The Platonic solids are among the most beautiful and, from a mathematical perspective, the richest spatial forms. They are based on a wealth of harmonious properties that do not occur together in any other geometric body. Through numerous figures and explanatory texts, the characteristics of the Platonic solids are highlighted, such as their perfect symmetry and the uniformity and regularity of their vertices and surface shapes. For some of these spatial figures, the relationship to the surrounding space is incorporated using projective geometry. The final chapters are dedicated to various approaches to the inversion of Platonic solids, starting with Paul Schatz's invertible cube.