Carving and breaking the Forma Aedificii Gatesensis

Photographs and text by Marc Levoy
Movies courtesy of Sha Sha Chu
June 6, 2001

Among the assumptions made by jigsaw puzzle solvers, whether human or computer, is that the pieces fit together well. In the case of the Forma Urbis Romae (properly called the Severan marble plan), many pieces of the puzzle are missing, and those that remain have been subjected to 2000 years of breakage and weathering. Based on the few dozen matches that have been found by scholars, it is evident that the fragments don't fit closely. A key problem, then, is to characterize the fits we can expect to find using computer matching algorithms, and to develop algorithms that are robust to poor fits. To gain an intuition for this problem, we decided to conduct an experiment in breaking and reconstructing a marble slab. Here's how we proceeded.

We first procured a slab of Carrara marble: 2 feet x 2 feet x 2 inches thick. Although not identical in type or thickness to the marble used in the Severan plan (see the discussion at the bottom of this web page), it was close enough to exhibit similar breakage patterns. As our map, we chose the third floor of the Gates Computer Science Building - home of the Stanford Computer Graphics Laboratory. If transferred to the slab at a scale of 140:1, the pattern and sizes of hallways and offices approximates the pattern of streets and shops on the Severan plan (which was carved at 240:1).

The first step was to transfer the plan to the slab. We did this by printing the plan on a stout sheet of paper, but reversed left-for-right. We then traced over the printed lines with a soft charcoal pencil, affixed the sheet face-down to the marble surface, and rubbed on the back of the sheet with an artist's burnishing tool. This technique for transferring a drawing to a stone surface has been known for millenia; a similar trick might have been used on the Severan plan.

Using the same chisels Marc Levoy employed during his carving apprenticeship in Italy two years ago, we incised the lines of the plan into the marble surface. The image at left shows PhD student Natasha Gelfand carving the walls of our video lab in stone. At right, Professor Jennifer Trimble (Stanford Classics Department) ponders the appropriate translation of "Gates Building" into Latin. See the next image for her solution.

Here are two views of the finished slab. In general we attempted to match the depth and shape of the incisions in the Severan plan, but variations were inevitable. As in the Severan plan, the hands of many artisans can be discerned. Our "carving guild" consisted of Natasha Gelfand, Jennifer Trimble, Marc Levoy, James Davis, Nicolas Scapel, and Steve Marschner. With two people working at the same time from opposite sides of the slab, the carving task took about two days. Carving letters is hard. Carving serifs is especially hard.

The finished slab, which weighs about 120 pounds, is wheeled outside to its rendez-vous with destiny. From left to right is Jennifer Trimble (with the 10-pound sledgehammer), Sha Sha Chu, Kekoa Proudfoot, Natasha Gelfand, Steve Marschner, Szymon Rusinkiewicz, Ravi Ramamoorthi, Daniel Russel (with the 20-pound sledgehammer), and Sean Anderson. Why are these people smiling?

At left, chief executioner Natasha Gelfand poses with the unfortunate slab. It lies face down beside her, supported on its corners by wooden blocks to promote breakage in the center. To prevent the hammer blow from shattering the marble unnecessarily finely, we padded it with cloth. Unfortunately, we padded it too well, as you can see from the video at right (available as .mpg, .rm, or .mov).

In fairness to Natasha, she failed the first time not because she is weak, but because she tried being gentle, in an effort not to create too many pieces. On her second attempt, this time raising the (20-pound) hammer higher, she succeeded in breaking the slab neatly into four quarters.

Turning the pieces over, we were astonished to find that the fracture lines followed the incisions we had carved, even though these incisions were only 1mm deep! In retrospect this shouldn't have surprised us; the slab's carved side faced down, and it was struck from above, so the incisions operated like scoring lines on plate glass.

Looking back at Carretoni's 1960 annotated photographs of the Severan Plan, we can find several fragments that broke along incision lines. Two examples are shown at left. However, not many fragments broke this way. Presumably, most of the fragments fell forward from the wall and were shattered by impacts to their front surface when they hit the ground, rather than to their back surface. Also, most slabs used in the Severan plan were thicker than ours.

At this point, we distributed each quarter of the slab to a different student, to break as they saw fit. Sean threw his piece down violently (.mpg, .rm, or .mov). It landed on a corner, which broke off neatly, then pivoted to strike another corner, which shattered into many tiny fragments. These fragments are smaller than any that are preserved from the Severan plan. Many such fragments must therefore be lost forever.

Kekoa Proudfoot uses the 10-pound hammer to dispatch a piece.

In the sequence above, the incisions were face down, and once again the piece broke along the incisions (left image). For his next piece, Kekoa placed the incisions facing up, and this time the slab broke in a classic starburst pattern. (right image).

Looking at Carretoni's photographs, we can find several matched sets of fragments (left image) that suggest the simple four-way breakage of Kekoa's first experiment, although the fractures do not appear to follow incision lines. We can also find several sets that suggest a starburst pattern (right image). Note how much marble is missing from between the fragments in this example.

So how representative was our experiment? Only moderately so. First of all, our slab was only 2 inches thick (50mm). At right is a histogram (courtesy of David Koller) of the thicknesses of the fragments of the Severan plan. Some were 50 mm thick, but most were between 60mm and 70mm.
Secondly, Carrara is not a good match for the Hymettian and Pentelic marbles thought to have been used on the Severan plan. The image at right shows closeups of a freshly broken surface (not by us!) of a Severan plan fragment (at top) and a Gates slab fragment (at bottom). The scale is in millimeters and centimenters. The average crystal size in the Severan plan marble appears to be about 1mm, versus about 0.5mm for the Carrara marble. (Photo courtesy of Steve Marschner.)
Finally, although some of the Severan plan fragments are known to have been broken up using tools (e.g. to build a garden wall in the Farnese Palace), most of them probably shattered when they fell from the wall on which the map was mounted. Since the map was tall - the top being nearly 20 meters above the ground - this fall was larger than those used in our experiments. We considered dropping our slab from a third-floor window of Gates Hall, but rejected this scheme in deference to the landscaping below!

So what's next? We'll scan the map and try using our matching algorithms to reconstruct it. Then we'll "erode" the pieces (suggestions are welcome on how best to do this), scan it again, and try again to reconstruct it. We'll repeat this process several times, with the goal of gaining an intuition for how much erosion our algorithms can tolerate before they fail. For a more detailed analysis of how the slab broke, and for closeup photographs of the individual pieces, look at Natasha Gelfand's web page analyzing the fragments of the Forma Aedificii Gatesensis.

levoy@cs.stanford.edu

	We first procured a slab of Carrara marble: 2 feet x 2 feet x 2 inches thick. Although not identical in type or thickness to the marble used in the Severan plan (see the discussion at the bottom of this web page), it was close enough to exhibit similar breakage patterns. As our map, we chose the third floor of the Gates Computer Science Building - home of the Stanford Computer Graphics Laboratory. If transferred to the slab at a scale of 140:1, the pattern and sizes of hallways and offices approximates the pattern of streets and shops on the Severan plan (which was carved at 240:1).
	The first step was to transfer the plan to the slab. We did this by printing the plan on a stout sheet of paper, but reversed left-for-right. We then traced over the printed lines with a soft charcoal pencil, affixed the sheet face-down to the marble surface, and rubbed on the back of the sheet with an artist's burnishing tool. This technique for transferring a drawing to a stone surface has been known for millenia; a similar trick might have been used on the Severan plan.
	Using the same chisels Marc Levoy employed during his carving apprenticeship in Italy two years ago, we incised the lines of the plan into the marble surface. The image at left shows PhD student Natasha Gelfand carving the walls of our video lab in stone. At right, Professor Jennifer Trimble (Stanford Classics Department) ponders the appropriate translation of "Gates Building" into Latin. See the next image for her solution.
	Here are two views of the finished slab. In general we attempted to match the depth and shape of the incisions in the Severan plan, but variations were inevitable. As in the Severan plan, the hands of many artisans can be discerned. Our "carving guild" consisted of Natasha Gelfand, Jennifer Trimble, Marc Levoy, James Davis, Nicolas Scapel, and Steve Marschner. With two people working at the same time from opposite sides of the slab, the carving task took about two days. Carving letters is hard. Carving serifs is especially hard.
	The finished slab, which weighs about 120 pounds, is wheeled outside to its rendez-vous with destiny. From left to right is Jennifer Trimble (with the 10-pound sledgehammer), Sha Sha Chu, Kekoa Proudfoot, Natasha Gelfand, Steve Marschner, Szymon Rusinkiewicz, Ravi Ramamoorthi, Daniel Russel (with the 20-pound sledgehammer), and Sean Anderson. Why are these people smiling?
	At left, chief executioner Natasha Gelfand poses with the unfortunate slab. It lies face down beside her, supported on its corners by wooden blocks to promote breakage in the center. To prevent the hammer blow from shattering the marble unnecessarily finely, we padded it with cloth. Unfortunately, we padded it too well, as you can see from the video at right (available as .mpg, .rm, or .mov).
	In fairness to Natasha, she failed the first time not because she is weak, but because she tried being gentle, in an effort not to create too many pieces. On her second attempt, this time raising the (20-pound) hammer higher, she succeeded in breaking the slab neatly into four quarters.
	Turning the pieces over, we were astonished to find that the fracture lines followed the incisions we had carved, even though these incisions were only 1mm deep! In retrospect this shouldn't have surprised us; the slab's carved side faced down, and it was struck from above, so the incisions operated like scoring lines on plate glass.
	Looking back at Carretoni's 1960 annotated photographs of the Severan Plan, we can find several fragments that broke along incision lines. Two examples are shown at left. However, not many fragments broke this way. Presumably, most of the fragments fell forward from the wall and were shattered by impacts to their front surface when they hit the ground, rather than to their back surface. Also, most slabs used in the Severan plan were thicker than ours.
	At this point, we distributed each quarter of the slab to a different student, to break as they saw fit. Sean threw his piece down violently (.mpg, .rm, or .mov). It landed on a corner, which broke off neatly, then pivoted to strike another corner, which shattered into many tiny fragments. These fragments are smaller than any that are preserved from the Severan plan. Many such fragments must therefore be lost forever.
	Kekoa Proudfoot uses the 10-pound hammer to dispatch a piece.
	In the sequence above, the incisions were face down, and once again the piece broke along the incisions (left image). For his next piece, Kekoa placed the incisions facing up, and this time the slab broke in a classic starburst pattern. (right image).
	Looking at Carretoni's photographs, we can find several matched sets of fragments (left image) that suggest the simple four-way breakage of Kekoa's first experiment, although the fractures do not appear to follow incision lines. We can also find several sets that suggest a starburst pattern (right image). Note how much marble is missing from between the fragments in this example.