The French mathematician Nicolas de Condorcet was a passionate believer in democracy, but he also discovered a frustrating paradox within it: A majority can sometimes vote itself into a logical loop. Mathematicians have now found that this “logical loop” maps onto a strange topological shape: the Klein bottle, a surface with no distinct inside or outside. On this episode of The Quanta Podcast, host Samir Patel speaks with writer Alex Stone about this new connection between this centuries-old voting paradox and the Klein bottle — and how it illustrates just how difficult it is to translate a group of individual preferences into a single choice. This topic was covered in a recent story for Quanta Magazine.  Each week on The Quanta Podcast, Quanta Magazine editor-in-chief Samir Patel and Senior Editor Hannah Waters speak with the people behind the award-winning publication to navigate through some of the most important and mind-expanding questions in science and math.