Abstract: Matter can arrange itself in the most ingenious ways. In addition to the solid, liquid and gas phases that are familiar in classical physics, quantum mechanics enables the existence of electronic phases of matter that can have both exotic and useful properties. In the last century, the thorough understanding of the simplest quantum electronic phase - the electrical insulator - enabled the development of the solid state electronics technology that is ubiquitous in today's information age. In the present century, new "topological" electronic phases are being discovered that may enable future technologies by allowing the seemingly impossible to occur: indivisible objects, like an electron or a quantum bit of information, can be split into two, allowing mysterious features of quantum mechanics to be harnessed. Our understanding of topological phases, which was celebrated by the 2016 Nobel Prize in physics, builds on deep ideas in mathematics. We will try to convey that they are as beautiful as they are fundamental.