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Figure 1 shows one scan of the middle toe of David's right foot, taken from our full-resolution dataset. The X-Y resolution of our scanner is 0.29mm (290 microns), and its Z-resolution is 0.05mm (50 microns). Figure 2 shows a corresponding scan acquired by the National Research Council of Canada's ultra-high-resolution laser triangulation scanner. The X-Y resolution of their scanner is 0.05mm (50 microns), and its Z-resolution is 0.01mm (10 microns). Figure 3 is a photograph for comparison. Figure 4 shows a closeup of the central portion of the toenail, and figure 5 is a plot of depth as a function of position along the red line in figure 4. The grid in figure 5 is marked in 10th's of a millimeter and is distorted vertically to improve readability.
Our reason we asked the NRC to make these scans was to give us a gold standard for our own data. Another reason was to help us understand the noise we were seeing in our data. As figure 2 shows, although their resolution is better than ours, their scans also exhibited unexpected noise. In the images above, the few large pits in the toenail, which measure about 0.5mm across, are real, but the fine-grain roughness, which measures 50-150 microns peak-to-peak in our scan and 10-30 microns in theirs, is not. Polished marble is smoother than this. Our hypothesis is that this noise arises from scattering of the laser within the crystal structure of the marble, not from noise in our camera or scanning systems. This suggests that we have reached the limit of laser scanning technology, at least for marble.
Much as we might like to digitize Michelangelo's statues at the extraordinary resolution shown in figure 2, it is currently impractical. Aside from the large size of their scanner, which is mounted on an optical bench, and the difficulty of holding such a scanner steady at a height of 24 feet, the datasets would simply be too large. Digitizing the David at at this resolution would produce 50 billion polygons and, with equivalent-resolution color data, would occupy over a terabyte.
Click here to read a technical paper giving some preliminary results of our study of the effect of subsurface scattering on laser scanning of marble surfaces.
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When people think of Michelangelo's David, they often think of his slim figure and tight abs (figure 1). Actually, his belly bulges considerably, as figure 2 shows. This bulge is missing from most handmade replicas of the statue, making them look too muscular and "tucked in." To be precise, the width of David's waist is 75 cm left to right at a in this annotated front view, and 53 and 57 cm front to back at b and c, respectively in this annotated side view. (The height of the David is approximately 497 cm from head to toe (i.e. without the carved rock on which he is standing.)
These renderings were made from a simplified polygon mesh model computed by Paolo Cignoni (CNR/Pisa) from the artificially colored 1.0 mm model of the statue from our archive.
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The ankles of the David contain several hairline cracks. For a brief period in the 19th century, while the David still stood outdoors in the Piazza Signoria (the statue now standing in that plaza is a replica), there was settling of the ground beneath the statue. This caused it to tip forward by several degrees, alarming the authorities and precipitating its removal to the Galleria dell'Accademia, where it now stands. The hypothesis has been advanced that this leaning, and the stresses it induced in the marble, is sufficient to explain the cracks we see. This would be preferrable to a modern cause, which would be of great concern. Therefore, prior to the 2002-2004 restoration of the statue, the museum asked us to estimate whether there is torque (and therefore stress) on David's ankles as he now stands. In this context torque means a tendancy of the statue to overturn. At the same time, many people have asked us how much marble is in the David, or how much he weighs. Since we have an accurate geometric model of the statue, it is possible to answer the question posed by the museum, as well as producing these other statistics. Thanks to Sean Anderson for these calculations.
Volume. There are many ways of computing the volume enclosed by a polygonal boundary mesh. Unfortunately, our mesh of the David is not watertight. In fact, it has many topological problems. We therefore adopted the approximate method of (1) connecting each triangle in our 1 million polygon model to an arbitrary point in space, thereby forming a million long skinny tetrahedra, (2) using standard formulae to compute the volume of each tetrahedron, and (3) adding or subtracting these volumes depending on which way each triangle faces relative to our point in space. Using this method, we estimate the volume of the statue to be 2.16 cubic meters (76.3 cubic feet).Weight. Multiplying this volume by an assumed density for marble of 2700 kg per cubic meter (168 pounds per cubic foot), we arrive at a weight of 5800 kg (6.4 tons) for the statue. Like our volume estimate, this figure includes the ground on which David stands, which is part of Michelangelo's original block, but does not include the 19th-century pedestal on which the statue is mounted.
Center of mass. By computing and averaging the centroids of each tetrahedron in our volume analysis, we estimate that the center of mass of the statue is inside David's left thigh, about 5 cm below his scrotum, just to its left (from his point of view), and about midway between the front and back of the leg.
Torque on the ankles. Removing that portion of the model below a horizontal plane located at mid-shin (just above the cracks), we repeated our center of mass calculation. The new center of mass was about 5cm above David's navel, just on its right, and midway between the back and front of his abdomen. Dropping a line from this point to our horizontal plane, we found that it fell very close to a line connecting the two shins and about 8 cm away from the right shin. This suggests that there is little torque on David's ankles as he now stands, although huge compressive forces of course. It also suggests that Michelangelo was extraordinarily adept at balancing the huge weight of his marble figure over two skinny columns (David's legs). Finally, since the center of mass falls closest to the right leg, as would be expected for a classic contraposto pose, Michelangelo was wise to buttress this leg with a carved tree stump.
While the museum accepted our conclusions, it spurred their interest in computing the magnitude and location of stresses that would have been present when the statue was leaning. For this purpose they engaged A. Borri, A. Grazini, and L. Marchetti to perform a finite element structural analysis of the statue using our geometric model. Figure 1 above, reproduced from the book Exploring David, published in 2004 by Giunti Press, show tensile stresses in the left leg with the statue tilted 3 degrees forward, as it was in 1871. Figure 2 shows tensile stresses in the tree trunk supporting the right leg under the same conditions. These images show that the stresses would indeed be highest (red shading) in the vicinity of the observed cracks (red lines). Their analysis also confirmed our conclusion that there is currently no torque on the ankles, and therefore the cracks are unlikely to get worse. Whew!
By the way...
Height. As we discuss elsewhere in these web pages, the David is 517 cm tall without the 19th-century pedestal, not 434 cm tall as given in many history books. This error still exists in many current sources, including a web page at the Galleria dell'Accademia itself.
In a letter to the editor of the New England Journal of Medicine, oncologist James Stark and art historian Jonathan Nelson have hypothesized that several visually obvious lumps and depressions on the left breast of Michelangelo's famous statue of Night represent an advanced stage of female breast cancer. Since we have digitized this statue, they asked for our help visualizing these lumps and measuring them. The image above shows a rendering of a low-resolution computer model of the breast. The view is looking down from above, i.e. from the viewpoint of Night's head. Two depressions are indeed visible, to the left and right of the nipple.
With the help of Marc Levoy, Stanford freshman May Allen took some measurements from this computer model wrote them up as a project for Marc's freshman seminar course CS 99D - The Science of Art. Here is her online report. Click here to read Nelson and Stark's letter.
Notice: The images of Michelangelo's statues that appear on this web page are the property of the Digital Michelangelo Project and the Soprintendenza ai beni artistici e storici per le province di Firenze, Pistoia, e Prato. They may not be copied, downloaded and stored, forwarded, or reproduced in any form, including electronic forms such as email or the web, by any persons, regardless of purpose, without express written permission from the project director Marc Levoy. Any commerical use also requires written permission from the Soprintendenza.
Notizia: Questi modelli elaborati al computer, immagini computerizzate, e fotografiche sono proprietà del Progetto Digitale Michelangelo e la Soprintendenza Per I Beni Artistici e Storici per le Province di Firenze, Pistoia e Prato. Non possono essere copiati, scaricati da internet su un file, inviati, o riprodotti in nessuna forma, incluso la posta elettronica o il web, da nessuna persona per nessun motivo, senza un permesso scritto da Marc Levoy, il direttore del progetto. Eventuali usi commerciali esigono anche il permesso scritto dalla Soprintendenza.
