The Tokyo Olympic debate: squashed cycling helmet or architectural gem? In Kon Ichikawa's great film of the 1964 Olympic Games, the first thing you see is a wrecking ball, smashing into the concrete facade of an old industrial building. Then the director shows you what replaced that building and the others that had once stood around it: the giant bowl of a new sports stadium, rimmed with flagpoles, fit for a modern age. If the organisers of the 2020 Games are planning a new version of Tokyo Olympiad, the cameras of Ichikawa's successor will be trained on that same stadium early next year, watching the construction crews move in to create the space for a replacement already surrounded by a controversy that intensified this week. In its elegant simplicity and satisfying symmetry, and its sense of being open to the elements of nature in which the Games" track and field events take place, Mitsuo Katayama's stadium, opened in 1958, echoed the places where the citizens of ancient Greece assembled to watched their sporting heroes. But probably more has happened to architecture in general, and to the design of sports stadiums in particular, in the past half-century than in the 1,900 years between the building of the Panathenaic Stadium in Athens and the first Tokyo stadium. That's certainly how it feels when you look at the artist's impression of Dame Zaha Hadid's design for 2020, a swoopy dome slashed with vents, which has been compared many times to a squashed cycling helmet - the kind with a small peak worn by mountain bikers - but from above also resembles an inert spermatozoon. Like everything designed by Hadid, an Iraqi-born architect trained and based in London, it is a statement of the possibilities of architecture as much as a reaction to the requirement of the brief, which also requires it to be ready to welcome 80,000 spectators for the final of the 2019 Rugby World Cup. Since she succeeded in overcoming early resistance to her work and finally got her projects built, she has shown that sometimes the result can achieve the desired transcendence. In the past few years she has done well in the field of sport. Her ski-jump tower and ramp at Innsbruck, host to the annual Four Hills championship, and the railway stations that link to it, manage to leap out of the Tyrolean landscape while seeming integral to it - "like ice dribbles," she has said. Her spectacular aquatics centre was part of the presentation that won the 2012 Olympic bid for London, although its aesthetics were severely compromised during the event by the need to add temporary grandstands. Its reception was also cooled by the knowledge it had run massively over budget - from an original figure of £75m to a final total of £269m, not all of it the result of specification changes. Her critics claimed the increased cost was also generated by her habit of dreaming up the shape of her building first and worrying later about how the structural engineers were going to get it built. Unfavourable comparisons were made with London's velodrome, which came in on budget at £90m and made at least as satisfactory an aesthetic impression. Since the aquatics centre's unsightly temporary structures were removed, however, and the swimming pool was opened to the public, its users now have the benefit of a exhilarating facility whose shape, like that of a dolphin arrested in mid-leap, is an adornment to the landscape of the Olympic Park. And for locals there is the benefit of a 10-lane 50m pool and a 25m diving pool - with its three platforms like matching electric toothbrush chargers in his, hers and child's sizes - for use at less than a fiver an hour for adults at peak times. So, for all the controversy that surrounded a cost increase that seemed to symbolise the uncontrolled spending on London 2012 as a whole, Hadid's project turns out to have a satisfactory legacy. That may be of some comfort to the organisers of the 2020 Olympics as they contemplate the row that blew up this week when an eminent 83-year-old Japanese architect, Arata Isozaki, issued fierce criticisms of Hadid's design for Tokyo, calling for something more appropriate to its setting. Other leading Japanese architects, led by the 86-year-old Fumihiko Maki, had already gathered 32,000 signatures on a petition of protest. Given that Japan's architects have a considerable reputation to protect, the obvious countercharge is one of jealousy. Hadid's commission came as a result of her victory in a competition judged by a 10-man panel of whom the only non-Japanese members were two Britons: Sir Norman Foster - the designer of the new Wembley - and Sir Richard Rogers. A round of cost-cutting has already forced Hadid to modify her design in order to reduce the budget from an extraordinary - let's say obscene - £1.8bn down to £970m, which is still almost twice that of London's flat-pack 2012 main stadium. The revised design, although lower than the original 70m height, still mocks the 15m limit that is supposed to govern the Meiji Jingu Park area, which includes a famous 100-year-old Shinto shrine. It is hard to take an unambivalent view of Hadid's work. Her most breathtaking building, the Heydar Aliyev Centre in Baku, was built to the glory of a thoroughly unpleasant regime. Her addition to the Serpentine Sackler Gallery in Hyde Park is like something scrawled on the back of an envelope and tossed to an assistant for completion via parametric design software. Big architecture is seldom neutral in impact - when it is, it usually isn't much good - and the stadiums for Olympic Games and World Cups occupy a particularly exposed place in our world. However many years" notice the planners are given, they are seldom able to make accurate budget projections or to devise a long-term plan for future use. Amid the gnashing of teeth over the extra expenditure incurred in converting the London 2012 stadium before making a virtual present of it to a Premier League football club, perhaps we should be grateful it has not shared the fate of Santiago Calatrava's lovely Athens 2004 stadium, now a wasteland, or even Beijing's 2008 Bird's Nest, designed by Herzog and de Meuron in collaboration with the artist Ai Weiwei but today not much more than a tourist attraction. Once you get past the stuff about "perceptual density" and "vectors of transformation," Hadid's Tokyo stadium raises all the old questions about finding the right balance between human ambition and the proper use of resources. And by insisting on infusing her designs with such blatant drama, she forces us to join a worthwhile debate.