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Caravaggio's "Supper at Emmaus" (1601-2): Problems in refocusing, problems in the studio, problems with illumination
Caravaggio figures prominently in Hockney's theory, yet this painting exposes numerous awkward implications of the theory related to refocusing, moving the canvas, and illumination.
We now must be just a bit quantitative. I measured the size of the images of Christ's right hand, Peter's left, and a few other objects in the painting and made some assumptions about the actual subjects such as Peter's arm span. I then used standard techniques described by Hockney and Falco to estimate the focal length of a concave mirror purportedly used in the creation of this painting. For simplicity, let's round my result up to f = 1 m, a value somewhat larger than that found by Hockney and Falco for other paintings. (Mirrors with shorter focal lengths yield problems even more severe as will be mentioned below.)
Now let us imagine Caravaggio uses this mirror to project an image of Peter's left hand, as shown in the bird's eye view diagram. We let ds denote the distance from the subject (hand) to the mirror and di the distance of the projected image to the mirror. The magnification turns out to be the ratio of these distances, i.e., M = di/ds. Since the magnification is M=1.0, then di = ds (regardless of the focal length of the mirror). Thus the figure shows a plausible arrangement of subject, mirror and projected images; the canvas would be placed at the image, just in front of the table.
If there had been no refocusing, the image of Peter's right hand would be about half as large as his right (because his right hand is nearly twice as far from the mirror as his left). Let us consider now Mr. Hockney's suggestion that Caravaggio refocused on Peter's right hand. If the image in the painting of his right hand were the same as that of his left (i.e., magnification M = 1.0), then the mirror and canvas would have to have been moved forward by 2 meters (Peter's arm span), duplicating the separations and relationships used in the projection for the left hand. But because the image of the right hand is just a bit smaller (by 10%), the mirror is just a little bit farther from that hand. It turns out from basic optics (see discussion of optics of concave mirrors below) that the distances must be about ds = 2.11 m and di = 1.90 m, as shown.
These distances imply that after refocusing, the canvas must be on top of Christ's head, that Caravaggio must stand in the middle of the table, that the light can no longer reach Peter's right hand because it is blocked by Caravaggio, the canvas and the easel. This is an major disruption to the studio that would have impeded Caravaggio's ability to paint. Note especially that an even more severe disruption would occur if the mirror had a shorter focal length, say f = 50 cm -- the mirror and Caravaggio and the canvas would be in the position of the table. Note especially that an even more severe disruption would occur even if the mirror had a shorter focal length, say f = 50 cm -- the mirror and Caravaggio and the canvas would be in the position of the table.
The top figure shows the placement of the canvas while Caravaggio was painting the left subject, A. The inverted image of A appears at the right of the upside-down canvas. (Later, the canvas will be rotated rightside-up thereby placing the image of A on the left, where it belongs.) We imagine that Caravaggio painted or traced A, then reoriented the mirror somewhat and then painted B in this setup.
When Caravaggio tries to paint subject C, there is a problem. Concave mirrors (and indeed all simple lenses) suffer from off-axis aberrations -- distortions and degradation of images produced at a large angle from the axis of the mirror itself. The aberrations most relevant here are curvature of field (the image is curved in space, so cannot be all in focus at the same time on the canvas), coma (the image of a point is spread out, like the tail of a comet), and distortion (the image is stretched out of shape, as if on a rubber sheet). The fourth off-axis aberration is astigmatism, which is not quite the same as the familiar vision problem corrected by eyeglasses, but nevertheless also leads to an out-of-focus image.
The theory of such aberrations is a bit tricky and well beyond us here. Nevertheless, it is a simple matter to observe the resulting image degradation by projecting an image far from the axis of the mirror, such as that as shown by the red lines in the middle figure. I did this with a concave makeup mirror (f = 70 cm, mirror diameter = 6 cm), and observed the degradation of the image for an angle of about 20 degrees away from the axis, or 40 degrees total spread between the incoming and outgoing directions. These aberrations were far more serious than depth of field problems that Hockney and Falco mention in several paintings. Most importantly, unlike the blurring due to depth of field, such off-axis aberrations cannot be "focused away" by moving the mirror forwards or backwards.
Thus, when Caravaggio sought to paint subject C he would have experienced these off-axis aberrations because the relevant angle is quite large -- nearly 90 degrees. His only recourse would then be to move the canvas to the right, reorient the mirror and then paint C and D, as shown in the bottom figure. Furthermore, if he wanted to alternate between painting A and D, he would have moved his canvas back and forth, back and forth -- an extremely awkward and unnatural process.
Now we turn to the third problem: the crucial issue of illumination. The images produced by mirrors are dim because the light from the subject that strikes the (small) mirror is then spread out over the canvas -- only a portion of the light ultimately reaches the artist's eye. The reduction in brightness depends upon the focal length and the diameter of the mirror (whose ratio photographers call its f-number).
Optical scientists have a technical jargon of luminance, illuminance, foot-candles, lux, and much else, but fortunately for our case we can colloquially refer to light intensity without doing an injustice to the true science. From basic optics, it turns out that the ratio of the intensity of an image to the intensity of its source subject is the area A of the concave mirror divided by the mathematical square of the mirror's focal length f, that is A/f2. I calculated this ratio for my concave makeup mirror whose disk had a radius of 2.5 cm and focal length f = 70 cm and found the ratio to be 0.004. That is, the real scene is 1/0.004 = 250 times as intense as the projected image. I checked this experimentally the following way. I went into a darkened room and closed the curtains but for a small opening. I the pointed a standard photographer's light meter out the opening to measure the intensity of the light of the sunlit scene outside. I express my result in standard photographic terms: f/11 @ 1/60 second. I then held the mirror within the room to project an image of the outdoor scene onto a white piece of paper. I then measured the intensity of this projected image and found f/2.8 @ 1/2 second. The difference between these two measurements corresponds to 9 f-stops or 9 powers of 2, and means that the ratio in their intensities is 29 = 512. This means the intensity I actually measured is about half that given by theory (i.e., 1/512 rather than 1/250). I attribute this to the fact that measurement of the outside scene captured the light actually reflected toward the mirror while the measurement of the image captured only a portion of the light scattered by the canvas -- much was scattered into different directions. Regardless, below I will in every case use the values that are most favorable to the Hockney/Falco theory, e.g., a ratio of 250 rather than the more pessimistic 512. (Note that this ratio is independent of the overall intensity outside. Had the light outside been half as intense, then the projected image would have been half as intense too.)
The bottom line: For my mirror, if you can just see the real scene with a certain number of candles (say), then to see the projected image you must illuminate the scene with roughly 250 times as many candles. For the mirror calculated by Hockney and Falco for the Lotto "Husband and wife," that ratio of intensities turns out to be 645.
Hockney (p. 231) reminds us that Caravaggio " ...worked in dark rooms -- cellars -- very common in those days... He used artificial lighting " (emphasis added) and indeed there are many reasons we can reject the proposal that sunlight was used in Caravaggio's studio. For instance in "Supper at Emmaus," had the sun been the source of the the low light from the left it would have been available for only a very short period in the early morning or late afternoon on cloudless days, unless someone held a plane mirror and tracked the sun. In fact, art historians believe Caravaggio illuminated his subjects by candles and oil lamps, and this was part of the source of drama in his paintings. (Incidentally, two candleholders, an oil lamp and a con vex mirror -- not a con cave mirror required by the Hockney/Falco theory -- were found in Caravaggio's estate in 1605. There were no large caches of candles.)
If Caravaggio had employed the methods of Hockney and Falco, how many candles would he have needed? Imagine you are in Caravaggio's cellar, and it is totally dark. You stand in the dark for a half hour so your eyes become sensitive or dark adapted. Then imagine an assistant brings in lighted candles, one by one, placing them around the subjects until you can just distinguish the figures, their contours, and so forth -- that is, just bright enough to be visible during drawing. I tried a version of this experiment and estimated that at minimum Caravaggio would need 5 candles for "Supper at Emmaus." But then of course the image projected by the mirror is far too dim. From the ratio calculation above, in order to make the projected image usable, Caravaggio needs to illuminate the scene with 5 x 250 = 1250 candles or roughly 200 oil lamps. Even if we imagine that Caravaggio had a larger mirror and somewhat shorter focal length mirror, the number of candles he would need is still several hundred.
My conservative estimate of 1250 candles is quite a surprisingly large number, and would have been expensive and dangerous in a cellar or indeed anywhere else. Such a large number means the candles must be spread apart somewhat, producing diffuse illumination, thereby precluding the sharp shadows that pervade the painting and for which Caravaggio is justly famous.
Even if we acknowledge there might be slight inconsistencies in lighting in a la Tour painting, it is extremely difficult and highly implausible that he obtained such illumination with sunlight, as can be deduced from the shadows and much else.
Summary: The "Supper at Emmaus" presents a number of difficulties for the specific explanations of Mr. Hockney and for the theory more generally. The refocusing he suggested would have led to an extremely awkward and severe disruption of the studio; avoiding off-axis aberrations demands an awkward reshuffling of the studio; the artificial illumination in the cellar must have had at least 1250 candles (or roughly 200 oil lamps), been extremely awkward, dangerous and expensive and would impede Caravaggio's ability to get the dramatic shadows that empower his art. All these consequences of the Hockney/Falco theory could only impede rather than aid Caravaggio in his quest to portray his models "optically." Nor are these problems unique to "Supper at Emmaus"; virtually any large painting would produce such refocusing and rearrangement problems; virtually any painting executed solely by artificial light would require an implausibly large number of candles or other artificial sources of light.