Cells tell the tale of a hunch

Be that as it may. The forehead and cheeks of the head are smooth. It sports a mustache and beard, is not linked, nor is it particularly thick. The hair on top is styled almost like a close-fitting helmet, which extends down the back to the neck. Some of this makes the Head unusual for Greek sculpture of the time. Men were usually sculpted with long hair. While beards were common, what is unusual is that the mustache does not match the beard.

For these reasons, the Sabouroff Head has been the subject of a great deal of speculation. Specifically, it was apparently part of a larger statue. What was that idol?

A few years ago, some researchers suggested an answer to that question. What led him to this was… yes, math.

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

Take a blank sheet of paper. Using a pen, mark two points on it—for now, keep one near the corner and the other near the opposite corner. Now draw a line that divides the sheet into two parts and each has a point. But make it such that in each segment, any point you choose is closer to that point than another point in that segment. When you have marked such a line, you have divided the sheet into two parts – two so-called Voronoi cells.

Once you understand that notion, you’ll know that the dots don’t have to be near corners. Wherever they are on the sheet, you can find a suitable line that divides the sheet into a pair of such cells. And then you can extend the idea to more than two points. mark several points on the sheet, fairly randomly; 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 whatever point you choose is closer to this seed than to any other point. That’s just how they are defined: a Voronoi cell is a shape around a seed that has all its points closer to it than to any other seed. So, if you have many seeds on your sheet of paper, and you mark each seed’s Voronoi cell, you will have divided the sheet into a grid – known mathematically as a “tessellation” – of these cells. This grid, this division of the sheet into Voronoi cells is called a Voronoi diagram.

Incidentally, cells are named after another Russian, Ukrainian-born mathematician called Georgy Feodosevich Voronoy (1868–1908), who defined them. But they were used earlier also in some form or the other.

An example: during the worldwide cholera epidemic of 1846–1860, there was a sudden outbreak of cholera on Broad Street in London in September 1854. At the same time, 616 people have died. Later, physician John Snow investigates the outbreak. He was already doubting the prevailing theory at the time, that cholera was spread by “miasma”, or bad air. They believed that it was the water supply that was responsible, as it was contaminated.

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

When Snow made his findings public, officials disabled the pump by removing its handle. Although there is some evidence that it was not particularly effective, simply because the outbreak had largely subsided by then. Nevertheless, Snow’s work with his Voronoi chambers 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 through the flight in case an emergency landing needs to be made. Meteorologists examine precipitation data and the stations that record it through Voronoi diagrams.

Then there are archaeologists.

In 2020, an archaeologist, a mathematician and a computer scientist from the University of Heidelberg in Germany published the results of their detailed examination of the Sabouroff Head. Essentially, he took individual points on the head and mapped the distance from those points to anywhere else on it. Of course, he gave them the Voronoi diagram of the sculpture. He repeated these distance measurements, moving the points 1 cm at a time. This produced a set of lines on the head, which apparently showed something surprising: “the middle of the neck does not line up well with the middle axis of the face.” That is, the face is slightly asymmetric. Voronoi cells show the same slight asymmetry.

Overall, the asymmetry suggests that the head was originally tilted slightly to the left. Given other, more complete statues that survive from the same period in Greek history, this tilt is a “strong indication that the head is that of an equestrian statue.”

Think about that. The ancient Greeks sculpted the faces of men in a different way, geometrically speaking, than they were doing in sculpture. At least with the Sabouroff Head, you and I can’t look that difference in the face.

But it is enough for modern scientists using Voronoi diagrams to figure it out. Mathematics at your service, again. (Here’s a fascinating video explaining Sabouroff Head’s analysis: https://tinyurl.com/SabouroffHead,

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

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