ARGOS PANOPTES IN THE MYTH OF ZEUS AND IO
The distribution of points in the sphere
In the Greek Mythology, Argos Panoptes was a giant whose body was covered by one hundred eyes. This made him a perfect guardian: he could look in every direction even if many of his eyes were closed.
A famous open mathematical problem is related to this myth: if one has to allocate a number of points in some given set, where should these points be positioned in order to maximize/minimize some given function?
In the myth above, we have 100 eyes to be distributed in the body of Argos and the function would be the solid angle covered by the eyes’ field of view after randomly closing a number of them.
There are several classic choices for the function to be optimized:
Thomson (the discoverer of the electron) suggested to minimize the electric potential of the points, seen as charged particles of the same sign; Tammes (a botanist studying pollen particles) tried to find points with the property that the minimal separation distance between any pair of them became maximal; Shub and Smale demand a computational algorithm to approximate points that maximize the product of their mutual distances.
This last problem (computationally approximate points that maximize the product of the mutual distances) is one of the most popular choices. It is known as Smale’s 7th problem, and it is included in the Field medallist’s famous list of «Mathematical Problems for the Next Century».
This beautiful problem is open even for the most simple cases. For example, if the body is just a sphere and the number of points is 7, the best distribution is believed to consist of two opposite points (say, the North and South poles) and the verteces of a regular pentagon in the equator. But no proof of this fact has been found yet!
Greek artists representing Argos many centuries ago allocated his eyes in some reasonable fashion all over his body. A 5th-century BC Athenian vase in the Kunsthistorisches Museum (Vienna) depicts a moment in the myth of Zeus and Io where Argos shows up.
We tell you the history in this animated video.
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