The Schwarz Lantern
It was yet known to Archimedes that the circumference of a circle could be approximated by the perimeters of inscribed (and also circumscribed) regular polygons. However, "proximity" of two curves does not always insure that their lengths are also close. There is an almost trivial example of a broken line approximating a square, with no convergence of their lengths.
In 3D the situation is analogous. It is quite natural to define the areas of curved surfaces as limits of the polyhedral surface areas. For example, the surface area of a cylinder can be defined as the limit of the surface areas of inscribed regular prisms. But, as in the 2D case, the caution must be exercised. For, not every such surface approximation ensures convergences of areas.
The surface illustrated by the applet below is variously referred to as the Schwarz surface or Schwarz lantern after the German mathematician Karl Hermann Amandus Schwarz (1843-1921), or, with no attribution, the Chinese lantern (Berger, p. 263). Scwartz published the example in 1880.
The Schwarz lantern is a polyhedral surface inscribed into a cylinder. The surface depends on two parameters N and M.
M is the number of (equal) rings - "short" cylinders - cur off the cylinder lengthwise.
Each of the rings is triangulated into 2N congruent isosceles triangles. This is achieved by dividing the two circular boundaries of a ring into N parts and then shifting the division points on one circle relative to the points on the other, and joining the points with a zig-zag line.
By the construction, the vertices of the Schwarz lanterns lie on the cylinder. Furthermore, as N and M both grow, the surfaces approach the cylinder pointwise and in the Hausdorff metric. The behavior of the areas is rather less certain. Depending on the relative magnitudes of N and M as they grow, the areas of Schwarz lanterns may have a limit anywhere between the area of the cylinder and infinity (inclusive).
To approximate the area of the cylinder, the triangles forming the lantern need to approach the planes tangential to the cylinder. This only happens when, as both N and M grow,
The surface in the applet is controlled by dragging the mouse. With the ALT key pressed, the shape will zoom in or out. Otherwise, it will rotate. A right click on the front side of a triangular piece will remove (or return) the piece.
|What if applet does not run?|
It is worth noting that, as N grows, the volume of the lantern does approach that of the cylinder. This is so because the volume of the lantern exceeds that of a smaller cylinder inscribed in it. The same cylinder is inscribed into the right regular prism with N sides whose base is inscribed into the base of the original cylinder. As N grows, the volumes of the two cylinders become arbitrarily close.
- M. Berger, Geometry I, Springer-Verlag, 1994, p. 263