Schwarz lantern

Summary

In mathematics, the Schwarz lantern is a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the areas of polyhedra.[1] It consists of a family of polyhedral approximations to a right circular cylinder that converge pointwise to the cylinder but whose areas do not converge to the area of the cylinder. It is also known as the Chinese lantern, because of its resemblance to a cylindrical paper lantern, or as Schwarz's boot. The "Schwarz lantern" and "Schwarz's boot" names are from mathematician Hermann Schwarz.

Schwarz boot on display in the German Museum of Technology Berlin.

The sum of the angles at each vertex is equal to two flat angles ( radians). This has as a consequence that the Schwarz lantern can be folded out of a flat piece of paper. The crease pattern for this folded surface, a tessellation of the paper by isosceles triangles, has also been called the Yoshimura pattern,[2] after the work of Y. Yoshimura on the Yoshimura buckling pattern of cylindrical surfaces under axial compression, which can be similar in shape to the Schwarz lantern.[3]

ConstructionEdit

 
Animation of Schwarz lantern convergence (or lack thereof) for various relations between its two parameters

The discrete polyhedral approximation considered by Schwarz can be described by two parameters,   and  . The cylinder is sliced by parallel planes into   circles. Each of these circles contains   vertices of the Schwarz lantern, placed with equal spacing around the circle at (for unit circles) a circumferential distance of   from each other. Importantly, the vertices are placed so they shift in phase by   with each slice.[4][5]

From these vertices, the Schwarz lantern is defined as a polyhedral surface formed from isosceles triangles. Each triangle has as its base two consecutive vertices along one of the circular slices, and as its apex a vertex from an adjacent cycle. These triangles meet edge-to-edge to form a polyhedral manifold, topologically equivalent to the cylinder that is being approximated.

As Schwarz showed, it is not sufficient to simply increase   and   if we wish for the surface area of the polyhedron to converge to the surface area of the curved surface. Depending on the relation of   and   the area of the lantern can converge to the area of the cylinder, to a limit arbitrarily larger than the area of the cylinder, to infinity or in other words to diverge. Thus, the Schwarz lantern demonstrates that simply connecting inscribed vertices is not enough to ensure surface area convergence.[4][5]

History and motivationEdit

In the work of Archimedes it already appears that the length of a circle can be approximated by the length of regular polyhedra inscribed or circumscribed in the circle.[6][7] In general, for smooth or rectifiable curves their length can be defined as the supremum of the lengths of polygonal curves inscribed in them. The Schwarz lantern shows that surface area cannot be defined as the supremum of inscribed polyhedral surfaces.[8]

Schwarz devised his construction in the late 19th century as a counterexample to the erroneous definition in J. A. Serret's book Cours de calcul differentiel et integral,[9] which incorrectly states that:

Soit une portion de surface courbe terminee par un contour  ; nous nommerons aire de cette surface la limite   vers laquelle tend l'aire d'une surface polyedrale inscrite formee de faces triangulaires et terminee par un contour polygonal   ayant pour limite le contour  .

Il faut demontrer que la limite   existe et qu'elle est independante de la loi suivant laquelle decroissent les faces de la surface polyedrale inscrite'.

In English:

Let a portion of curved surface be bounded by a contour  ; we will define the area of this surface to be the limit   tended towards by the area of an inscribed polyhedral surface formed from triangular faces and bounded by a polygonal contour   whose limit is the contour  .

It must be shown that the limit   exists and that it is independent of the law according to which the faces of the inscribed polyhedral surface shrink.

Independently of Schwarz, Giuseppe Peano found the same counterexample. At the time, Peano was a student of Angelo Genocchi, who already knew about the difficulty on defining surface area from communication with Schwarz. Genocchi informed Charles Hermite, who had been using Serret's erroneous definition in his course. Hermite asked Schwarz for details, revised his course, and published the example in the second edition of his lecture notes (1883). The original note from Schwarz was not published until the second edition of his collected works in 1890.[10]

Limits of the areaEdit

A straight circular cylinder of radius   and height   can be parametrized in Cartesian coordinates using the equations

 
 
 

for   and  . The Schwarz lantern is a polyhedron with   triangular faces inscribed in the cylinder.

The vertices of the polyhedron correspond in the parametrization to the points

 
 

and the points

 
 

with   and  . All the faces are isosceles triangles congruent to each other. The base and the height of each of these triangles have lengths

 

respectively. This gives a total surface area for the Schwarz lantern

 .

Simplifying sines when  

 .

From this formula it follows that:

  1. If   for some constant  , then   when  . This limit is the surface area of the cylinder in which the Schwarz lantern is inscribed.
  2. If   for some constant  , then   when  . This limit depends on the value of   and can be made equal to any number not smaller than the area of the cylinder  .
  3. If  , then   as  .

See alsoEdit

ReferencesEdit

  1. ^ Zames, Frieda (September 1977). "Surface area and the cylinder area paradox". The Two-Year College Mathematics Journal. 8 (4): 207–211. doi:10.2307/3026930. JSTOR 3026930.
  2. ^ Miura, Koryo; Tachi, Tomohiro (2010). "Synthesis of rigid-foldable cylindrical polyhedra" (PDF). Symmetry: Art and Science, 8th Congress and Exhibition of ISIS. Gmünd.
  3. ^ Yoshimura, Yoshimaru (July 1955). On the mechanism of buckling of a circular cylindrical shell under axial compression. Technical Memorandum 1390. National Advisory Committee for Aeronautics.
  4. ^ a b Dubrovsky, Vladimir (March–April 1991). "In search of a definition of surface area" (PDF). Quantum. 1 (4): 6-9 and 64.
  5. ^ a b Berger, Marcel (1987). Geometry I. Universitext. Springer-Verlag, Berlin. pp. 263–264. doi:10.1007/978-3-540-93815-6. ISBN 978-3-540-11658-5. MR 2724360.
  6. ^ Traub, Gilbert (1984). The Development of the Mathematical Analysis of Curve Length from Archimedes to Lebesgue (Doctoral dissertation). New York University. p. 470. MR 2633321.
  7. ^ Brodie, Scott E. (1980). "Archimedes' axioms for arc-length and area". Mathematics Magazine. 53 (1): 36–39. doi:10.1080/0025570X.1980.11976824. JSTOR 2690029. MR 0560018.
  8. ^ Makarov, Boris; Podkorytov, Anatolii (2013). "Section 8.2.4". Real analysis: measures, integrals and applications. Universitext. Springer-Verlag, Berlin. pp. 415–416. doi:10.1007/978-1-4471-5122-7. ISBN 978-1-4471-5121-0. MR 3089088.
  9. ^ J. A. Serret, Cours de calcul differentiel et integral, Vol. II, page 296 of the first edition or page 298 of the second edition
  10. ^ Schwarz, H. A. (1890). "Sur une définition erronée de l'aire d'une surface courbe". Gesammelte Mathematische Abhandlungen von H. A. Schwarz (in French). Verlag von Julius Springer. pp. 309–311.

External linksEdit

  • Bogomolny, Alexander. "The Schwarz Lantern Explained". Cut-the-knot.