This article was originally published with the title “ Non-Euclidean Geometry before Euclid ” in Scientific American Magazine Vol. 221 No. 5 (November 1969), p. 87 doi:10.1038 ...

Euclidean geometry is the branch of mathematics describing shapes and the spatial relationships between objects in two-dimensional (2D) and three-dimensional (3D) settings. The rules of Euclidean ...

However, the spin-controlled topological phase transition in non-Euclidean space has not yet been explored. In work, titled "Spin-controlled topological phase transition in non-Euclidean space" ...