PAINTINGS exist in two dimensions and aspire to three. Sculptures exist in three and sometimes, insofar as they may either refer to another time and place or refute time and context entirely, in four. But the paradox of the living arts -- textiles, ceramics, jewelry, furniture, architecture -- is that their purpose, their essential interaction with humans, imbues them with additional tactile and sensory dimensions; and when they are relegated to, or even revered as, museum exhibits, they are unavoidably diminished.

This dismemberment of the man-made environment, this dislocation of the arts-in-service, haunts the small but fine exhibit of medieval Hispanic artifacts at the Arthur M. Sackler Gallery, "Caliphs and Kings: The Art and Influence of Islamic Spain." It requires the visitor to make an effort not to view the objects only in isolation, but to imagine them fitted together in a whole, to multiply the images like a kaleidoscope -- and to recognize the depth and complexity of their cultural references.

Consider the 15th-century sanctuary doors, cleverly distressed to seem even older and, though wrought by Muslim artisans, carved with Catholic entreaties; the intricate tapestries showing two young women toasting as if from an ancient Greek amphora; the carved and gilded marble capitals that recycled not only old Roman ideals of classic grandeur but actually revived their ancient quarries. Track a Byzantine Greek book of morals, translated in the 9th century by a Christian physician into Arabic and then in the late 12th or early 13th century into Hebrew and Catalan.

Study the mural-size photo of the Alhambra, that jewel box of a palace with its fretwork balconies and overhangs like paper snowflakes. Now imagine the feverish richness of a world in which the shadows of those eaves, black in the blinding Spanish sun, fell upon brilliant mosaics whose own designs were only partially covered by tapestries and carpets. It was a culture of such visual extravagance and such restless imagination that it sent explorers far into the unknown -- carrying astrolabes from Jewish astrologers and compasses acquired from the Chinese -- and saw them return with maps of dragon-infested forests, expanses of Africa and Asia populated by camels and elephants, and cities and palaces of gold. It's this extravagantly vivid context the visitor must supply.

Al-Andalus, as the Iberian peninsula was known for nearly eight centuries, was among the greatest crossroads of medieval culture, an arguably unequaled artistic and scientific crossroads, where Muslims, Jews and Catholics lived and worked and exchanged ideas in a harmony unequaled before or likely since. That it ended so harshly, with the expulsion or forcible conversion of the Jews and Muslims by Isabel and Ferdinand -- ironically, in the same year as their emissary Columbus "discovered" the continent that was to idealize universal freedom -- has obscured its many achievements.

Nevertheless, the advances made in agriculture, mathematics and astronomy, cartography and navigation, architecture, silk weaving, ceramics and marble carving, leather- and metalworking, and bookmaking were of such note and in such widespread demand that they are still hallmarks of quality: Toledo steel, Cordoban leather, Damascene silver, Granadan silk, Seville oranges, etc. Not surprisingly, perhaps, the phrase that appears repeatedly in the tapestries and ceramic designs is "prosperity," a neat joining of God and Mammon.

"Caliphs and Kings" includes 89 rarely exhibited pieces from New York's Hispanic Society of America, considered the finest collection of Islamic Hispanic art outside Spain itself, with a few pieces from the Freer Gallery of Art and National Museum of Natural History. (Serendipitously, 2004 marks the centennial of the Hispanic Society's founding by Archer M. Huntington, and this is the first time it has lent any of its holdings.)

One of the most winning pieces in the entire exhibit is a map of the world attributed to Giovanni Vespucci, nephew to our eponymous Amerigo, in 1526; believed to be a copy of the Spanish government's secret military/commercial charts, it places Spain squarely at the center of the universe, with ships headed off in all directions and its imperial eagles -- the two-headed symbol later adopted by the Hapsburgs, as King Philip II was both Holy Roman emperor and king of Spain -- dominating North America. In intriguing contrast, an earlier map by the 12th-century Judeo-converso (converted Jew) Mallorcan cartographer Pere Rosell, tellingly focuses on the wind patterns and port-to-port distances of the Mediterranean only to the Old World, north, south and east.

Other select but real pleasures include a fabulously inscribed Hebrew Bible manuscript completed only two decades before the expulsion; a 13th-century Latin translation of the groundbreaking al-Khwarizmi 9th-century treatise on the use of zero; the Freer's decapitated but still lovely lusterware urn with bronze and gold glazing, near cousin to a landmark piece in the Alhambra; and a fantastically carved ivory perfume case designed to suggest a young woman's breast (and so inscribed) probably presented by Caliph al-Hakam II in the mid-10th century to his favorite concubine.

"Caliphs and Kings" was primarily sponsored by the Washington-based Mosaic Foundation, a charitable organization of spouses of Arab ambassadors to the United States dedicated to improving the lives of women and children and to promoting a greater understanding of Arab culture. The Mosaic Foundation is also sponsoring a free screening of Egyptian director Youssef Chahine's 1998 film "Destiny (Al-Masir)" Tuesday at 7 at the Avalon Theatre, 5612 Connecticut Ave. NW.

CALIPHS AND KINGS: THE ART AND INFLUENCE OF ISLAMIC SPAIN -- Through Oct. 17 at the Arthur M. Sackler Gallery, 1050 Independence Ave. SW. 202-633-1000 or www.asia.si.edu. Metro: Smithsonian. Open daily 10-5:30.

A carved perfume case, circa 966, is made of ivory with silver-gilt mounts.A map of the world attributed to Giovanni Vespucci, nephew of explorer Amerigo Vespucci, centers on Spain. At left, a Hebrew Bible.