‘Look at these,” says Waqas Khan. “I call them Jali. They’re Islamic windows.” I look, as he urges, at the unglassed stone windows from Agra that are installed all around us in the Victoria & Albert Museum’s South Asia gallery. If you have ever visited it, you will have seen them, but not everyone looks as closely as Khan. “Look at the repetition,” he says. “You can see it in all these patterns.” He means this as praise – and it’s easy to see the beauty of that apparently infinite repetition. The yellowish sandstone window grilles have a dazzling range of designs – floral, crystalline, triangular – in every window, interlocking and extrapolating and expanding with a complex regularity that looks as if it may go on for ever, far beyond these cooling lattices. The first time I saw the work of Waqas Khan was in the Jameel prize here at the V&A in London. His microscopically precise yet vastly conceptualised abstract drawings took the art heritage of the Islamic world into a modern, ethereally beautiful new vision. Since then, he has expanded his art into big abstract canvases, adding a painterly feel for colour to his graphic brilliance. He is one of the most exciting abstractionists of our time. He’s also an artist with rich historical roots. We are here to look at the V&A’s remarkable collection of art from India and Pakistan ahead of The Great Mughals, a blockbuster exhibition that overlaps with Khan showing his own work in the UK, at Purdy Hicks in London. The Great Mughals aims to unveil the art and architecture of the early Mughal empire, from 1560 to 1660. If that sounds specialist, think again. In fact, think Taj Mahal. That unique building in Agra, where all the V&A’s Jali come from, is indisputably among the world’s most beautiful. It was raised by the Mughal emperor Shah Jahan in 1631 and its famous story is wildly romantic: he built this filigree weightless wonder as a tomb for his favourite wife, Mumtaz Mahal. Its delicate embodiment of passion is typical of the early Mughals, who liked to have themselves portrayed literally smelling the flowers. “They loved the beauty around them,” says Khan. We move on to look at a fantastical painting of a prince fighting a demon under a brilliantly flowing, joyously unreal pattern of blue sky and white clouds. “This is from the Hamzanama,” says Khan. It’s one of the superb illustrations commissioned by the emperor Akbar in the mid-16th century for a book about the life of Muhammad’s uncle. As well as building delicately gorgeous pavilions, palaces and tombs in Agra, Lahore, Delhi and more – even their massive forts are ravishing inside – the early Mughals sponsored a golden age of miniature painting, the figurative Muslim court art that had originated in 14th-century Persia. What does he think of so much art from Pakistan and India being owned by the V&A, which inherited much of it from the East India Company’s India Museum? “At least it’s preserved,” he says. “Now we have museums, now we are working on those things, but at least it has been preserved.” Khan in fact trained as a miniaturist. He explains how you learn to paint a Mughal miniature, by imitating one put in front of you: “They give you this and you have to copy it. It’s done with a squirrel brush.” He was trained to do this at the National College of Arts in Lahore, Pakistan. He loves this city, one of the great centres of the early Mughals, and still lives there. But he was born in the Punjab countryside: “For me, going to that school was a cultural shock – the guy from the village going to this art college. None of the family knows about art, ‘What is art all about?’” He stresses that he didn’t just study miniature painting: it was a module of a much broader course. I have visited the college with him to see where he worked on miniaturism as a student. It is a 19th-century building whose founding principal was John Lockwood Kipling, father of Rudyard. Like much of the British imperial architecture in Lahore, the National College in fact emulates Islamic art, creating a neo-Mughal style. So you pass through sunny courtyards and shady arcades to find a calm white-walled workspace with a high gallery reached by an old staircase. Here, Khan showed me the small space where he first fell in love with the intricate, meditative practice of miniaturist work. “I loved the space,” he says. But Khan applies Mughal skills of precision, of calligraphy even, and turns them into abstract art. He spends many hours a week drawing in his studio in what he says is not just work but a “meditative” practice. The lines he weaves are free and abstract. He’s not drawing princes and demons but lattices and labyrinths, exact yet seemingly infinite, shimmering circles and lines of unreadable script. He feels a true affinity, he says, for the abstract patterns of the big rugs hanging above the V&A’s Mughal miniatures: he wants me to look at their complex repeating designs. We follow the magic carpet ride through from the South Asia gallery to admire abstract designs from across the Islamic world, gazing at the interlaced wood-and-ivory stars on a 15th-century minbar (a stepped pulpit from Cairo) and at the vast Ardabil Carpet, woven in 1500s Persia by “the slave of the portal, Maqsud Kashani” (as it is signed), with its stupefyingly subtle blend of realistic flowers and algorithmic self-reproduction. All the “repetitions” Waqas points to in Mughal, Persian and North African art have the unstoppable energy of an algorithm that will never stop or end. The Persian word algorithm is named for Muḥammad ibn Mūsā al-Khwārizmī, who wrote on these computational instructions in the ninth century as Islamic art was taking shape. Khan seems to channel this algorithmic beauty. Afterwards, we walk to the nearby Purdy Hicks gallery where he has some work in a group show. They are brilliantly enigmatic blue canvases in which white unreadable script – like ghost calligraphy – and interlaced cosmic circles like the Jali from Agra, hold your eye, mind and soul. He has recently looked hard at Mark Rothko’s paintings: we talk about the Rothko Chapel in Houston, where you can read the Qur’an, Torah, Christian Bible – or simply nothing – as you contemplate his art. There is an affinity between these blue canvases and Rothko: they are vertical layerings of rectangular shapes that create a hypnotic aura. It comes partly from the amazing colour built up with extreme subtlety to generate depth. Over this deep colour are his painstakingly penned sequences of dots, forming bubbles and wordless writing. This is art to transport you. To his Mughal heritage, Khan has added the majesty of Rothko. Muslim history merges with American modernism. Lahore meets Manhattan. Khan’s work is uplifting and entrancing. He is an artist of peace in a world at war. The Great Mughals opens at the V&A, London, on 9 November. Waqas Khan’s work is at Purdy Hicks, London, until 16 November.