Classical architecture forms such an integral part of our national identity that it is easily forgotten that the style was once a foreign import. The first example on these shores was the house that Inigo Jones designed for James I’s wife, Anne, at Greenwich in 1616 – more than a century after the classical language of architecture had begun to be revived in renaissance Italy.

Before his elevation to the position of court architect, Jones had travelled in Italy but the new style asserted its influence on him, above all, by way of a publication. First printed in 1570, I quattro libri dell’architettura (The Four Books of Architecture) is a manual that lays out a comprehensive set of rules for building in the classical language and, particularly significantly, establishes how a vocabulary that had originally been developed for use in the religious architecture of the ancient world might be deployed on modern buildings of secular function. It is focused around a sequence of drawings of the many houses that its author, the architect Andrea Palladio, designed in and around his native Vicenza – each one an exquisite variation on persistent themes.

With its rigorously square plan, symmetrical facades and use of the classical orders, Jones’s house at Greenwich is essentially an extension of that series. It would prove far from the last. In the centuries since, Palladio has influenced the design of structures ranging from banks to parliament buildings, hotels to shopping malls, on sites across the western world and beyond.

A compact but compellingly curated survey of this vast legacy is provided by Palladian Design: The Good, The Bad and the Unexpected, which opens at the Royal Institute of British Architects, this week. It begins with a selection of Palladio drawings purchased by Jones in Italy that now form one of the principal treasures of the RIBA drawings’ collection. They include original designs but also survey drawings of the ruins of ancient Rome. A drawn reconstruction of the Portico of Octavia (AD 203) illustrates one of the architect’s key sources, subsequently redeployed in a number of his most celebrated villa designs.

However, the exhibition’s narrative moves speedily on to address Palladio’s influence: first on Inigo Jones’s work and then on the great Palladian revival of the early eighteenth century. Spearheaded by the connoisseur and amateur architect the 3rd Earl of Burlington, this manifested itself in countless country houses and many of the era’s principal buildings of crown and state. It coincided with the formation of the first Whig government in 1715 and was guided by the ambition of the new ruling class to reinvent Britain in a new image. William Kent even developed designs for a new parliament building on Palladian lines – a scheme that was only abandoned after Robert Walpole’s government ceded power to the Tories in 1742.

While the British ultimately opted for a gothic seat of power, the Palladian language would go on to form the basis of government buildings the world over, including the United States Capitol and Viceroy’s House in New Delhi – an extraordinary synthesis of classical form and hindu motifs, completed to designs by Sir Edwin Lutyens in 1932. Some of the later – highly abstracted – examples of Palladian influence gathered in the exhibition are no less surprising, including a postmodern ski-lodge in the mountains of Canada and a largely subterranean house designed for the new Mongolian city of Ordos.

The show has been beautifully designed by Caruso St John Architects with many of the exhibits simply laid out on unpainted softwood benches. Beautiful as they frequently are, they are shown not as artworks, but as the working tools of architects practicing over a span of almost 500 years but united by a common obsession.

Until January 2016. Free entry. Details: architecture.com