At some point in your teenage years, you probably kept a compass and straightedge in your backpack, learned the ways to prove two triangles are congruent, and knew what a secant was. It all had a taste of the classical about it: Euclid, Archimedes and Pythagoras had figured everything out and passed it down to us. But geometry may be more democratic than it seems. As a group of native Amazonians showed, you don't need to go to school to explain Euclid.
French researcher Veronique Izard and her colleagues wanted to know if an understanding of Euclidean geometry is intuitive. It makes sense for humans and other animals to have a basic sense of shapes and distances, so we can find reachable fruits and flee approaching predators. But our eyes often deceive us. So do children, or remote tribespeople, instinctively understand that two parallel lines never cross? Or how many points define a line?
The researchers traveled to the Amazon and recruited children (ages 7 to 13) and adults from a group called the Mundurucu. They had no education in geometry, and their language doesn't include any words to describe concepts such as parallel lines or right angles. But the Mundurucu face challenging navigational tasks every day, just moving around their environment. The researchers quizzed them on basic Euclidean tenets.
Instead of points and lines, researchers described villages and straight paths. They asked two sets of questions, one concerning the geometry of a plane (described as a flat world that extends forever) and the other about a sphere (a "very round world"). For a visual aid, they used either a tabletop or a half a calabash.
Participants were also shown two corners of a triangle and asked to demonstrate, with their hands, what the missing corner would look like and where it would be.
The Mundurucu did great on their geometry quiz. The children performed just as well as the adults, and overall the Mundurucu did almost as well as American adults and French children that took the same quiz. All groups did better on questions about a flat plane than questions about the surface of a sphere, maybe because the former is more similar to what we observe in our daily lives.
To find out whether this kind of knowledge is truly innate, or something that develops over time, the researchers repeated the quiz with American kids just 5 and 6 years old. The kids did OK, but not as well as older children or adults. They especially had difficulty completing the triangles.
The results suggest that we're not born with an understanding of geometry. Rather, we learn as we grow how angles and lines work in the world. It would be interesting to see how another untrained group, one with less navigational experience than the Mundurucu, would handle the same questions. If a person grows up in a static and unchallenging environment, does he or she have a less intuitive grasp of distances and perspectives? Might the laws of the world be a little more mysterious?
Some of the questions the Mundurucu correctly answered had to do with abstract ideas, such as infinitely extending lines. This showed that they weren't just describing basic physical relationships they'd observed, but extending their knowledge of the world to larger mathematical concepts. Euclid may have come up with the terms and the postulates, but the Mundurucu show that anyone at all, using their eyes and their understanding, could have invented geometry.
Izard, V., Pica, P., Spelke, E., & Dehaene, S. (2011). From the Cover: Flexible intuitions of Euclidean geometry in an Amazonian indigene group Proceedings of the National Academy of Sciences, 108 (24), 9782-9787 DOI: 10.1073/pnas.1016686108