Its critical reception was mixed, with some reviewers praising it for its accessible and thorough approach to its material and for its many inspiring illustrations,[2][3][4][5] and others complaining about its inconsistent level of difficulty,[6] overuse of neologisms, failure to adequately cite prior work, and technical errors.
[8][9] It includes the first published classification of four-dimensional convex uniform polytopes announced by Conway and Richard K. Guy in 1965, and a discussion of William Thurston's geometrization conjecture, proved by Grigori Perelman shortly before the publication of the book, according to which all three-dimensional manifolds can be realized as symmetric spaces.
[2] However, reviewer Robert Moyer finds fault with its choice to include material at significantly different levels of difficulty, writing that for most of its audience, too much of the book will be unreadable.
[3] Jaron Lanier calls it "a plaything, an inexhaustible exercise in brain expansion for the reader, a work of art and a bold statement of what the culture of math can be like", and "a masterpiece".
These include the unnecessary use of "cute" neologisms for concepts that already have well-established terminology, an inadequate treatment of MacBeath's and Andreas Dress's contributions to the book's notation, sloppy reasoning in some arguments, inaccurate claims of novelty and failure to credit previous work in the classification of colored plane patterns, missing cases in this classification, likely errors in other of the more technical parts, poor copyediting, and a lack of clear definitions that ends up leaving out such central notions as the symmetries of a circle without providing any explanation of why they were omitted.