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# The cells tell the story of a tilt

## The Sabouroff Head has raised many doubts, in particular about the rest of the sculpture

The Sabouroff Head is a life-sized marble sculpture of a man’s head from ancient Greece. It has been dated to the 6th century BCE and was most probably sculpted in Athens. The nose and lips are damaged, but otherwise, it is a remarkably well-preserved artifact, about nine inches tall. In case you’re wondering, you’re right: Sabouroff is not a Greek name. The bust is named for Peter Alexandrovich Saburov, a 19th century Russian diplomat, amateur chess player and art collector.

Be that as it may. The Head’s forehead and cheeks are smooth. It sports a moustache and beard, not connected, and neither is particularly thick. The hair on top is styled almost like a close-fitting helmet, extending down to the neck at the back. Some of this makes the Head unusual for Greek sculpture of the time. Men were typically sculpted then with long hair. While beards were common, what’s uncommon is that the moustache doesn’t meet the beard.

For these reasons, the Sabouroff Head has been the subject of a fair deal of speculation. In particular, clearly it was part of a larger sculpture. What was that sculpture?

Some years ago, some researchers suggested an answer to that question. What led them to it was...yes, mathematics.

First, though, let me tell you about Voronoi cells.

Take a blank sheet of paper. Using a pen, mark two dots on it—for now, put one near a corner and the other near the opposite corner. Now draw a line that divides the sheet into two sections with a dot in each. But draw it such that in each section, any point you choose is closer to the dot in that section than it is to the other dot. When you’ve marked out such a line, you have divided the sheet into two parts—two so-called Voronoi cells.

Once you’ve grasped that notion, you’ll know that the dots don’t have to be near the corners. Wherever they are on the sheet, you can find an appropriate line that divides the sheet into a pair of such cells. And then you can extend the idea to more than two points. Mark a number of points on the sheet, quite at random; mathematicians who work with Voronoi cells call these seeds. Take each seed in turn, and think about this: there is a part of the sheet, containing the seed, inside which any point you choose is closer to this seed than to any other seed. This is just how they are defined: A Voronoi cell is the shape around a seed which contains all the points closer to it than to any other seed. So, if your sheet of paper has a number of seeds, and you mark out each seed’s Voronoi cell, you will have divided the sheet into a grid—known mathematically as a “tesselation"—of these cells. This grid, this partition of the sheet into Voronoi cells, is called a Voronoi diagram.

Incidentally, the cells are named for another Russian, the mathematician of Ukrainian descent called Georgy Feodosievych Voronoy (1868-1908) who defined them. But they were used in some form even earlier.

One example: During the worldwide cholera epidemic of 1846-1860, there was a sudden outbreak of cholera on Broad Street in London in September 1854. The famous nurse Florence Nightingale actually volunteered to help with the rush of cases from there. Still, 616 people died. Later, the physician John Snow investigated the outbreak. He was already sceptical of the prevailing theory of the time, that cholera was spread by “miasma", or bad air. He believed it was the water supply that was responsible, because it was contaminated.

Snow plotted the cases of illness and death on a map of the area. He also marked on the map where the water pumps—residents’ main source of water—in the area were located. Using this, he mapped cells around each pump that contained the cholera cases closer to that pump than any other: Voronoi cells, of course, even if he couldn’t have called them that. This exercise immediately showed Snow that the largest number of cases were in the cell surrounding a particular pump on Broad Street. And that pump was supplying contaminated water from the Thames River to houses in the area.

When Snow made his findings public, authorities disabled the pump by removing its handle. Though there’s some evidence that was not particularly effective, only because by then, the outbreak had considerably eased. Still, Snow’s work with his Voronoi cells had a great impact on public health and policy issues, leading to widespread efforts to improve sanitation facilities.

So yes, Voronoi cells have their uses. Flight planners use them to find the nearest airfields all through a flight, in case there’s a need to make an emergency landing. Meteorologists examine rainfall data and the stations that record it, via Voronoi diagrams.

Then there are archaeologists.

In 2020, an archaeologist, a mathematician and a computer scientist at the University of Heidelberg in Germany published results of their detailed examination of the Sabouroff Head. Essentially, they took different points on the head and mapped the distance to those points from elsewhere on it. These gave them, of course, a Voronoi diagram of the sculpture. They repeated these distance measurements, moving the points 1cm at a time. This produced a set of lines across the head, which clearly showed something surprising: “the middle of the neck doesn’t correspond well with the middle axis of the face." That is, the face is slightly asymmetric. The Voronoi cells show the same slight asymmetry.

All in all, the asymmetry suggests that the head was originally tilted slightly to the left. Given other, more complete statues that have survived from the same period in Greek history, this tilt is a “strong hint that the head belongs to an equestrian statue."

Think of that. Ancient Greeks sculpted faces of men differently, geometrically speaking, according to what those men were doing in the sculpture. At least with the Sabouroff Head, you and I can’t see that difference in the face.

But it’s enough for modern scientists, using Voronoi diagrams, to detect. Mathematics at your service, again. (A fascinating video explains the analysis of the Sabouroff Head here: https://tinyurl.com/SabouroffHead)

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.

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Updated: 20 Jan 2023, 12:22 AM IST
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