For centuries, Muslims have travelled to Mecca for the hajj. Paupers have saved for a lifetime in order to make the journey and princes have travelled across the desert in palanquins.

Today, nearly 3 million Muslims make the pilgrimage every year. In a cynical, disbelieving world, many still describe the experience as transformative.

For non-Muslims, however, the hajj is out of bounds, spiritually and literally. The aim of the British Museum’s new exhibition, Hajj: journey to the heart of Islam, therefore, is to provide a concentrated, facsimile version of the pilgrim experience.

There are three parts to the hajj and, thus, to the exhibition: the journey to Mecca, Mecca itself and the aftermath.

In the era before mechanised transport, the journey to Mecca could take months. One 12th-century pilgrim set out from Andalucia in February. His slightly circuitous route - avoiding Crusader Jerusalem - took six months.

A colourfully detailed map traces the 14th-century journey of Mansa Musa, a Malian king. He dispensed so much gold during a stop in Cairo that he depressed the Egyptian economy for a decade afterwards.

This part of the exhibition is, essentially, a bigger, shinier, grown-up version of one of its own exhibits: a 19th-century toy theatre, entitled The Caravan to Mecca. In front of a desert landscape, cardboard figures in fezzes erect tents and tend camels; other cut-out characters kneel in prayer.

The highlight, however, is the section describing the five days of rituals that pilgrims undertake in Mecca. It is impossible to recreate once-in-a-lifetime religious exultation in a museum. But, watching film footage of the hajj rituals, one begins to get a sense of the collective euphoria that dominates the experience.

And there are quotes from those who have experienced it themselves. “The Kaaba is that world’s sun, that attracts you into its orbit,” says a 20th-century pilgrim, referring to the... Actually, this is the only - but nonetheless key - problem with the exhibition. It does not explain what the Kaaba is.

Here is what the exhibition does say: the Kaaba is a large cuboid in the centre of Mecca. Although it predates Islam, it is the religion’s holiest site. Muslims believe it was built by Adam after he was expelled from paradise, and later resurrected by Abraham and Ishmael. Hajj pilgrims circumambulate it seven times. It is covered by an intricately detailed black and gold curtain, replaced annually.

But nowhere are we told what, under its curtains, behind its myths, the Kaaba is. And - significantly - what it was, before Islam appropriated it. Given that it is the point of those arduous, desert-traversing journeys, this seems a not-inconsiderable omission.

Possibly the lack of faith-free analysis is attributable to the museum’s need for Saudi cooperation. It is the only disappointment in an otherwise instructive, thoughtfully constructed exhibition.