The translucent orb in the Salvator Mundi (ca. 1500, attributed by some experts to Leonardo da Vinci) shows no inversion of the robe behind it, the passage of the robe folds across the orb boundary largely without visible break, a darkened interior, and three enigmatic white spots. A physically based rendering study concluded that the orb is most likely a hollow glass sphere with wall thickness at most about 1.3 mm (for a 6.8 cm radius), while a 2026 crystallographic study counters that the speckles are fluid inclusions characteristic of rock crystal (quartz) and the painting contains no significant optical errors, so no hollow sphere is needed. The open problem is a rigorous quantitative determination of which historically plausible orb hypotheses (solid quartz, calcite, or glass; hollow thin-walled glass) can reproduce the observed optical appearance, and which are excluded.

Background

The Salvator Mundi is a painting of Christ dated to about 1500, attributed by some experts to Leonardo da Vinci (authorship is still debated; the rendering literature writes "Leonardo?" to mark this). Since its 2017 sale for $450.3 million, a scientific debate has focused on the transparent orb in the subject's left hand. A solid transparent sphere optically denser than air acts as a thick convex lens: it should invert and magnify the image of the robe behind it and obscure the palm. The painting shows no such distortion, and the orb carries three small white spots. Martin Kemp (Nature, 2011) argued the orb depicts solid rock crystal whose cavity inclusions produce the spots; André J. Noest and Martin Kemp replied in a Nature correspondence titled 'No refraction in Leonardo's orb' (2011) that the absence of refraction is inconsistent with a solid orb, and the debate continued without resolution.

Liang, Goodrich, and Zhao (arXiv 1912.03416, 2019) approached the question with physically based rendering (PBR) — a computer-graphics technique that simulates light transport in a virtual 3D scene to produce optically accurate images, here using the Mitsuba renderer. They built a virtual scene with a relief proxy for the subject, an external hand model, and an orb of 6.8 cm radius — their value, scaled from the subject's left hand to match the relative size and position in the painting — 25 cm in front of the subject, lit by a strong directional light plus dim ambient fill. Comparing renderings, they concluded that the orb was most likely hollow, conditional on the painting following standard light transport: a hollow sphere (a thin glass shell, as produced by glass blowing) does not invert the background. They further found that of the five prominent folds crossing the orb's upper-right boundary, four are painted so that their edges, extended as straight lines, converge at the orb's center from the viewing position — the one geometry in which lines cross a hollow sphere's boundary without visible break — while the leftmost fold's extended edge misses that convergence point and is blurred in the painting; and they determined that for a hollow glass orb (refractive index 1.51714, the ratio of light's speed in vacuum to that in the material) of 6.8 cm radius with the folds following lines approaching the orb's center, the wall thickness must be at most about 1.3 mm to avoid visible fold distortion; a 2.6–2.7 mm wall already introduces a dark reflected layer. They also tested Kemp's solid-calcite theory, rendering birefringence (calcite's property of splitting light into two rays with different effective refractive indices, 1.658 ordinary and 1.486 extraordinary) and finding the solid-sphere distortion would dominate any double-contour effect. Their journal version (Leonardo, 2020) extended the analysis to the interior glow, the orb anomalies, and the three white spots, again supporting the painting's optical accuracy.

These studies were qualitative forward renderings with manually tuned parameters (gamma correction, adjusted transmission, chosen illumination), a proxy relief rather than reconstructed geometry, and no parameter estimation from imaging data of the actual painting, so their numerical bounds are conditional on the modeled configuration. In 2026, García-Ruiz and Modestini published a crystallographic and mineralogical study arguing the opposite: the millimeter-sized speckles are anisotropic fluid inclusions characteristic of rock crystal (quartz), quartz's birefringence is too weak to matter, a solid sphere placed very close to the robe avoids the inverted-image problem, and the painting "contains no significant optical errors" — so no hollow sphere or artistic-license excuse is needed. They estimate the painted orb's diameter at ca. 18 cm from the handspan, far larger than the crystal or glass balls known in 1500 (the largest attested comparanda being ca. 5–6 cm), and conclude that there is no evidence that a crystal sphere like the one painted existed in Leonardo's time. They also report that the paint layers at the upper part of the orb, where the diagnostic distortions would be evident, have completely perished due to harsh cleaning in the past, so that effect cannot be discerned in the present painting, although the Barberini painting and the Hollar engraving — both copies of it — do show it. A further speculative note (Sivak, 2025) asks whether Leonardo may have been depicting the optical quality of the spherical lens of a fish eye, a lens with a gradient refractive index; it is a speculation about depiction and intent, offered without quantitative test. The literature thus contains two incompatible conclusions — rendering-based evidence for a hollow thin-walled orb and crystallographic evidence for a solid rock-crystal orb — and no study has performed the quantitative optical analysis that could discriminate between them.

Problem Statement

Determine, by quantitative optical analysis of the painting, which historically plausible hypotheses about the orb of the Salvator Mundi are consistent with the painting's optical appearance, and which are excluded. The appearance to be explained comprises: the absence of inversion of the robe behind the orb (a solid orb optically denser than air acts as a convex lens and would invert and magnify that image); the behavior of the robe folds where they cross the orb boundary, including the four folds whose extended edges converge at the orb's center from the viewing position and cross smoothly, and the leftmost fold whose extended edge misses that point and appears blurred; the darkened interior and glow; the three white spots; and the millimeter-sized speckle field on the lower right of the orb, which grounds the crystallographic fluid-inclusion evidence. Because the paint layers at the upper part of the orb have perished, a complete answer must explain all these features and state, for each of them, which datum it is based on — the current post-restoration state, pre-restoration documentation, or the copies (the Barberini painting, the Hollar engraving) — since the fold-boundary behaviour is assessable only in the pre-restoration documentation and the copies, whereas the speckle field is on the current painting. The hypothesis space to be covered must include at least: a solid sphere of rock crystal (quartz), calcite, or glass at historically plausible size, placement (including placement close to the robe), and refractive behavior (including birefringence); and a hollow glass sphere with wall thickness up to a few millimeters. A complete answer must address each of these hypotheses, at least to include or exclude it. The orb's size is itself a quantity the analysis must estimate or constrain from the painting: Liang et al.'s bound of about 1.3 mm on the wall thickness of a hollow orb holds only under their modeled 6.8 cm radius, refractive index 1.51714, and placement of the folds on lines approaching the orb's center, whereas García-Ruiz and Modestini estimate the painted diameter at ca. 18 cm and report that the largest crystal or glass spheres attested for 1500 are ca. 5–6 cm. Any inclusion or exclusion claim, and any rescaled wall-thickness bound, must therefore state the size convention and the historical-plausibility evidence it relies on; likewise, the criterion for similarity between a rendering and the painting, and the admissible historically plausible class, must be defined and justified from evidence rather than assumed. A complete answer must state, on the basis of this quantitative analysis, which hypotheses can and cannot reproduce the observed appearance under historically plausible conditions (materials, lighting, and optical knowledge available ca. 1500), reconciling the rendering-based hollow-orb conclusion and the crystallographic solid-rock-crystal conclusion.

References

  1. Marco (Zhanhang) Liang, Michael T. Goodrich, Shuang Zhao, 'On the Optical Accuracy of the Salvator Mundi', arXiv:1912.03416 (2019), DOI 10.48550/arXiv.1912.03416, https://arxiv.org/abs/1912.03416
  2. Marco (Zhanhang) Liang, Shuang Zhao, Michael T. Goodrich, 'Inverse-Rendering-Based Analysis of the Fine Illumination Effects in Salvator Mundi', Leonardo 53(4):380-386 (2020), DOI 10.1162/leon_a_01923, https://doi.org/10.1162/leon_a_01923
  3. Juan Manuel García-Ruiz, Dianne Modestini, 'A crystallographic approach to the orb of the New York Salvator Mundi', npj Heritage Science 14:287 (2026), DOI 10.1038/s40494-026-02558-9, https://doi.org/10.1038/s40494-026-02558-9
  4. Martin Kemp, 'Art history: Sight and salvation', Nature 479:174-175 (2011), DOI 10.1038/479174a, https://doi.org/10.1038/479174a
  5. André J. Noest, Martin Kemp, 'No refraction in Leonardo's orb', Nature 480:457 (2011), DOI 10.1038/480457a, https://doi.org/10.1038/480457a
  6. J. Sivak, 'Was Da Vinci Thinking of the Spherical Lens of the Fish Eye in his Painting Salvator Mundi?', Hindsight: Journal of Optometry History 56(1) (2025), DOI 10.14434/hindsight.v56i1.41588, https://doi.org/10.14434/hindsight.v56i1.41588
  7. Nica Gutman Rieppi, Beth A. Price, Ken Sutherland, Andrew P. Lins, Richard Newman, Peng Wang, Ting Wang, Thomas J. Tague, 'Salvator Mundi: an investigation of the painting's materials and techniques', Heritage Science 8:39 (2020), DOI 10.1186/s40494-020-00382-3, https://doi.org/10.1186/s40494-020-00382-3