At first sight, the Louvre’s grand Leonardo da Vinci exhibition, the most keenly awaited event of his 500th anniversary year, might look like a wasted opportunity. Disputes with Italian museums – and a general resistance to letting the gallery that happens to own the Mona Lisa appoint itself Leonardo Central – mean there are fewer paintings here than in the blockbuster show at the National Gallery in 2011. But you soon realise this is no mistake – or it’s a very happy one. This is the great Leonardo show of our time because it reveals his true identity as a scientist, inventor, engineer and infinitely curious observer of life. It’s a blockbuster with a brain that reveals why we will never tire of the genius who dreamed of our future, five centuries ago.

The greatest Leonardo work that France owns is not the Mona Lisa. It’s a medium-sized notebook known as Manuscrit B that’s been lent by the Institut de France. It is displayed open on a double spread that had me inwardly whooping with delight. You go, Leonardo! On the left hand page, there’s a design for a helicopter that works like a giant screw drilling upward through the air, drawn precisely in firm ink lines. Above is a rapid sketch of what looks like a flying saucer but is an armoured car that he suggested might be fun for jousting.

Facebook Twitter Pinterest Takeoff! Drawings of one of Leonardo’s flying machines. Photograph: Benoît Tessier/Reuters

Leonardo’s windmill-punk visions get even bolder on the opposite page. A man is standing in the wooden cockpit of a flying machine as he uses his own physical strength to flap its wings up and down. While the chopper and the tank were daring fancies, this scheme for an ornithopter – a flying machine that imitates a bird – would obsess him all his life, becoming increasingly plausible as he improved it. That’s why this manuscript is more important than the Mona Lisa. It shows Leonardo, soon after he moved from his native Tuscany to the powerful military-industrial state Milan, turning from a painter into a polymath.

This is the Leonardo da Vinci the Louvre lets you see in an exhibition that lays out exactly why it’s worth making a fuss over his quincentenary. Leonardo died in France in 1519, after being invited to spend his last years as court artist to King Francis I. His biographer Giorgio Vasari said Francis treasured him as a “philosopher” and loved to talk to him about ideas – in other words, Leonardo was valued for his mind. Yet half a millennium later, the art world promotes a much more conservative image of this “old master”. The Da Vinci story of our time is Salvator Mundi, a forgotten picture of Christ holding a translucent orb that was restored, reattributed to Leonardo and auctioned for £342,182,751, setting a record as the most expensive artwork auctioned. Is it the real thing? And is it in this exhibition, as planned? We’ll get to that. But one thing needs saying: it does an injustice to Leonardo to fetishise his paintings when he was the first conceptual artist.

Facebook Twitter Pinterest Timed for selfies … the Mona Lisa at the Louvre. Photograph: Grzegorz Czapski/Alamy

As I waited for the press view I visited the heart of darkness of today’s painterly Leonardo cult: the Louvre’s permanent display of the Mona Lisa in its glass-fronted case. The hubbub to see it is so unmanageable that you are allowed what feels like only 30 seconds before being moved on. This is not long enough for anyone to engage with art. It’s timed for selfies, not the brain. This madness is where a conservative reverence for Leonardo’s shiny, finished oil paintings has led us.

The Louvre has redeemed itself with an exhibition that captures the real flow of Leonardo’s mind. Here are pages of the Codex Atlanticus from Milan’s Ambrosian Library that show him exploring geometry and optics, drawings of the moon and mountains, and always, his fascination with flight. Leonardo was convinced humans could learn how to fly like birds. Some of the most stunning drawings in the exhibition observe in astounding detail how birds use updrafts and prevailing winds to enhance their flight. His reasoning is that if a flying machine can work with air currents, it too might get the lift it needs.

Facebook Twitter Pinterest Dark eye pits and decay … Study of The Hanging of Bernardo di Bandini Baroncelli, by Leonardo. Photograph: Benoît Tessier/Reuters

These analyses anticipate, by almost four centuries, the first sequence of photographs of a bird in flight taken by Eadweard Muybridge in the 1880s. Leonardo was a camera; his ability to look and draw was so extremely developed he could record everyday life in a way that collapses time. We are beside him in Florence in December 1479 as he sketches the hanged body of Bernardo di Bandini Baroncelli, a participant in the Pazzi conspiracy against the city’s ruling Medici. The dark pits of his eyes and the emaciated sheen of his decaying cheeks are captured in a few pen strokes. Leonardo even lists the exotic Turkish garments Baroncelli was hanged in after being brought back from Constantinople to face Lorenzo de’ Medici’s revenge.

Leonardo is a movie camera, too. Drawings of babies playing with cats made in Florence around 1480 show motion as a blur of positions sketched as life happens – a multiple image unique to Leonardo. It also appears in his drawings of Saint Sebastian and a fight between a knight and dragon. These early sketches show him trying to draw as fast as life and somehow put the full complex flow of reality on paper. This becomes a delirious overload in his designs for The Adoration of the Magi, a painting he was never to finish. Here he tries to use vanishing-point perspective to contain an infinite vision of battles, buildings and nature receding into ambiguous chaos. It’s dazzling. This boy from the Tuscan countryside had a brain that couldn’t stop. He would spend the rest of his life applying it in a singlehanded attempt to understand nature and invent the technological age.

Leonardo’s attempts to reconcile science and art made him take on many projects and finish few. All the important drawings and copies from his unfinished painting The Battle of Anghiari are in the show. You also encounter the psychotic war leader Cesare Borgia, who employed Leonardo as a military engineer, looking pensive and even perversely Christ-like in a red chalk drawing – a part for Joaquin Phoenix.

Surrounded by images as exciting as these, it is hard to care about Salvator Mundi. The owner of the world’s most expensive painting has not lent it, after all – which can only add to growing speculation about its dodgy provenance. But there’s worse news: another version, credited to Leonardo’s studio, is here along with designs from the Royal Collection. They look so staid and rigid compared with the restless, dynamic images of laughing babies, flying birds and rearing horses that fill this show. You realise it doesn’t much matter how genuine Salvator Mundi is because it was his dullest design.

Facebook Twitter Pinterest Leonardo’s last and greatest surviving painting … The Virgin and Child With Saint Anne, 1508-10. Photograph: Alamy

These relics are shown in the same climactic space where the Louvre’s St John the Baptist looks out of the dark with a fleshy intensity that’s both spiritual and erotic. On the opposite wall hangs The Virgin and Child With Saint Anne, Leonardo’s last and greatest surviving painting. To add to this sublime gathering, the National Gallery has lent its full-size drawing of the Virgin and Saint Anne with Christ and Saint John the Baptist – known as The Burlington House Cartoon. These marvels push at the boundaries of the known. The Virgin and Saint Anne achieves what Leonardo tried to do when he was young: contain the world in a painting. In the foreground, rocks and strata reveal his interest in geology. In the distance, blue Alps melt into a crystal atmosphere. It’s like a model planet Earth. At the centre of it all are women and children – a mother, her son, his grandmother. Why are there two maternal presences? Freud observed that Leonardo had two mothers – his birth mother Caterina, a country girl his father never married, and his stepmother Albiera di Giovanni Amadori. Is Leonardo remembering his childhood in his last and most satisfying painting? In The Burlington House Cartoon, we see those two mothers but the older woman has a face as deathly as a hanged man’s. It is a vision of hope and despair.

But the Louvre can’t completely ignore Lisa. You can put on a virtual reality headset to meet the “real” Lisa del Giocondo, the Florentine silk merchant’s wife who posed for Leonardo in 1503. It’s freaky to encounter in full VR an actor whose face has been digitally altered to give her the Mona Lisa’s long fine nose and deep-set eyes.

What a gimmick – and how Leonardo da Vinci would have loved it. He’d probably prefer this techno Mona Lisa to his own painting. One thread that connects his experiments is the belief that the mathematics of perspective can mirror and make worlds. Virtual reality’s illusion of solidity is a Renaissance dream come true. It’s impossible to exhaust Leonardo’s science-fiction prophecies, and this exhibition joyously opens his notebooks to us all.