In 1974, Roger Penrose, a British mathematician, created a revolutionary set of tiles that could be used to cover an infinite plane in a pattern that never repeats. In 1982, Daniel Shechtman, an Israeli crystallographer, discovered a metallic alloy whose atoms were organized unlike anything ever observed in materials science. Penrose garnered public renown on a scale rarely seen in mathematics. Shechtman won the Nobel Prize. Both scientists defied human intuition and changed our basic understanding of nature's design, revealing how infinite variation could emerge within a highly ordered environment. At the heart of their breakthroughs is "forbidden symmetry," so-called because it flies in the face of a deeply ingrained association between symmetry and repetition.