# Rewriting the History of Geometry with Image Processing and Ancient Art

### Engineers Show the Use of Complex Curves Dates Back to the Bronze Age

Archaeological digs can reveal much about ancient civilizations. But while they may discover what those civilizations built, sculpted, or painted, those finds don’t necessarily reveal *how* they did it. A research team at the National Technical University of Athens (NTUA) has challenged that idea.

Led by NTUA Professor Constantin Papaodysseus, the team of engineers has developed image processing and segmentation techniques to tell the stories behind elaborate frescoes from the Minoan civilization. Over the project’s 20-year history, they have also, in parallel, rewritten the history of geometry. They have stretched the field’s historic timeline back over a thousand years by attributing the use of complex shapes and curves to Bronze Age artists.

### A History of Analyzing History

Papaodysseus has a history of using informatics and MATLAB^{®} to solve archaeological problems. At first, Papaodysseus collaborated with the distinguished archaeologist Professor Steven Tracy, then director of the American School of Classical Studies at Athens, in order to classify inscriptions according to their *lithioxus*, i.e., the writer who carved an inscription. Such a classification is of fundamental importance for history, given that a lithioxus did not date nor sign his work, thus making the time classification of an inscription practically impossible. However, when the entire work of a lithioxus is gathered, then, as a rule, careful examination of this ensemble makes it gain a date. Thus, Papaodysseus, using MATLAB and mathematics, managed to classify more than thirty inscriptions into nine writers, with 100% success.

Next, the Center for Hellenic Studies at Harvard University asked Papaodysseus to help identify whether the same hand penned two separate Iliad documents: one preserved in the Venice Museum and another preserved in the monastery of Escorial, outside Madrid. This would narrow down the dates of the artifacts and help trace their origins.

“It struck me that we could do this with modern mathematics, informatics, and MATLAB,” says Papaodysseus. He and his team, together with Professor of Classics Christopher Blackwell, showed that the same hand had written both documents. The Greek parliament has even enlisted his detecting skills to identify the writers of other documents unearthed by historians.

“With MATLAB, we can help archeologists form a more complete and accurate history of ancient artifacts,” says Papaodysseus.

The story of the fresco work begins beneath Akrotiri, a town in southwestern Santorini, a Greek island in the Mediterranean’s Aegean Sea. Around 1620 BCE, during the Minoan civilization, a massive volcanic eruption buried the island beneath layers of pumice and ash up to 20 meters thick. This created a rich trove of artifacts that volcanic rock preserved for millennia. Among the excavated discoveries were shattered remnants of elaborate wall paintings or frescoes.

In the late 1990s, Papaodysseus began using MATLAB to analyze ancient frescoes and develop methods for piecing together hundreds of wall painting fragments, including 2D and 3D reassembly. One night, while trying to match pieces of frescoes on his computer, he saw something unexpected on the monitor. The image was of a girl he had seen in another fresco, “The Crocus Gathering.” In this fresco, the girl is leaning toward a plant, but with this latest image of her now on his screen, Papaodysseus noticed the curve of her back looked like a hyperbola.

There was one problem—the first descriptions of hyperbolae and other conic sections were from remarkable thinkers of the Classical era, such as Menaechmus and Euclid, who lived around 350–250 BCE—over 1,300 years after the fresco’s creation. The notion was, quite possibly, “crazy,” Papaodysseus recalled.

“But inspiration comes without asking you, and I realized we could find out how the artist created the celebrated fresco,” says Papaodysseus. Like his realization with the Iliad documents, he knew mathematics and MATLAB would help.

### The Proof Is in the Painting

Papaodysseus wanted to see where this “hyperbola-curved back” might take him. After his fateful observation, Papaodysseus spent the next few months creating a method in MATLAB to test his hypothesis. He developed an image segmentation algorithm that could isolate the individual contours of the images in a chain of single pixels. The contours are unbroken, uniform curves or lines, so a given painting comprises many individual contours.

The next step was to determine whether the artist of “The Crocus Gathering” had painted a true hyperbola and, if so, how this was accomplished with such precision. Analyzing the girl’s back and other components of the fresco, Papaodysseus’ team tested the idea that the artists of the day used stencils to guide their brush strokes. So, they looked to see whether the contours aligned with certain shapes and curves: straight lines, circles, ellipses, parabolas, hyperbolae, more complex curves such as cycloids, and various spirals.

“We have dealt with shapes that either appear in nature or that have only been understood and extensively studied since the time of the great Classical-era mathematicians, such as Archimedes and Euclid,” says Papaodysseus. “At the time, I didn’t believe that the Minoans, the ancient forebears of Euclid and Archimedes, would have understood or worked with these shapes.”

Intuitively, the idea that these artists used guides or stencils to create these shapes made sense. At that time, artists painting on a wall required precision and speed to complete sections on wet plaster before it dried. Given the smooth, steady contours of the finished paintings, it seems more likely the artists used some sort of guide to paint on this rough surface.

With the image of the girl’s curved back in “The Crocus Gathering,” Papaodysseus and his team plotted curves with varying parameters and applied algorithmic schemes that could identify and compare drawn contours with the potential stencil shapes using MATLAB and its function minimization algorithms.

“In MATLAB, it’s easy to plot a hyperbola. Using two functions from Image Processing Toolbox™, you can superimpose the hyperbola on an image,” says Athanasios-Rafail Mamatsis, a researcher in Papaodysseus’ team.

Applying these functions to the girl’s image in “The Crocus Gathering,” they confirmed that her curved back aligned with a true hyperbola and that no other possible shape or stencil matched up (see Figure 1). As further proof, when he tried this method on another girl in the same fresco, he found contours corresponding to the same hyperbola (see Figure 2).

“The geometric guide and the actual drawing matched so closely that it couldn’t be accidental,” Papaodysseus says.

### It’s a Match

As the years went on, Ph.D. students working on the original project became colleagues, and the team further iterated on and improved Papaodysseus’ original algorithms to explore new ideas. “In particular, after the hyperbola finding, we had no hesitation in testing whether the Minoans had used linear spirals for the frescoes,” says Papaodysseus.

Further supporting the idea that Bronze Age artists used stencils, the paintings’ contours typically vary from the computer-generated guides by less than 0.3 millimeters but never more than 0.8 millimeters.

Their first instinct was to test spiral shapes that were simple or commonly found in nature. Prime candidates for analysis were the exponential spirals appearing in seashells with excellent precision or the spiral created from unwrapping a string wound around a peg. It seemed less likely that the artists would have stumbled across the harder-to-construct linear spiral, also known as Archimedes’ spiral. Yet, when they ran their comparisons, the swirls decorating frescoes were, in fact, these strange spirals (see Figure 3).

Upon analyzing more and more wall paintings, Papaodysseus’ group consistently found matches for six specific stencils (four hyperbolae and two linear spirals) in image contours. Examining the frescoes, they even spotted small holes in the plaster where the artists may have pinned their guides. Further supporting the idea that Bronze Age artists used stencils, the paintings’ contours typically vary from the computer-generated guides by less than 0.3 millimeters but never more than 0.8 millimeters. “This excludes randomness,” says Papaodysseus. Such close matches are unlikely to be a coincidence, especially for lengths of stencils that exceed 14 cm, 15 cm, 17 cm, 22 cm, etc.

In a 2022 study in the Journal of Cultural Heritage, Papaodysseus and his colleagues write, “in numerous wall paintings, preliminary curves engraved on the wall appear in parts of the fresco where the colors have faded out.” The artists could have completed painting their subjects by combining stencils or stencil segments. In the same paper, Papaodysseus’ team also showed that their stencils don’t match the contour parts drawn by conservators to fill in missing parts of the fresco, suggesting that the Bronze Age artists used a specialized, precise technique. Furthermore, it seems that the same technique of using the very same stencils for drawing frescoes has been applied to Minoan Crete (see Figure 4).

### Lessons in Geometry

More evidence that the artists, or stencil-makers at least, had an advanced geometric understanding comes from the possible ways they created the hyperbola and spiral stencils. Consider a hyperbola: It isn’t simply a curved line that happens to match the shape of a curved back. All points on the curve have a constant difference in distance from two fixed points on the same plane, which are now called *focuses*. The researchers suggest that the artists could have constructed such curves by drawing two circles of different radii, each with a defined center point such that the circles have a portion of overlap, like a lopsided Venn diagram. By drawing concentric circles around the original two, each time increasing both circles’ radii by the same amount, one can draw a hyperbola by connecting the intersection points between each corresponding circle (see Figure 5).

For the linear spiral, the researchers believe the artists could have again used concentric circles. The team’s previous work had shown that Akrotiri’s Bronze Age inhabitants could construct central angles of sequences of regular polygons. One could draw a linear spiral by connecting the points where the corresponding straight lines from the polygons intersect with concentric circles (see Figure 6).

Throughout the 20-plus years Papaodysseus and his team have conducted this work, they always used MATLAB. “I wouldn’t trade MATLAB for anything, even though I know many programming languages,” says Papaodysseus. For tasks where he needs to write in C/C++, he calls the code from MATLAB and receives instant results. He says the same task would take triple the time without this MATLAB wrapping availability. The usability, efficiency, and quality of the computing environment are indispensable. “It reduces the number of lines of code you must write, and the clarity of results is incredible.”

### Far-Reaching Findings

It’s not just Santorini that has these impressive frescoes. As previously mentioned, in the fresco of Figure 4, which once decorated a wall in the Palace of Knossos—but now resides in museums—a man rides a bull upside down while two women on either side prepare to help. When analyzing this painting from Crete, Papaodysseus’ group saw that the back of the bull corresponded uniquely to a hyperbola the team had seen before—in Santorini’s frescoes—while various other contour parts of this fresco optimally fit corresponding parts of hyperbolae and spiral parts spotted at Akrotiri, Thera.

So far, the researchers have found supporting evidence for their stencil theory in over two dozen frescoes from different Aegean islands and centuries.

The more the team looked for similarities in frescoes across different Aegean islands, the more they found. So far, the researchers have found supporting evidence for their stencil theory in over two dozen frescoes from different Aegean islands and centuries. The various wall paintings have wide-ranging years of origin, spanning the 550 years between 1650 and 1100 BCE.

“In Santorini, Crete, Mycenae, and Thebes, the uncovered wall paintings all have contours that uniquely correspond to four hyperbolae and two linear spirals with an exceptionally low matching error,” says Papaodysseus. This suggests that during the Bronze Age, artists around the Aegean Sea had a common training in using stencils of complex shapes well before the Classical-era mathematicians described them.

Now, the group is teaming up with the University of Athens to expand the reach of their techniques and analyze more Minoan art. In doing so, they hope to illuminate geometry’s rich history further and give credit to the pioneering artists.

### Read Other Stories

AI / ACADEMIA

#### AI Unveils the Secrets of Ancient Artifacts

Using Deep Learning and Image Processing to Restore and Preserve Artwork