Non-Euclidean Geometry

Courtesy: National Science Foundation

Kuen’s Surface: A Meditation on Euclid, Lobachevsky and Quantum Fields
Credit: Richard Palais and Luc Benard, University of California at Irvine

Sketch a line and then draw a point off it. How many lines parallel to the first line can you draw through that point? The Greek mathematician Euclid said just one, but for more than 2,000 years after his death, mathematicians struggled to prove that he was right based on his other geometric rules. Then the 19th century Russian mathematician Nikolai Lobachevsky showed that you couldn’t: In some circumstances, you can sketch an infinite number of lines through that point and not violate any of Euclid’s other axioms. Mathematician Dick Palais of the University of California, Irvine, and digital artist Luc Benard wanted to convey the history of Lobachevsky’s solution to this mathematical puzzle with their illustration.

In this illustration, a sheet of paper shows sketches of one of these surfaces, called Kuen’s surface, and the expression, called a soliton, that describes it. “We wanted to talk about these equations in a way that non mathematicians could understand,” Palais says. “So we took a symbolic approach: The surface itself stands as a symbol for that equation.”


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