisse, Miro, Calder, and others, Picasso is paving the way for much of the biomorphic abstraction used by artists all over the world ever since. The busts of Marie-Thérèse are a real depiction of how love deforms. They’re the ones I’d want in my inner museum. They also lead directly to the desexualized screaming faces and tortured forms of Guernica (1937). The curators also provide a great insider finger-wag at and lesson about Picasso the psycho pirate. Installed in the center of this gallery is the bronze Reclining Bather (1931). This work comes directly out of all Matisse’s much-earlier nudes. Placed here, it is a reminder that, though Matisse was never this animally sexual, without his incredible lifelong influence, many reckon that there’d be no continually reforming (a.k.a. catching up) Picasso. That’s how thorough these curators are.

There’s so much more: galleries packed with uncovered information for artists and viewers, single works that can detonate into whole careers. Please visit this exhibit more than once; every gallery is a world unto itself that deserves attention. And before you leave the show, think about where it is in the museum. And why. All other rotating shows have been elsewhere at MoMA, in temporary galleries. While many museum visitors might protest the absence of the usual masterpieces on this floor, we must be grateful that MoMA undertook the gargantuan task of totally dismantling the fourth-floor permanent collection for the Picasso show. The ceilings and spaces of the second- and sixth-floor galleries would have overwhelmed, dwarfed, and distracted from this work. Other spaces at MoMA are too small. And composite cement floors would have been wrong. Temkin and Umland’s Picasso show points to a way through MoMA’s spatial problems, suggesting that these galleries don’t always have to be static with teleologically installed masterpieces from the enormous collection. Instead, they can be used for continual experimenting with the greatest collection of Modernism on Earth, opening that collection up, letting the world see just how incredible MoMA really is, and perhaps helping to heal a multitude of sins. This Picasso show lets us see things we hadn’t dreamed of before. All while seeing an artist edging into undiscovered spaces, interstices between flatness and dimensionality, spaces that are at once ancient Egyptian, clear, and Mesopotamian in insistence. Picasso lifts a veil. I might still want to live with a Matisse, but “Picasso Sculpture” lets us know that space — even MoMA’s — really has a wide wingspan.

*A version of this article appears in the September 21, 2015 issue of New York Magazine.