How many oranges can you fit in a box? Mathematicians are obsessed with perfecting their answer to this question in not just our familiar three-dimensional world, but in higher and higher dimensions beyond it. For several decades, they've made only minimal progress toward finding an optimal solution. Then, this past April, an outsider to the field named Boaz Klartag posted a proof that bested these previous records by a significant margin. In this episode of The Quanta Podcast, host Samir Patel and Quanta math staff writer Joseph Howlett discuss how Klartag resuscitated an old technique that experts had abandoned decades earlier to optimize sphere packing in any arbitrarily high dimension. This topic was covered in a recent story for Quanta Magazine.  Each week on The Quanta Podcast, Quanta Magazine editor in chief Samir Patel speaks with the people behind the award-winning publication to navigate through some of the most important and mind-expanding questions in science and math. Audio coda created by Daniel Simion