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 ...

Einstein's 1917 paper, "Cosmological Considerations in the General Theory of Relativity," applied general relativity to the universe as a whole, establishing a firmer basis for cosmology. To create a ...

In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight ...

Science advances by overturning theories, replacing them by better ones. Sometimes the old theories continue to serve as valuable approximations, as with Newton’s laws of motion. Sometimes the older ...