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Finite area, infinite perimeter - Colegiul National Mihail Eminescu (Satu Mare - Roumanie) Lycée Bellevue (Alès)

Titre du sujet: 
Finite area, infinite perimeter
Etablissement(s) jumelé(s):
Is it possible to draw a figure with a finite area and an infinite perimeter? Draw examples. And more specifically, can we modify a figure to widen its perimeter endlessly without changing its area? We can define the dimension of an object that way: when we make a homothety of ratio k of that figure, then its size increases of a ratio k^d, where d defines the dimension of the object. For instance, - If we have a curve of l length and we make a homothety of ratio 2, then the length of the curve doubles (it is equal to 2l=2^1l). So the dimension of the curve is equal to 1. - In the same way, if a surface has an area equal to a, then, after a homothety of ratio 2, the area of the surface so obtained is equal to 4a=2^2a, so the dimension of the surface is equal to 2. By repeating this method on the infinite perimeter of the surface with finite area, is it possible to calculate the dimension of the perimeter?
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Sujet | by Dr. Radut