Extra Miler: Back to the Future.

The sun rises on the streets of Nişantaşı, and so begins another day in the 21st century. This upmarket neighbourhood speaks of a sleek, modern Istanbul, a city of chic boutiques and orderly streets, where Zadig & Voltaire jostles for space with Alexander McQueen, and art nouveau apartment buildings are turned over to elegant eateries and hotels.

Walk a pedicured hundred yards here and you could be in Paris or Vienna. “But every street in this city behaves differently,” says Asli Altay. “One minute you’ll be in a contemporary glass-and-steel construct by the Bosphorus, the next at the bottom of a Byzantine alleyway, continually moving from future to past and back again.”

Time travel. It’s a fiction, surely? Not in Istanbul – a city that bridges eras, empires, religions, cultures and continents, where Greeks, Persians, Romans, Byzantines and Ottomans have come and scattered their keepsakes, where the historic Christian world met its Muslim counterpart, and where the skyline stops you in your tracks with its striking mix of neon-lit skyscrapers, 14th-century towers and sixth-century minarets.

Local couple Can and Asli Altay, he an artist and academic, she a graphic designer, are representatives of a design-savvy, forward-thinking Istanbul: living with their Persian cat, Petit Beurre, in polished Nişantaşı; working from a smartly renovated 19th-century workshop in buzzy Galata (the headquarters of Asli’s graphic-design studio, Future Anecdotes Istanbul); and collaborating on creative ventures that involve anything from installing mirror-ball sculptures in unsuspecting venues to using public art to contextualise the role of a city park.

Big ideas. Immense talent. Can and Asli are heralds of the city’s future. But one of their greatest sources of inspiration is its past. “On the weekend we often drive into the old city for a change of perspective,” says Can, “over the Galata bridge, the glittering Golden Horn, back through the centuries. Istanbul’s history is capti-vating. You can peel back the layers slowly, or encounter them suddenly and unexpectedly, but you’re continually experiencing different eras of time.”

Today their cheerful vehicle for time travel is their Ice Blue MINI Cooper S Cabrio, its roof rolled down, and after firing up the engine they motor away in the sunshine from the quirky concept shops of Galata towards the ancient marketplaces of the bazaar district.

Once over the Galata bridge – a link between present and past – Can and Asli enter a different world. It’s a world of mosaic-adorned Byzantine palaces and magnificent Ottoman mosques, where locals smoke hookah pipes and play backgammon in traditional tea gardens, and where the frescoed ceilings of old hammams have been dripping with steam for 600 years.

They pass beneath a Roman aqueduct, provider of water to the former Constan-tinople; close to the exotic Topkapi Palace, stalked by the ghosts of sultans, courtiers and concubines; beyond the splendid, architecturally arresting Aya Sofya mosque, unimaginable in its scale; and back towards the tranquil stone streets surrounding the Süleymaniye mosque and the colourful, chaotic Grand Bazaar.

Parking their MINI in a nearby street, Can and Asli wander through the bazaar’s labyrinthine network of covered streets, some dating back to the 1400s when the market was constructed for the trading of textiles. “I come here sometimes to buy materials for my work,” says Asli. “You can still find beautiful handcrafted products. But often you’ll see an old man weighing saffron next door to an iPhone repair guy. For me it’s the perfect illustration of how the city straddles old and new.”

Outside the bazaar the air is thick with the aroma of roasted corn and chestnuts. Street vendors peddle simit pastries and pickled cucumbers; others squeeze pomegranates by the dozen to satisfy the sunny city’s endless thirst. Stray cats slope silently across the rooftops. Workers haul merchandise on bent backs. And, beyond the racket of commerce and chatter, there’s the low, lingering sound of a muezzin’s call to prayer.

“Being inside the old city is an extraordinary cultural experience”, says Can, “and makes you think of the long centuries through which scenes such as these have played out.” It’s only when Can, Asli and the MINI Cooper S Cabrio sweep back over the Galata bridge, however, past the fishermen casting lines for bluefish, the boats on the Bosphorus, returning to the brave new world, that the sharp juxtapositions between old and new Istanbul begin to fully reveal themselves.

Immediately on the other side of the Golden Horn – in stark contrast to the old city – is Karaköy, the latest up-and-coming neighbourhood of the Beyoğlu district, where hardware stores rub shoulders with cultural institutions such as artSümer and Istanbul’74. A friend of Can and Asli’s recently reopened the 16th-century Kılıç Ali Paşa hammam here, sprucing it up with a state-of-the-art spa, while the nearby Istanbul Modern, daddy of the city’s contemporary movement, sits in a looming warehouse down at the docks.

Driving north up Bog˘azkesen Caddesi, the Altays swing right into the leafy and sun-dappled streets of the antique district, where hip bars are popping up among the piles of plates and tarnished Aladdin’s lamps. They stop for tea at the rooftop lounge of The House Hotel Galatasaray, one of the many establishments designed by the acclaimed Istanbul interior design studio Autoban, then journey north-east to the Santral campus in Eyüp, where Can is the director of the Department of Product Design for Istanbul Bilgi University, to pick up some books.

Today’s Istanbul has a hunger for putting old spaces to new use, and Santralistanbul – a former Ottoman-era power station turned university campus and cultural powerhouse – is a perfect example. Comprising an energy museum, amphitheatre, concert hall, modern art museum and library, the 118,000-square-metre site was exhaustively converted by cutting-edge architects Nevzat Sayin, Emre Arolat and Han Tümertekin. “They’ve preserved the historical integrity of the site,” says Can, “but it’s bravely contemporary, too.”

And, if Santral represents modernity in Istanbul, buildings such as the Kanyon shopping centre in the Levent business district, and the brand-new Varyap Meridian mixed-use development on the Asian side, are positively futuristic: the former a curvy masterpiece designed by the experiential Jerde Partnership, based in Los Angeles; the latter a group of zany blue and orange structures that would look more at home in Dubai. See the ancient minarets meeting space-age buildings such as these, and the notion of time travel in Istanbul begins to resonate.

Later, as the MINI glides effortlessly back along the Golden Horn, Can and Asli reflect on their day. “This is an ancient and primitive city, with a mighty memory, but it’s a contemporary creature, too,” says Can. “The face of the city changes constantly, with old buildings and neighbourhoods reinventing themselves, but while it was Galata yesterday, and Karaköy now, it will be Balat, probably, tomorrow.”

As evening draws in, the Altays stop at their apartment in Nişantaşı to feed a purring Persian cat, before taking a last drive over the Bosphorus to the Asian side, to visit their favourite place for dinner.

Suna’nin Yeri, in Kandilli, is a simple seafood restaurant set on the water, under the plane trees, with white paper tablecloths, twinkly-eyed old waiters and an unfussy menu of fresh mackerel, salad and bread rolls. “Here, the continual leaping back and forth from past to future stops momentarily,” says Asli. “Everything is still.”

But tomorrow is another day, and Istanbul will keep moving – an old friend, a new love, a bridge between what was and what will be.