New tendencies

Modern geometry has continued along this road and made it even more obvious, in both directions: more generality, more complexity. Correspondingly, mathematic techniques have become more and more abstract, to the point of making it difficult, if not impossible, to describe them even approximately. Since we don't want to completely renounce giving an idea of modern developments of curve geometry, we chose two examples that represent the two tendencies: geodetics and fractals.

On a flat surface, as for example in a city square, the shortest path between two points is a straight line. But if we move along a curved surface, like the surface of the Earth, it is not possible to go in a straight line, and in its place there is a curve, called geodetic, whose length is the minimum among all curves that touch both its ends. Just like as we pulled a string taut to get a straight line at the beginning of the exhibition, we could find the geodetic of a convex surface by pulling a string between two points. In the case of the Earth, which is more or less a sphere, the geodetics are the the maximum circles, those that divide the sphere in two equal parts. To go from one point to another it is best to move along the maximum circle that goes through the two points. This is the reason for polar routes in intercontinental air travel: on the maps they seem much longer, because they are drawn on a plane, but it is enough to pull an elastic band on a globe to see how actually the shortest route between Pisa and Los Angeles goes near the North Pole. The second example regards the very concept of curve. Those we have seen so far correspond to the intuitive idea of a curve that we all have; they are a single-dimensioned object, which we can think can be obtained bending a piece of string. If they have some special points, such as the edge of a polygon, these are finite and isolated. Around the end of the nineteenth century, pathological curves with surprising properties begin to pop up. One of which is Peano's curve, which fills a square and poses the problem of the meaning of dimensions: another is Koch's curve, which has no tangent in any point. Finally, in recent years, some new object have emerged, the fractals, which have a fractionary dimension and the surprising property that every part of them, no matter how small, is similar to the whole. These forms, whose generation is relatively simple, produce images of a remarkable beauty: images with which we conclude our visit in the world of curves.