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The Bizarre Object We Believed Was Impossible to Visualize

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Mathematicians have now visualized abstract mathematical objects called flat tori — items resembling donuts with corrugated, fractal surfaces. These were thought to be impossible to envision in ordinary 3-D space... until now.

To imagine a flat torus, imagine a video game with a wraparound screen — for instance, if you go up the top side, you emerge from the down side. In the 1950s, mathematicians Nicolaas Kuiper and the Nobel laureate John Nash demonstrated the existence of this object, called a flat torus:


Imagine taking a square flat torus, wrapping it into a tube, and then bending its ends so they met to form a ring.


Is it possible to visualize this object in 3-D space? You might not think so — after all, a globe cannot be flattened into two dimensions, without distorting the distance between points on it. However, researchers in France have now accomplished exactly that.

The key, they explain, is to use corrugations. They piled up corrugations, until the distances between points was accurate.


The resulting surfaces of the objects are what are known as smooth fractals, which the researchers say lie halfway between fractals and ordinary surfaces:


These findings are more than just beautiful — they could open up new avenues in applied mathematics. For instance, this work could help visualize differential equations that are encountered in physics and biology.

The researchers detailed their findings online April 20 in the Proceedings of the National Academy of Sciences.