Shape

Jordan Ellenberg writes with the enviable ease of someone who can make a theorem feel like a conversational aside. In Shape, he uses geometry as a kind of intellectual cartography: not merely the study of figures, but a way of thinking about limits, symmetry, dimension, and the peculiar stubbornness of reality when it is forced into mathematical form. At his best, he is brisk, witty, and exact without becoming clerical. I admire especially his instinct for choosing examples that reveal how geometry sits underneath apparently unrelated subjects.

The book is strongest when it stays close to actual mathematical content and resists the lure of mere anecdote. Ellenberg has a real gift for translating abstraction into something a non-specialist can grasp without being patronised. He understands that a popular mathematics book should not be a puddle of metaphors with a theorem floating in it; it should preserve the structure of the ideas. On that count, Shape is often excellent. I found his explanations of dimension and geometry’s role in modern thought genuinely stimulating, and the prose has a clean confidence rare in this genre.

Yet I should not overpraise it, because the book does sometimes spread itself thin. It has the characteristic weakness of many ambitious popular mathematics books: too much horizon, not always enough depth. Ellenberg is so alert to the metaphysical implications of geometry that he occasionally sacrifices rigour for a breezy sweep, and the transitions between philosophical reflection and technical discussion can feel lightly underexplained. For a reader already mathematically literate, the book may occasionally seem to circle the more interesting difficulties rather than entering them. For a lay reader, certain passages may be elegant but not entirely illuminating.

Still, these are the failures of abundance, not poverty. Shape is a polished and often delightful book that takes its subject seriously enough to be worth serious attention. It is not a masterpiece of mathematical exposition, because it prefers breadth to penetration, but it is a very good piece of intellectual writing: humane, agile, and mathematically alert. I came away grateful for its intelligence, even while wishing it had been a little less eager to charm and a little more willing to linger over the hard edges.