Math

Length and area are not what a shape looks like

An approximation can hug a curve as closely as you please and still have the wrong length: a staircase converging to a diagonal is 41 % too long at every refinement, for ever. One dimension up it is worse. Schwarz's lantern has every vertex on a cylinder and triangles as small as you like, and its total area tends to the true 2π, or to five times that, or to infinity - depending only on how its two refinements race each other. What decides is not the size of the pieces but their tilt, and the panel measures it.

By Mykhailo Makarov

Actual application image

What happens

The length of a curve is the limit of inscribed polygons - for the circle that works, and each doubling of the sides cuts the deficit by four. But a staircase drawn ever finer along a diagonal keeps its length 2 while it hugs a diagonal of length √2: the polygon must have short sides AND small angles. Area is worse. Schwarz's lantern inscribes triangles in a cylinder, and depending on how you refine it the total area converges to the cylinder's, to something larger, or to infinity - the triangles fold like a paper lantern. Refine the lantern along both knobs and watch where the area goes.

Equations used in the module

Equations
inscribed n-gon: L = 2nR sin(π/n) → 2πR, deficit ~ π³R/3n²
staircase to a diagonal: length 2 for every n, though the distance → 0
Schwarz's lantern, m rings × n sectors: A = 2mnR sin(π/n) √((H/m)² + R²(1 − cos π/n)²)
m = n: A → 2πRH;  m = n³: A → ∞

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