Barbican, London; Serpentine Gallery, London

This must be a first: an architectural installation which, at a total of 300g, weighs the same as its accompanying press pack. Also, an exhibition of architecture which is almost invisible, and beyond the power of photography to capture. Such is Junya Ishigami's Architecture as Air, just opened in the Curve at the Barbican.

Once you have taken your shoes off, and listened to the instructions to take extreme care, you see nothing except the curving gallery itself, its ventilation grilles, lights and electrical conduits. Then you see there are what you might assume are threads, at regular intervals, running from floor to ceiling, and after a while some diagonal threads, cobweb-thin, caught in the light, and that's about it. Emperor's new clothes in its purest form, you might think: nothing masquerading as something.

You will probably need to read the text on the wall, or notice a small, hand-written pencil annotation, to find that the verticals are not actually threads and do not reach the ceiling, but are columns. There are 53 in number, and they are hand-rolled from carbon-fibre sheet. They stand, not hang, braced by the cobwebby diagonals, of which there are 2,756. There are also beams running from column to column, making this the skinniest colonnade in history. "How much does your building weigh?" the American sage Buckminster Fuller used to ask, and he would never have got an answer like this. At the very least, the piece is an extraordinary piece of engineering, achieved with the help of Jun Sato Structural Engineers and, through testing in a full-size mock-up of the Barbican's gallery, erected in a shed in Yokohama.

"I want to make a new scale of architecture, a natural scale, an elemental scale," explains Ishigami, who is dressed in the traditional all-black of designers, but with added attitude. He has skinny jeans, ankle boots and hair escaping in all directions from under his black hat, like a stylish scarecrow, or the weird genius in a Manga story. The columns, he says, are about as thick as raindrops, and the bracing threads are as thick as drops of water vapour in clouds. Where classical architecture is based on the dimensions of the human, this is based on other sizes found in nature. He also says that "in nature structure and space are not divided. Air is space but it also has a structure. But architecture divides these things." He wants his structure, by being so thin, to become "like mist".

The Barbican says that the 37-year-old Ishigami is "internationally acclaimed", and there is certainly a buzz and fascination around him. Last year he won the Golden Lion, the highest prize at the Venice Architecture Biennale, for a structure that collapsed almost as soon as it was built, following an accident with a cat. Little was left but a scrawled note saying "Scusate, si è rotto. I'm sorry It's broken." (And as a result of this Kiplingian-feline experience, from disaster to triumph, from cat to Lion, extra-special care is being taken with the Barbican construction.)

Before that he charmed and intrigued his audiences with installations such as Balloon (2008), a box of aluminium the size of a four-storey house, and weighing a ton, which he filled with helium so that it floated in the galleries of the Tokyo National Museum of Modern Art, and could be moved at the touch of a finger. His most significant building to date is the University Project Space at the Kanagawa Institute of Technology, a single open space under a flat oblong roof, supported by irregular groupings of sapling-like columns which are mingled with real plants to create a forest-like effect. For the same campus he has designed a cafe, to be completed next year, which is the opposite – a wide, open, column-free space he describes as "a boundless landscape". His drawings of this project are like none seen from an architect before, with two zones of green-yellow scribble, in various tones, representing ground and ceiling, white silhouettes or people, and almost no sign of structure.

He likes certain themes, such as overlapping buildings and nature, like trees and columns, in such a way that they interact and the distinctions between them blur. He likes to make materials do things you don't expect, like the floating ton of aluminium and the impossibly thin columns. He is capable of doing things that you could not predict from his previous work. He has designed a home for old people with dementia, also to be completed next year, made up of fragments rescued from old houses that were being demolished all over Japan. The theory is that the residents "will naturally recognise the different features" of each structure, which will reduce the disorientation that comes with dementia. "I want to think of so many ways of making beautiful things," he says.

Ishigami used to work for the celebrated practice of SANAA. Kazuyo Sejima, co-founder of SANAA, once worked for the equally celebrated Toyo Ito. He is therefore in a lineage of architects who are preoccupied with what Sejima calls "atmosphere", with lightness and effects of light and appearance. He shares with them, and takes further, a mistrust of architecture's usual ponderousness, and a desire to make it disappear.

What makes Ishigami different are his conceptual leaps, like those of an artist. Much of his reputation so far is based on works in galleries, but he regards these and habitable buildings as all part of the same architectural enterprise, and there is something wonderful about his creative agility. His piece at the Barbican is one of his more cerebral works – you have to know what is going on to get it, and to see the technical achievement – and not the most engaging. But he looks like someone who, once he's sorted out his cat-proofing, really will achieve new ways of imagining and making buildings.

This year's Serpentine Gallery pavilion proceeds by negatives. It is not at first sight the English summer. It is not Wimbledon, not blazing white, not boaters, not Will and Kate. The thing it most resembles is the Chelsea flower show, but not quite like that either.

It presents a matt-black windowless wall, guarded from architecture fanatics before it actually opened to the public last Friday by a huge Slavic bouncer. It is finished in a rough, stained, textured surface, like a superior shed. Then you enter a dark oblong passage before you reach the point of the project, a rough-hewn but highly planned garden by designer Piet Oudolf.

This is the Chelsea bit, but different through its severity, pleasure held back until you see the plants. The blackness and plainness is intended to intensify the experience, including the sky above, the clouds, rain and sun, and other people. It is also unlike the Chelsea flower show in that it lasts longer, until October, and will, Oudolf says, "get better as long as it grows". It is unlike the Serpentine's usual pavilions in that, conceived in the era of spectacular corporate sponsorship, they are usually hurried things of partying and show. Peter Zumthor, the Swiss architect of the structure, usually likes to work slowly and intensely.

This means that there is – intentionally – not the density of detail that his work usually has, and there are things like fire hoses, highlighted as intensely as the sky, which he would rather weren't there. But there is also method in this combining of eminent architecture and plants and people.

Both Oudolf and Zumthor say the point of the space is to bring people together, which, to an unprecedented degree, it could do. And, with a magic rectangle of admittedly less vigorous action, it is like a slowed-down Centre Court after all.