Has the Royal Academy set itself an impossible task? When you shut your eyes and imagine Byzantium, the dome of your skull is roofed with a million glittering mosaic pieces. Your mind is suffused by a shimmering golden light.

Can a museum show capture that vision? Can we discover its glories through a series of historical artifacts when what we really hoped for was the Hagia Sophia, the epitome of Byzantine architecture in Istanbul?

The visitor must bring as much to this new Royal Academy exhibition as the curators have done. And they have brought a lot, gathering together almost 350 objects, many of which are only very rarely loaned from the museum collections and monastic treasuries to which they belong. Several will never before have been seen in this country and will no doubt, in your lifetime, never come here again. These are the myriad pieces that, like mosaic fragments, must be carefully fitted together to form the glimmering visions of your imagination.

Together they must capture the story of a vast empire that, ruled from its fabled capital of Constantinople (modern-day Istanbul), encompassed at its most extensive a long belt of North Africa, Egypt, the Holy Land, Italy and Greece, large parts of the Balkans and the southern regions of Spain. They must capture a sense of its shifting fortunes as it waxed and waned, diminished by plague, ransacked by barbarians, besieged by Muslims or enlarged and enriched by plundering conquests. They must convey its colourful, turbulent and often sensational history as a succession of 90 emperors fought (often viciously) for power, as well as the richness and diversity of its artistic output as it subsumed myriad cultures into its melting pot. Lastly the artefacts must highlight the power and importance of its spiritual vision as, with the conversion of Constantine the Great to Christianity on his deathbed, the religion that he had so radically legalised in 313 spread.

This must be one of the most ambitious and complex shows that the Royal Academy has taken on. It does not flinch from its task. Covering the period from 330, when Constantine inaugurated his “new Rome” with sumptuous festivities and chariot races, to 1453, when the glittering capital of Christendom finally fell to the Ottoman Turks, it takes nothing less than the entire 1,100-year history of the Byzantine civilisation as its time span. And on top of that it asks us to wonder how it is that this period has been so unjustly neglected, why it is that we leap from the splendours of classical antiquity to Renaissance glories, barely noticing this era, which we dismiss as the “Dark Ages”, along the way - and not least when, as the curators are keen to suggest, so many of the foundations of our own modern civilisation were laid in this era.

The show, assembling an extraordinary jumble of treasures, from coins to cutlery to processional crosses, through fresco paintings, wood carvings and embroidered silks, to ivory diptychs, brooches, icons (such as this one of St Theodore Tero) and bracelets, evokes an intriguing sense of the sprawling diversity of this civilisation. It sweeps through its history following the broadest of courses, but helping us to make a bit more sense of it along the way by ushering us into thematically organised galleries. There are sections that look at iconoclasm, for instance, or domestic lifestyle or the luxuries of court.

Unless you are already an expert, dip into a history beforehand (I loved Judith Herrin's delightfully lively and manageably brief Byzantium), invest in the splendidly illustrated catalogue, or maybe (I didn't listen to it) use the audio tour, because the printed labels will tell you little. It is all the more irritating that they are in the display cases so that exquisitely detailed objects that require minute examination are so far away that you can't see them properly, especially when the lighting is so gloomy. The truth is, many objects can be seen only in the catalogue.

But keep your focus. Each of the objects can become a metonym for something far bigger. Look at the fantastically fine ivory carving of the Veroli casket, for instance, and see it as a legacy of the luxury of the imperial court, where the Byzantine emperor sat on a gold throne, flanked by golden lions that roared and golden birds that twittered in jewel-encrusted golden trees. Astoundingly beautiful illuminated manuscripts speak of an articulate, highly literate culture that built Christian teachings on to sound classical foundations. Coins evoke a strong economy that maintained a gold standard unchanged for almost a millennium. Fabulous jewellery serves as more than mere decoration. It creates a shimmering aura of power that establishes authority throughout an empire.

Sometimes you can almost hear the music of a society whose long banquets were enlivened by dances performed to organs operated by water power, whose church services were accompanied by choirs of castrati, by unearthly chanting and the sounds of bells. You can almost smell the headily exotic scents that rise smoking from braziers or uncoil from the lips of a fish-shaped perfume flask. Sometimes, for a moment the shimmering eternity of Yeats's poem Byzantium becomes almost real again as we perceive its precious marvels. For a few moments, perhaps, we can almost stand, as a pair of 10th-century Slavic ambassadors once stood in the Hagia Sophia, and report: “We knew not whether we were in Heaven or Earth.”

And then we come back down to Earth. Thanks to the historical legacy of Montesquieu, Voltaire and the historian Edward Gibbon, the stereotype of Byzantium is of a despotic and often corrupt government of ambitious tyrants and creepy eunuchs, of a society mired in intrigue and obsessed with empty rituals and bureaucracy. It represents “the triumph of barbarism and religion”, Gibbon says.

Byzantium certainly has its brutal and barbarous moments. Here are stepmothers boiled alive in overheated baths, an emperor with his nose and tongue cut off to prevent him ever ruling again (he survived and returned to power wearing a golden nose patch and using an interpreter to speak) and an empress who, in her determination for power, blinds her own son in the same purple chamber in which she had given birth to him 26 years earlier.

Such grisly details seize the imagination. But they upset a balance which this exhibition now sets out to redress. Byzantium was not, as Gibbon might have us believe, a civilisation that replaced the glories of Ancient Greece with a barbarous primitivism. This show invites us to follow carefully the many ways in which an empire founded on the principles of tolerance of the Roman Emperor Constantine, who ended the persecution of Christians, adapted and incorporated the culture of the classical pagan world into its new land of Christendom. Here in a new Christian world with its saints are images of the hunt or the hippodrome races that the classical world loved. These were, after all, people who were brought up on the stories of Homer, familiar with the tales of pagan mythology. Here is the sleeping Endymion reawakened as the biblical Jonah, the beautiful Apollo resurrected as Christ.

But even more importantly, how can we see the glimmering icons of the Byzantine culture merely as a deteroriation of Classical acquirements? The icons are a numinous presence in this show. Gaze at the treasures that, for the first and perhaps last time, have travelled from the monastery of St Catherine in Mount Sinai to Britain. How vivid is the Icon of the Heavenly Ladder of St John. Here is a sort of divine conveyor belt up whose rungs the faithful ascend, constantly in peril of missing their step, falling prey to the demons who hook, shoot and pincer them from their heavenly path. Look how casually one demon strolls off with his prey. Notice how frantically the monk kicks as he plunges headfirst into a pot of hellish pitch. Feel how thankfully the topmost climber reaches out his hands to a receiving God. It may be a schematised (and apparently much restored) image, but its message is strikingly real.

As you gaze into the great all-seeing eyes of the saints (one sternly condemning, one benignly forgiving), you are looking at a form of religious expression that set out to prove itself superior to the paganism of Classical predecessors, to replace literal lifelikeness with a spiritual presence that would outlast the years. These art works become less replicas than living things.

In the 1930s, even Soviet commanders, ordered to stamp out the influences of Christianity, understood their power. They didn't simply smash them. They lined them up, sentenced them to death and then shot them. These are works that for two millennia have kept their religious power.

Sadly, the earliest ever icon of Christ is too fragile to travel from Sinai. But as we look into the faces of his saintly ambassadors, we are linked in an unbroken chain to the foundations of our Christian culture. Here are the roots of many ceremonial, diplomatic, legal and economic traditions that still survive in our society. And here too can be found lessons that are still relevant today.

How is it, for instance, that the Christian monastery of St Catherine survived intact through the centuries, unharmed by Arab raiders? The answer was simple. A mosque was built alongside it. Muslim and Christian cohabited in peace. The Byzantines could see the larger picture. It shimmered like their visions of glittering gold.

Byzantium: 330-1453, Royal Academy of Arts, Piccadilly, London W1 (www.royalacademy.org.uk ; 0870 8488484), from Sat until March 22

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