The researchers call these objects "multi-dimensional sandcastles," which does a decent job of describing a twisted complex shape that's somewhat fleeting. "We found a world that we had never ...

The existence of these shapes, mathematicians have shown ... these points will typically form an 11-dimensional manifold. Now consider all the different points on your smaller sphere. Roughly speaking ...

But to mathematicians, shapes encompass a vast universe of surprising forms, from one-dimensional loops to polytopes (geometric objects with flat sides that can exist in any desired dimension).

Proteins begin as strings of chemical compounds, then twist and fold into three-dimensional shapes that define how these molecules bind to others. If scientists can pinpoint the shape of a ...