In the next section, the author deals with manifolds, which he characterizes as follows: “A form is called a manifold if it has no special points: no end points, no intersection points, no edge points, no branch points. It must be the same everywhere.” From this colloquial description, it quickly becomes clear that only the circle and the infinite line can be considered as one-dimensional manifolds.
Analogous to this, one finds the sphere and the infinite plane as two-dimensional manifolds. As a further variety, Beckman then introduces the torus, a kind of donut whose characteristic feature is the hole in the middle. This shape can then be further expanded into an infinite family of tori with more holes. While this can still be represented well, the view of so-called real projective planes is missing. Nonetheless, Beckman describes how one might “make” such a shape, namely by sewing a disk together with a Mobius strip.
Unfortunately, the author avoids mentioning the name of Möbius (or Listing) in this context – as does Poincaré and Perelman below; Rather, he confines himself to the rather casual statement: “The third dimension (…) is now fairly well researched, even if it took a hundred years and a million dollars in prize money to get there.” He also addresses the Klein bottle , without attempting to outline its main feature. It’s a pity, these retrospectives and outlooks on the historical development of the theory would have fitted well into the chapter.
The author then deals with a generalized notion of dimension applied to everyday situations (flavors are five-dimensional, colors are three-dimensional, and so on) and concludes with four-dimensional space-time, the topological form of which is as yet unclear.