Schwarz lantern
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.
[5][6] 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.
[1] However, for this to work correctly, the vertices of the polygonal chains must lie on the given curve, rather than merely near it.
Otherwise, in a counterexample sometimes known as the staircase paradox, polygonal chains of vertical and horizontal line segments of total length
can lie arbitrarily close to a diagonal line segment of length
[1] German mathematician Hermann Schwarz (1843–1921) devised his construction in the late 19th century[a] as a counterexample to the erroneous definition in J.
A. Serret's 1868 book Cours de calcul differentiel et integral,[12] which incorrectly states that: Soit une portion de surface courbe terminée par un contour
; nous nommerons aire de cette surface la limite
vers laquelle tend l'aire d'une surface polyédrale inscrite formée de faces triangulaires et terminee par un contour polygonal
existe et qu'elle est indépendante de la loi suivant laquelle décroissent les faces de la surface polyedrale inscrite.Let a portion of curved surface be bounded by a contour
tended towards by the area of an inscribed polyhedral surface formed from triangular faces and bounded by a polygonal contour
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.
[10] At the time, Peano was a student of Angelo Genocchi, who, from communication with Schwarz, already knew about the difficulty of defining surface area.
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).
[13][14] An instructive example of the value of careful definitions in calculus,[5] the Schwarz lantern also highlights the need for care in choosing a triangulation for applications in computer graphics and for the finite element method for scientific and engineering simulations.
[6] The failure of Schwarz lanterns to converge to the cylinder's area only happens when they include highly obtuse triangles, with angles close to 180°.
The Schwarz lantern's example shows that, even for simple functions such as the height of a cylinder above a plane through its axis, and even when the function values are calculated accurately at the triangulation vertices, a triangulation with angles close to 180° can produce highly inaccurate simulation results.
), the resulting surface consists of the triangular faces of an antiprism of order
vertices of the Schwarz lantern are spaced equally, forming a regular polygon.
These triangles meet edge-to-edge to form the Schwarz lantern, a polyhedral surface that is topologically equivalent to the cylinder.
[16] Ignoring top and bottom vertices, each vertex touches two apex angles and four base angles of congruent isosceles triangles, just as it would in a tessellation of the plane by triangles of the same shape.
As a consequence, the Schwarz lantern can be folded from a flat piece of paper, with this tessellation as its crease pattern.
Combining the formula for the area of each triangle from its base and height, and the total number
The Schwarz lanterns, for large values of both parameters, converge uniformly to the cylinder that they approximate.
, and to examine the limit as both parameters grow large simultaneously, maintaining this relation.
Different choices of this relation can lead to either of the two behaviors described above, convergence to the correct area or divergence to infinity.
Any desired larger area can be obtained by making an appropriate choice of the constant
