THIS work will be found really valuable by all students of geometry, especially by those who know little or nothing of the non-Euclidean theories. First of all we have a discussion of the elementary ...

Virtual reality can take you to some far-out places — mountaintops, distant cities and even fantastical game worlds. A team of artists and mathematicians is now adding to that list: universes where ...

This is a preview. Log in through your library . Journal Information PNAS is the world's most-cited multidisciplinary scientific serial. It publishes high-impact research reports, commentaries, ...

When a team of astronomers in 1992 released the first full-sky map of the cosmic microwave background —also known as the afterglow of the big bang—George Smoot, one of the group’s leaders and later a ...

In the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation ...

THE philosopher Kant declared that Euclidean geometry was inherent in the human mind and expressed the truth about space. We now recognize that non-Euclidean geometry is equally valid as an abstract ...