Liang and co begin by re-creating a virtual version of the painting. “We depict the scene geometry using a rough approximation for the subject’s body along with more detailed representations for the orb and the hand holding it,” they say.

By comparison with the hand, they estimated the diameter of the orb to be 6.8 cm and its distance from the body to be 25 cm. They also refined the geometry of the orb-holding hand to make it touch the orb softly, using Maya, a type of 3D modeling and animation software.

By studying the shadows in the painting, the team concluded that the subject was lit by a strong directional light source from above as well as by a general diffuse illumination. At the same time, they estimated that the viewpoint in the picture is about 90 cm away from the subject.

“With the virtual scene ready, we tested whether the orb was solid by comparing renderings of a solid and a hollow orb,” say Liang and co.

A. Rendering of a solid sphere B. Rendering of a hollow sphere

The results make for interesting reading. The only way the team is able to reproduce the original painting is with a hollow orb. Furthermore, a hollow orb distorts the background in a specific way. For example, a straight line that passes through the center of the orb is not distorted. By contrast, straight lines that do not pass through the center of the orb are distorted in a way that creates a discontinuity at its edge.

In the painting, Christ’s robes are folded so that five lines appear to pass behind the orb. However, four of the lines have a fan-like arrangement that converges at the orb’s center. Consequently, there is no discontinuity visible in the reconstructed image or in the original.

However, the fifth fold does not follow this pattern, and the reconstructed image shows a clear discontinuity. The artist blurred this part of the painting where the fold enters the orb. This strongly suggests that he was aware of the way a hollow sphere distorts straight lines that pass behind it.

The team also experimented with varying the hollow orb’s thickness, the results suggesting that it cannot have been thicker than 1.3 mm.

An interesting question is whether Leonardo would have had access to the materials, light sources, and knowledge of optics that the new work suggests he must have had. On the point of optics, Liang and co have studied Leonardo’s notes and think this knowledge must have been well within his grasp. Hollow glass balls were well known at the time and appear in many paintings of the era. And Renaissance artists were experts in re-creating certain lighting conditions.

So Liang and co are sure of their conclusion: “Our experiments show that an optically accurate rendering qualitatively matching that of the painting is indeed possible using materials, light sources, and scientific knowledge available to Leonardo da Vinci circa 1500,” they say.

Of course, the team are not the first to suggest that the orb is hollow—Leonardo’s 2017 biographer Walter Isaacson makes a similar suggestion, and others have discussed it too. However, Liang and co are the first to show that the painting is a physically realistic rendering of a hollow orb and not a solid one.

That will help settle at least some of the controversy over the picture and its whopping price tag.

Ref: arxiv.org/abs/1912.03416 : On the Optical Accuracy of the Salvator Mundi