Schwarz lantern

In mathematics, the Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the areas of polyhedra. It is formed by stacked rings of isosceles triangles, arranged within each ring in the same pattern as an antiprism. The resulting shape can be folded from paper, and is named after mathematician Hermann Schwarz and for its resemblance to a cylindrical paper lantern. It is also known as Schwarz's boot, Schwarz's polyhedron, or the Chinese lantern. As Schwarz showed, for the surface area of a polyhedron to converge to the surface area of a curved surface, it is not sufficient to simply increase the number of rings and the number of isosceles triangles per ring. Depending on the relation of the number of rings to the number of triangles per ring, the area of the lantern can converge to the area of the cylinder, to a limit arbitrarily larger than the area of the cylinder, or to infinity—in other words, the area can diverge. The Schwarz lantern demonstrates that sampling a curved surface by close-together points and connecting them by small triangles is inadequate to ensure an accurate approximation of area, in contrast to the accurate approximation of arc length by inscribed polygonal chains. The phenomenon that closely sampled points can lead to inaccurate approximations of area has been called the Schwarz paradox. The Schwarz lantern is an instructive example in calculus and highlights the need for care when choosing a triangulation for applications in computer graphics and the finite element method. Archimedes approximated the circumference of circles by the lengths of inscribed or circumscribed regular polygons. More generally, the length of any smooth or rectifiable curve can be defined as the supremum of the lengths of polygonal chains inscribed in them. However, for this to work correctly, the vertices of the polygonal chains must lie on the given curve, rather than merely near it.