But then Polunin is no ordinary pierrot, no low-rent Biffo. He not only filled the 1000-seater auditorium but stilled the audience with his bittersweet series of loose-linked sketches, Snowshow. In fact, he quite literally blew them away with his piece de resistance, a tumultuous King Lear-style arctic blizzard that roared off the stage, pinning those lucky enough to be in the stalls back into their seats. Afterwards, the crowd refused to leave, batting around around huge balloons, while the king clown sat at the bar.
There are - to Polunin's regret - no such balloons in Edinburgh. The chandelier in the Assembly Rooms' Music Hall is too precious for that. The rest of Snowshow is, however, intact and Polunin, widely acknowledged as the world's greatest clown, is having his first crack at the world's greatest arts festival. He has installed himself and his family in a lavish flat a mile from the city centre, where he holds court to the press. In Russia, it is said, Polunin has the fame of a rock star. I may be a clown, his festival accommodation says but, where money is concerned, I am no fool.
Without his make-up, Slava resembles Professor Branestawm. He looks older than his 46 years. His hair has beaten a retreat straight down the centre of his domed head, leaving spindly grey tufts of gorse above his ears. Only when he smiles does he look younger - and in conversation he doesn't smile often. Mr Polunin - as the interpreter formally addresses him - is a very serious man, it seems - a philosopher clown. He is also, you soon suspect, a little obsessive.
Polunin owns, he reveals, 91 videotapes of Charlie Chaplin. In the two years before he went professional (in 1978), he went to the library most days from eight in the morning to seven at night and read every book about his profession that he could lay his hands on. "As soon as I wake up in the morning," he says, "it's as though I'm in the theatre. When I'm asleep, I'm living in the theatre. For over 15 years, I haven't had a holiday. It's only in the last couple of years that I can occasionally take a day off."
This indifference to designated leisure time seems to be more than just common-or-garden workaholism. It has philosophical roots. Polunin believes that it's important to believe that the idea of play-acting should not be confined to the stage. His favourite book, not surprisingly, is Don Quixote. Life should be a game; it "should take place in accordance with the laws of art". If you believe that, holidays seem rather irrelevant.
"The concept of play is such a huge one," Polunin adds, warming to his subject, embroidering the conversation with references to Stanislavsky and the commedia dell'arte. "There's a whole philosophy enclosed within it: Homo ludens. Play was a very powerful idea because in our highly regulated Soviet society, sincerity and naturalness, the openness of natural feeling and demonstration of your personality was extremely rare."
For his first two years as a professional, Polunin worried away at the whole idea like a dog at a bone. The result was a piece like The Fantasisers, in which four adults in short trousers act out extravagant daydreams - space travel, running away to sea - in the courtyard of a block of flats. "There were no games associated with everyday life in this piece, only romantic dreams," he says. "This theme became the leitmotif of all my later productions: to retain one's childlike dreams throughout adult life. A human being is happy when he follows his childhood dream right to the end."
And - though you would never believe it from the straight-backed, serious figure on the sofa opposite - Polunin has done exactly that, like some Slavic Quixote, reaching a peak of dreaminess in 1989 with his Caravan of the World. An implausibly gigantic, utterly impractical, travelling theatre that took three years to organise, and meandered from Red Square in Moscow to Paris's Tuileries Gardens.
It was, Polunin says, an unrepeatable experience. So now he is simply trying to "restore the art of clowning in the 20th-century". With the arrival of Cirque du Soleil and Archaos, the circus has reinvented itself. "They are contemporary. The only thing these new theatres lack is contemporary clowns. I'm now looking for the aesthetics and philosophy of a new clown language." And, for the moment, he has come to live here to do it.
Not that he'll probably stay here long. The Polunins have no permanent home. They live, he says, "on wheels", clocking up thousands of miles around Europe each year in a mobile home. Slava hates that most magical of experiences, flying. If they do settle down, it's unlikely to be in this country. Somewhere in central Europe, Polunin thinks, so he can nip with ease from country to country. And even then one suspects that life in the Polunin household will never be entirely ordinary. "We approach everything in our lives as though we're putting on a new show. We fantasise that, yes, we need our own home. It will have the bathroom here" - the clown moves his hand above an imaginary plan - "and it has to have lots of sea shells. You'll be able to hear the sound of the seas from above, and when you enter the bathroom some spray ought to hit you in the face."