Imagine a world where buildings don’t exist. A place with no rustic cottages nor gleaming skyscrapers, no classicism nor modernism, no preconceived idea of what a house or an office block should be. How would we make spaces from scratch if anything was possible? Freed from the conventions of architecture and construction, what would this world look like?

It’s a question Japanese architect Junya Ishigami has been trying to answer for the past decade, dreaming of structures that are as light as a cloud, as vast as the sky, as random as the trees in a forest or the stars in the sky. Many architects are fond of using nature metaphors, of attempting to blur the inside and out, but Ishigami is determined to create another nature altogether: by designing buildings that are as primitive and strange as natural phenomena.

As one of the most experimental of Japan’s younger generation of architects, his poetic visions have often seemed too radical to be realised, destined to be confined to the realms of books and installation art. In a gallery in Tokyo, he made a five-storey cuboid metal balloon that actually floated; and, at the Barbican in London, he installed a white frame so thin and delicate that visitors could only see it when invigilators dressed in black stood behind its columns. A similarly slender structure for the 2010 Venice Biennale collapsed just before the opening, leading sceptics to question if it was ever really there. Its title, Architecture as Air, didn’t help.

So it comes as a surprise to learn that this structural alchemist and defier of physics is now erecting substantial structures around the world, from a museum in Moscow to a monumental arch in Sydney. And, as a new exhibition of his work at the Cartier Foundation in Paris shows, they are buildings but not as we know them. “If we forget everything we know,” says the 43-year-old, standing in the foundation’s airy glass galleries, “just imagine how many more kinds of architecture there could be.”

Facebook Twitter Pinterest Junya Ishigami at the Fondation Cartier. Photograph: Renaud Monfourny

He’s surrounded by fragments from his fairytale worlds: ribbons of looping white steel dangle from the ceiling, above models of gossamer-thin rooftops draped over sunken spaces, like clingfilm stretched over a bowl. A gnarled concrete cave stands on knobbly legs, next to a menagerie of animals fashioned from kitchen foil. It is part scientific laboratory, part creche.

Children, and how they see the world, are a constant inspiration to Ishigami, in his mission to free himself from preconceived notions. “Imagine how the world would seem if you were a really tiny person, like an elf,” he says, leading me over to his collection of foil creatures. “Animals would seem like huge landscapes.”

In his design for an indoor children’s park in Japan, a dog becomes a big roof, a bear forms a dome, while the gaping mouth of a hippo makes a cave. He has deployed a similar approach for a kindergarten in China, where the plan of the building has been made by collaging together animal silhouettes. You can make out the tentacles of an octopus, the belly of a whale, the jaws of a crocodile – all merged to form an undulating landscape where floors become roofs and some spaces are so low-level only children can access them. Others, even smaller, are only for animals. Forming a loosely defined patchwork of ramps, plateaux, steps and caves, these spaces are whatever the kids and their imaginations want them to be.

Facebook Twitter Pinterest ‘Like a forest punctuated with clearings’ … the Kait Workshop, in Kanagawa, Japan. Photograph: Edmund Sumner/VIEW Pictures/Alamy

Ishigami is a product of the Pritzker prize-winning practice SANAA, where he worked for four years under Kazuyo Sejima, 61, who is herself a product of Toyo Ito, 76. The latter’s genre-defying buildings – from the coral-like columns of the Sendai Mediatheque in Japan to the labyrinthine cave of his opera house in Taiwan – are clearly a touchstone for Ishigami . Each new generation of this architectural family tree seems to strip back a layer, shedding the ballast of their forebears to float ever closer to the clouds. Ishigami has come the closest to weightlessness yet.

In 2010, several years after setting up his own office, he was catapulted to international recognition with his first building: a workshop for the Kanagawa Institute of Technology in Japan. Rather than erect the usual big shed, he created a glass-sided building containing hundreds of slender steel columns arranged randomly. It was like a forest punctuated by small clearings, for chairs and tables. “A lot of students go there to work alone,” he explains, “so I wanted to create a feeling of enclosed personal space, while keeping the sense of an open hall. No matter how big a space has to be, it should always be designed from the individual scale.”

Facebook Twitter Pinterest The planned Sydney Cloud Arch. Photograph: junya.ishigami+associates

His 60-metre-high Cloud Arch for Sydney was developed on the same principle and looks a little like something Salvador Dalí applied his melting powers to. Its drunkenly swaying silhouette (“moving like a cloud”) is designed to form a different shape from every possible viewpoint around the city.

Next door to the Kanagawa workshop, meanwhile, Ishigami is about to begin construction on his most ambitious project yet: a multipurpose covered square for student sports, parties and barbecues that promises to be one of the most unlikely structures ever built. “I made the forest,” he says, “Now I want to make the sky.”

His sky will take the form of a continuous plane of steel, just 12mm thick, hanging over a space 110m long and 70m wide – without a single supporting column in sight. Due to steel’s expansion and contraction, the height of the covered space will vary from 2.3m to 3m, depending on the weather, while a scattering of square holes will mottle the interior with pockets of sun and shade – and allow rain to fall inside. The shallow bowl-like floor means you’ll never see any end walls: they simply merge into one, in a continuous interior landscape boasting its own clouds and horizon. A 20m-long model in the exhibition gives you an idea of what a miraculous feat this will be, what an unusual space he will create beneath the huge drooping meniscus.

When Ishigami is not busy creating his own version of nature, he tries to preserve or amplify what is already there. On a rocky hillside on the edge of Dali in southwest China, he has designed a series of eight holiday homes in an area rapidly being denuded of its countryside. Rather than remove the big boulders and start with a clear site, as any sane architect would, he has meticulously arranged the rooms and furniture of the homes around the great rocks, all to be topped with a 300m-long wafer-thin roof. The natural boulders become structural pillars, forming a megalithic indoor landscape.

Facebook Twitter Pinterest Ishigami’s new glass pavilion for Park Groot Vijversburg, in the Netherlands. Photograph: junya.ishigami+associates

It comes as no great surprise to learn that Ishigami is obsessed with plants. In his beautiful pencil drawings, he often pays more attention to petals and leaves than he does to the building they surround. He says you should approach designing a house “as if you were putting together a garden, a little at a time”.

For a park in the grounds of a historic building the Netherlands, where no trees could be removed nor paths altered, he built a sinuous glass pavilion, again without columns, whose walls follow the line of the existing paths, disappearing into the trees. “I just turned the footpath into a space,” he says, making the elaborate feat sound effortless .

As he takes me around the show just days before it opens, armies of assistants are busy with tweezers, meticulously placing tiny pieces of coloured tissue paper on the models, to give them the desired hand-rendered finish. He doesn’t talk of designing buildings but of “making scenery”. And the simplicity with which Ishigami explains his projects – a style echoed by the exhibition catalogue, which is designed like a large children’s storybook – belies the fiendish complexity behind the facade.

He is currently building a cave-like restaurant in Yamaguchi, Japan, that looks like something tunnelled into being by giant worms. It was created by “digging holes in the ground, filling them with concrete and burrowing out the soil around them”. The resulting fat curvy columns, which spread out as they rise, required seven years of planning before construction could even begin. “The client wanted an old-feeling space, like a wine cellar,” he says, “but I didn’t want to do a pastiche. So I thought about how to make something feel truly archaic by digging it straight out of the ground.”

Facebook Twitter Pinterest The strange, burrowing construction technique being used to build a cave-like restaurant in Yamaguchi. Photograph: junya.ishigami+associates

It is easy to leave this exhibition, appropriately titled Freeing Architecture, feeling liberated from the shackles of architectural history, from the messiness of cities, and even from the laws of physics. It is a tranquilliser dart of pure poetry. Some might find it all too cute, the childlike naivety a little cloying. Free of context, it can all seem a little too perfect: only when these buildings are finished will it be possible to assess how Ishigami’s dream worlds perform in reality. For now, though, it’s tempting to close your eyes and fall under the spell of one of the most original minds in a generation – and be transported to a place where other architectures are possible.