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Photo by Spiros Kakos from Pexels

Atomic interactions in everyday solids and liquids are so complex that some of these materials’ properties continue to elude physicists’ understanding. Solving the problems mathematically is beyond the capabilities of modern computers, so scientists at Princeton University have turned to an unusual branch of geometry instead.

Researchers led by Andrew Houck, a professor of electrical engineering, have built an electronic array on a microchip that simulates particle interactions in a hyperbolic plane, a geometric surface in which space curves away from itself at every point. A hyperbolic plane is difficult to envision — the artist M.C. Escher used hyperbolic geometry in many of his mind-bending pieces — but is perfect for answering questions about particle interactions and other challenging mathematical questions. (1)

Draw a line on the paper.

Heat waves. Like… sound waves?

Look at the circle on the sand.

A teardrop falling on water.

The moon circling the Earth.

A circle turning into a square.

Sun turning into darkness.

The ink is blurring now.

The line is fading.

And with strange aeons…

Even the paper will reduce into dust.

Your geometry will be lost. Along with everything reminding it. You will be alone at the end. And your tears will fall in the water. And they will create circles again. Don’t cry. Just take the pen. Don’t wander whether you can draw one on paper. You know you can…