Schwarz lantern

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

In mathematics, the Schwarz lantern (also known as the Chinese lantern or Schwarz's boot, after mathematician Hermann Schwarz) 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.

The polyhedral surface bears resemblance to a cylindrical paper lantern. The sum of the angles at each vertex is equal to two flat angles (${\displaystyle 2\pi }$ 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]

Construction[]

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, ${\displaystyle m}$ and ${\displaystyle n}$. The cylinder is sliced by parallel planes into ${\displaystyle 2n}$ circles. Each of these circles contains ${\displaystyle 2m}$ vertices of the Schwarz lantern, placed with equal spacing around the circle at (for unit circles) a circumferential distance of ${\displaystyle \pi /m}$ from each other. Importantly, the vertices are placed so they shift in phase by ${\displaystyle \pi /2m}$ 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 ${\displaystyle m}$ and ${\displaystyle n}$ 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 ${\displaystyle m}$ and ${\displaystyle n}$ 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 motivation[]

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 as a counterexample to the erroneous definition in J. A. Serret's book Cours de calcul differentiel et integral, second volume, page 296 of the first edition or page 298 of the second edition, in which it is said:

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

Il faut demontrer que la limite ${\displaystyle S}$ 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 ${\displaystyle C}$; we will define the area of this surface to be the limit ${\displaystyle S}$ tended towards by the area of an inscribed polyhedral surface formed from triangular faces and bounded by a polygonal contour ${\displaystyle \Gamma }$ whose limit is the contour ${\displaystyle C}$.

It must be shown that the limit ${\displaystyle S}$ 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 while a student of his teacher Angelo Genocchi, who already knew about the difficulty on defining surface area from his communication with Schwarz. Genocchi informed Charles Hermite, who had been using Serret's erroneous definition in his course. After requesting details to Schwarz, Hermite 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.^[9]

Limits of the area[]

A straight circular cylinder of radius ${\displaystyle r}$ and height ${\displaystyle h}$ can be parametrized in Cartesian coordinates using the equations

${\displaystyle x=r\cos(u)}$

${\displaystyle y=r\sin(u)}$

${\displaystyle z=v}$

for ${\displaystyle 0\leq u\leq 2\pi }$ and ${\displaystyle 0\leq v\leq h}$. The Schwarz lantern is a polyhedron with ${\displaystyle 4mn}$ triangular faces inscribed in the cylinder.

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

${\displaystyle u={\frac {2\mu \pi }{m}}}$

${\displaystyle v={\frac {\nu h}{n}}}$

and the points

${\displaystyle u={\frac {(2\mu +1)\pi }{m}}}$

${\displaystyle v={\frac {(2\nu +1)h}{2n}}}$

with ${\displaystyle \mu =0,1,2,\ldots ,m-1}$ and ${\displaystyle \nu =0,1,2,\ldots ,n-1}$. All the faces are isosceles triangles congruent to each other. The base and the height of each of these triangles have lengths

${\displaystyle 2r\sin \left({\frac {\pi }{m}}\right){\text{ and }}{\sqrt {r^{2}\left[1-\cos \left({\frac {\pi }{m}}\right)\right]^{2}+\left({\frac {h}{2n}}\right)^{2}}}}$

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

${\displaystyle S(m,n)=4mnr\sin \left({\frac {\pi }{m}}\right){\sqrt {4r^{2}\sin ^{4}\left({\frac {\pi }{2m}}\right)+\left({\frac {h}{2n}}\right)^{2}}}}$.

Simplifying sines when ${\displaystyle m\to \infty }$

${\displaystyle S(m,n)\simeq 4\pi nr{\sqrt {\left({\frac {\pi ^{2}r}{2m^{2}}}\right)^{2}+\left({\frac {h}{2n}}\right)^{2}}}=2\pi r{\sqrt {\left(\pi ^{2}r{\frac {n}{m^{2}}}\right)^{2}+h^{2}}}}$.

From this formula it follows that:

1. If ${\displaystyle n=am}$ for some constant ${\displaystyle a}$, then ${\displaystyle S(m,am)\to 2\pi rh}$ when ${\displaystyle m\to \infty }$. This limit is the surface area of the cylinder in which the Schwarz lantern is inscribed.
2. If ${\displaystyle n=am^{2}}$ for some constant ${\displaystyle a}$, then ${\displaystyle S(m,am^{2})\to 2\pi r{\sqrt {\pi ^{4}r^{2}a^{2}+h^{2}}}}$ when ${\displaystyle m\to \infty }$. This limit depends on the value of ${\displaystyle a}$ and can be made equal to any number not smaller than the area of the cylinder ${\displaystyle 2r\pi h}$.
3. If ${\displaystyle n=am^{3}}$, then ${\displaystyle S(m,am^{3})\to \infty }$ as ${\displaystyle m\to \infty }$.

See also[]

• Runge's phenomenon, another example of failure of convergence

References[]

External links[]

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