Mathematics without numbers book review

In the next section, the author deals with manifolds, which he describes as follows: “A model is called a manifold if it has no special points: there are no end points, no intersection points, no edge points, and no branch points. It must be the same in every place. “

Similar to this, one finds the sphere and the infinite plane as two-dimensional manifolds. As an added variety, Beckmann serves up the torus, a type of donut that has a distinctive feature with the hole in the middle. This shape can then be expanded into an endless family of torii with more holes. While this can still be well represented, the view of the so-called true projective planes is missing. However, Beckmann describes how one can “make” such a figure, that is, by stitching a disk with Möbius tape.

Unfortunately, the author avoids mentioning the name Möbius (or List) in this context – as Poincaré and Perelman do below; Instead, he limited himself to the unofficial phrase: “The third dimension (…) is now well thought out, even if it takes a hundred years and a million dollars in prize money.” Klein also addresses the bottle, without trying to identify its main advantage. It is a pity, however, that these surveys and predictions about the historical development of the theory would have fit so well with the chapter.

The author then deals with the generalized concept of dimension applied to everyday situations (five-dimensional flavours, three-dimensional colors, etc.) and concludes with a four-dimensional space-time, the topological form of which is as yet unclear.