High-dimensional spheres can have a much wider variety of structures than mathematicians thought possible. If you’ve ever been stuck in traffic on a rainy afternoon, you’ve probably watched raindrops ...

In our mind’s eye, the universe seems to go on forever. But using geometry we can explore a variety of three-dimensional shapes that offer alternatives to “ordinary” infinite space. When you gaze out ...

A massive feature on the moon formed due to lunar rifts, in a surprise revision to earlier theories, research shows. Previously, scientists thought the moon's Ocean of Storms was a round crater left ...

How do astronomers describe the shape of the universe, and what could it mean for the fate of the cosmos? When you purchase through links on our site, we may earn an affiliate commission. Here’s how ...

This media is in the public domain (free of copyright restrictions). You can copy, modify, and distribute this work without contacting the Smithsonian. For more information, visit the Smithsonian's ...

When most people think of shapes, they imagine a triangle, a rectangle, or maybe even a fancier-sounding rhombus or trapezoid. But to mathematicians, shapes encompass a vast universe of surprising ...

This is a preview. Log in through your library . Abstract We give an extensive discussion of sphere-preserving maps defined on subdomains of Euclidean n-space, and their relationship to Möbius maps ...

When you purchase through links on our site, we may earn an affiliate commission. Here’s how it works. A massive feature on the moon formed due to lunar rifts, in a surprise revision to earlier ...