They are not invisible (but they seem to be): so mirrors and mathematics can deceive you

No object covered with a mirrored surface is completely invisible. It does not matter the size of a tennis ball or an airplane. This was demonstrated by a Russian mathematician who has just published a study to clarify the extent to which this phenomenon can be deceived.
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They are not invisible (but they seem to be): so mirrors and mathematics can deceive you
No object covered with a mirrored surface is completely invisible. It does not matter the size of a tennis ball or an airplane. This was demonstrated by a Russian mathematician who has just published a study to clarify the extent to which this phenomenon can be deceived.

For a while, at least once a year (if not more) headlines announce that scientists have created an invisibility cloak like that of Harry Potter. But most articles actually deal with inventions that have little or nothing to do with the fabric used by the young wizard: most are made of materials that deflect light producing an optical effect that deceives the eye but nothing of textile fibers.

The main problem of this type of substances, also known as metamaterials, is its complexity and the high cost of its manufacture. But the truth is that there is a much cheaper and easier alternative to produce. If we coated an object with a mirrored surface, would we stop seeing it?

The mathematicians Alexander Plakhov and Vera Roshchina, of the Portuguese universities of Aveiro and Évora respectively, have been studying this question for several years. In 2012 they managed to demonstrate that, even if we make a body - be it a plane or a small ball - completely with mirrors, it will never be completely invisible.


Mathematics has a lot to say in making invisibility cloaks | Jannis I Flickr
But not content with giving him several twists to this phenomenon, Plakhov has published another study on him in 'Proceedings of The Royal Society A'. While the Russian and his colleague made it clear that it is not possible to completely obliterate an object by covering it with mirrors, this time they have estimated to what extent we could hide it from sight.

Putting value to invisibility

Based on mathematical and optical notions, it has shown that the level of invisibility depends on the volume of the object and the radius of an imaginary sphere that could contain it. To quantify this degree of concealment, Plakhov has defined the visibility index, whose value is zero for an invisible body and increases the better it is seen.

To conceive this factor has used what in mathematics is known as billiard theory, being understood as billiards a dynamic system like that of (you can imagine) the billiard tables. Only in this case the rays of light that reflect the surfaces of a mirrored object replace the balls that hit the board walls of the popular game.

Thus, the visibility index is determined by the angles with which the rays of light deviate when they collide against a body. If it were totally invisible, the angle would be zero because the light would pass through it and, therefore, would not change its trajectory. On the contrary, if we could see it, it would mean that it diverts light in different directions.

According to billiard theory, the visibility index can never be zero, although it does reach a very small positive minimum value that indicates how closely an object of being is invisible. At least in theory.

Much remains to be discovered

And if you think that enough is enough and that the mathematics of invisibility do not give more, you're wrong because Plakhov has stacks for a while. In fact, the index he has conceived is, for the moment, a mere estimate that he plans to convert into a more accurate value in the future.


Military aircraft or submarines could be covered with mirrored materials to keep them hidden | Panoramas I Flickr
On the other hand, the Russian wants to find out if there is a series of objects of constant volume but with increasing diameter (which tends to infinity) whose visibility index can almost fade. Furthermore, since a body can be invisible only when viewed from certain directions, it would be interesting to study how the index varies according to the position of the observer.

The subject of invisibility by mirrors is not just a mathematical curiosity. This optical effect has applications in the military field (to hide submarines or airplanes) or to manage the operation of small electronic devices controlling the flow of light and heat they receive. Perhaps the desired layer of (almost) invisibility of the future is made of mirrors.