#topology

#topology

Cosmologists are now fairly certain that our universe is flat. But that doesn't explain the exact shape of space. It could extend infinitely along the three spatial dimensions or resemble a three-dimensional generalization of a donut's surfaceor take on even wilder forms. The mathematics of flat space is astonishingly versatile, and new research is upending the traditional thinking about the layout of our cosmos.

Hugo Steinhaus formulated a problem in 1938: Is it always possible to bisect three solids by one plane? He illustrated this with a sandwich example.

Any object or concept can be represented as a form, a topological surface, and consequently any process can be regarded as a transition from one form to another. If the transition is smooth and continuous, there are well-established mathematical methods for describing it. In nature, however, the evolution of forms usually involves abrupt changes and perplexing divergences, or transformations. Because these transformations represent sudden disruptions of otherwise continuous processes, Rene Thom of the Institut des Hautes Etudes Scientifiques in France termed them elementary catastrophes.

Standing in the middle of a field, we can easily forget that we live on a round planet. We're so small in comparison to the Earth that from our point of view, it looks flat. The world is full of such shapes-ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space.