Walter Isaacson is chief executive of the Aspen Institute and a professor of history at Tulane University. His latest book is “Leonardo da Vinci.”

Leonardo da Vinci’s portrait of Jesus Christ, “Saviour of the World,” sold for $450.3 million on Nov. 15 at a New York auction. (Reuters)

The excitement surrounding this week’s auction of Leonardo da Vinci’s painting “Salvator Mundi” raises the question of what makes his work so special. Why do we consider him, rightly, a genius, rather than merely a really great painter? What could possibly justify a $450 million price tag?

Part of the answer, I believe, is evidenced in “Salvator Mundi” (“Savior of the World”): his ability to connect science and art. He was interested in everything that could possibly be known about creation, including how we fit into it. With a passion that was both playful and obsessive, he pursued studies of anatomy, geology, math, engineering, water flows, architecture, theater, optics and dozens of other disciplines. In so doing, he was able to see the patterns that were woven into what he called “the infinite works of nature.”

Those who regard him primarily as an artist lament that he was too often distracted from his paintbrush by these diverse pursuits. “It was a variety of employment which Leonardo enjoyed, but which left posterity the poorer,” the 20th-century art historian Kenneth Clark wrote. Had he not “squandered” his time dissecting bodies or trying to build flying machines, he might have finished more paintings. That is probably true, but it also would have meant that he would not have been Leonardo da Vinci.

His artistic and scientific endeavors are profoundly interwoven in “Salvator Mundi.” One of fewer than 20 finished works, it was painted in the later years of his life, after 1500, when he was doing his most intensive work on anatomy, perspective, geology and optics. In conjunction with his painterly skills, these studies serve to give “Salvator Mundi” its qualities of genius. Like Leonardo’s beloved intricate knot patterns that adorn the garments of Jesus in the painting (and the “Mona Lisa”), the disparate strands of his interests were inextricably linked.

Leonardo’s feel for the patterns and connections of nature helped him to create, in “Salvator Mundi,” a face that manages to be at once both reassuring and unsettling. Jesus’ stare follows us as we move, and it seems to interact with us. Is he staring at us or into the distance? Move from side to side and ask again. Is there a hint of a smile? Look again. Is his blessing hand moving out at us?

Leonardo combined optics with perspective to convey a three-dimensional image on a two-dimensional panel. This ability to “make a flat surface display a body as if modeled and separated from this plane,” he said, was “the first intention of the painter.”

To pursue this goal, Leonardo in his notebooks drafted a long treatise on perspective. In it he explored the math of linear perspective — how objects get proportionally smaller as they get more distant — that had been honed by his predecessors Brunelleschi and Alberti. But Leonardo also wrote a long section on another aspect of perspective, which he called acuity or sharpness perspective. “You must diminish the sharpness of those objects in proportion to their increasing distance from the eye of the spectator,” he instructed. “The parts that are near in the foreground should be finished in a bold determined manner; but those in the distance must be unfinished, and confused in their outlines.”

From his optics studies he knew that light does not register in a single point in the eye but instead is perceived along the entire retina. Combined with the fact that we have two eyes, this means that we perceive the edges of objects as slightly blurred rather than sharply delineated. This led to his mastery of sfumato, the gentle blurring of lines in his paintings. We see that technique in blurred outlines of Jesus’ face.

But Leonardo also described in his optics notebooks how objects two or three feet away from our eyes look more sharply delineated, because that is our eyes’ focal point. This explains why Leonardo made the blessing hand of Jesus in “Salvator Mundi” so much sharper. Look at it carefully. It seems to be closer to us than the rest of the painting, as if it is coming toward us in blessing. It is a technique that Leonardo used in a similarly alluring late painting of John the Baptist, whose pointing hand seems to be coming out at us.

When painting “Salvator Mundi,” Leonardo was spending his nights in the morgue of the hospital of Santa Maria Nuova in Florence, peeling the flesh off the faces of cadavers and drawing the muscles and nerves underneath. He became fascinated by how a smile begins to form, and he instructed himself to analyze every possible movement of each part of the face and determine the origin of every nerve that controlled each facial muscle.

One page of anatomical drawings from around 1508 shows a pair of lips in an open-mouthed grimace and then drawn pursed. The muscle that purses the lips is the same muscle that forms the lower lip, he discovered. Move your lower lip and you can see that this is true; it can pout on its own, with or without the upper lip, but it is impossible to pout the upper lip alone. Then he drew a row of lips with the skin layer peeled off. At the top of this page is something delightful: a simpler drawing of a gentle smile, sketched lightly in black chalk. Even though the fine lines at the ends of the mouth turn down almost imperceptibly, the impression is that the lips are smiling. It was the beginnings of the smile he would use on all three of his late-life portraits: “Salvator Mundi,” “St. John the Baptist,” and the “Mona Lisa.”

Leonardo Da Vinci’s “Mona Lisa" on display in the Louvre museum in Paris. (Pascal Le Segretain/Getty Images)

"St. John the Baptist" by Leonardo Da Vinci presented at Milan’s Palazzo Reale museum as part of a 2015 exhibition. (Giuseppe Cacace/Agence France-Presse via Getty Images)

All three of these similar smiles are made elusive and eerie by Leonardo’s study of light rays and the retina. The central area of the retina, known as the fovea, is best at seeing color and small details; the area surrounding the fovea is best at picking up shadows and shadings of black and white.

When we look carefully at the smiles of Leonardo’s three portrait subjects, especially the left sides (from our vantage) of each set of lips, there are tiny black-and-white details that turn downward. Yet the shadows and colors turn upward. If you stare directly at the mouths, your retina catches the tiny details, making the smiles elusive. But if you move your gaze slightly away from the mouths, the shadows make the lips seem to turn upward into a subtle smile. The result are smiles that are interactive, augmented realities that flicker brighter the less you search for them.

The final scientific and artistic mystery involves the crystal orb that Jesus is holding in “Salvator Mundi.” It is rendered with beautiful scientific precision. There are jagged bubbles visible on the bottom right that have the irregular shape of the tiny gaps in crystal that are known as inclusions. Around that time, Leonardo was evaluating rock crystals for the wealthy art collector Isabella d’Este.

But Leonardo failed to paint the distortion that would occur when looking through a solid clear orb at objects behind it. Solid glass or crystal, whether shaped like an orb or a lens, produces magnified, inverted or reversed images. In “Salvator Mundi,” even the tiniest details and lines of the robe when closely inspected show no hint of diffraction.

Why did Leonardo not show the orb producing any distortion? One explanation is that he had not noticed how light is defracted in a solid sphere. That’s wrong. He was, at the time, deep into his optics studies, and the ways light reflected and refracted was an obsession. Scores of his notebook pages are filled with diagrams of light bouncing around at different angles.

Another explanation, more plausible, is that he knew the orb would distort, but he chose not to paint it that way because he thought it would be a distraction. It would indeed have looked weird. But I prefer a third explanation: I think that he knew how the orb would distort and that some of his more sophisticated viewers would also know, but what he is doing in his portrayal of Christ as savior of the world is trying to convey the miraculous quality of his stewardship, that nothing he touches is distorted.

Leonardo’s knowledge of science and engineering did not make him less open to mystery. Quite the contrary. Appreciating nature’s patterns allowed him to combine imagination with reality. It is true that he might have been more productive if he had been — as we sometimes push ourselves and our children to be — more focused and more specialized. But when you interact with the face of Salvator Mundi or gaze into his crystal ball, you get a sense of why that would have made him less of a creative genius — and why his works command the astronomical amount reached in Wednesday’s auction.