The malaria paradox is a mathematical paradox that states that a figure with an infinite surface area can be painted with a finite amount of paint .
Consider an infinite stepped plate consisting of rectangles: the first of them is a square with a side of 1 cm, the second has dimensions of 0.5 × 2 cm, and each next is twice as narrow and twice as long as the previous one. The area of each rectangle is 1 cm 2 , and the total area of the plate is infinite.
To paint it all, you need an infinite (in volume or mass) amount of paint. Consider the body obtained by rotating the plate around its direct infinite edge. The vessel consists of cylinders . The height of the kth cylinder is 2 k −1 cm, the radius is 2 1− k cm, and therefore its volume is cm 3 . Thus, the volumes of the cylinders form a decreasing geometric progression , their sum is finite and equal to 2π cm 3 .
Fill this vessel with paint. Immerse this endless plate in it and pull it out; it will be painted with a final amount of paint on both sides.
Paradox Resolution
The statement "in order to paint a figure of infinite area, you need an infinite amount of paint" comes from the fact that the figure is covered with a layer of paint of the same thickness.
The proposed painting method assumes that each successive segment will be covered with an increasingly thin layer, so that the infinite amount of ink expended on each segment with an area of 1π cm 2 converges to the final value.
It should be borne in mind that the proposed mathematical solution does not take into account the physical fact that the paint layer cannot have a thickness less than the size of one paint molecule . Since the vessel constructed in the described way will taper down to infinitely small diameters , when the paint is “poured” into such a vessel, this paint simply will not “flow” into those areas whose diameter is less than the diameter of the paint molecule. Nevertheless, from the point of view of a mathematical model that does not take into account the physical aspects of the structure of our world, the described solution is true, despite the paradox.
Perhaps it may seem absurd that a vessel of infinite length can have a finite volume (in this case 2π), and even contain a plate whose area is infinite . But the fact is that length, area and volume are different values . In mathematical models, figures having an infinite area with a finite volume (or infinite length with a finite area) are quite possible.
Links
A. Panov ,. Painting paradox // Quantum. - 1986. - No. 8 . - S. 13 .
See also
- Geometric progression