The superelipse of Hein | Science

The departure of Hex on a reduced 5 x 5 board that we saw last week was not plausible, because in the previous move Black could have prevented White's chain from being completed while creating a winning position.

It is easy to find a winning strategy for the first player in reduced boards. For example, in the 5 x 5 board the first player easily wins by occupying the center square in his initial move (although not in the way you see in last week's example). But on the 11 x 11 board there is such an astronomically large number of possibilities that a winning strategy is not known. However, in 1949 John Nash, which apparently reinvented the game independently of Piet Hein, showed that such a strategy has to exist; summarized (the rigorous demonstration is somewhat more complicated), his reasoning is as follows: suppose that the second player has an advantage; in that case, the first player has only to make an irrelevant initial play and then adopt the strategy of the second, since a chip of his own on the board can not be an inconvenience.

John Nash walks through the Plaza del Obradoiro, in Santiago de Compostela, in 2007.
John Nash walks through the Plaza del Obradoiro, in Santiago de Compostela, in 2007.

Nash's "proof of existence" is based on the assumption that one of the players must necessarily win, and, in fact, the tie is inconceivable in practice; but is it theoretically impossible?

From superelipse to superhuevo

As is well known, the circumference is a closed curve whose points equidistant from another, which is the center. And in the case of the ellipse, it is the sum of distances to two other points, called foci, which remains constant for all points of the curve.

The formula of the ellipse is xtwo/totwo + andtwo/ btwo = 1. If a = b, the formula becomes xtwo + andtwo = atwo, and the ellipse, in a circle of radius a. And in the same way that the circumference can be considered a particular case of the ellipse, we can consider that the ellipse is a particular case of a family of curves of the form xn/ton + andn/ bn = 1. In fact, this was considered by the French mathematician Gabriel Lamé, who named these curves (also called superelipses) and studied them in the mid-nineteenth century.

One hundred years later, the Danish writer and engineer Piet Hein, the inventor of the Hex, studied a curve of Lamé in particular: the one of exponent n = 2.5, with a = 4 and b = 3, and applied it to the design of tables and other furniture , as well as the layout of a roundabout in a rectangular square in Stockholm. The conventional ellipse would seem the first option, for a simple rule of three: circumference is to square as ellipse to rectangle. What is the advantage of Hein's superelipse over the ellipse? What happens as the exponent n increases?

Rotating its superelipse around its major axis, Hein obtained an interesting superelipsoid, called "superhuevo", which has been reproduced in very different sizes, as a gift and as a sculpture, and which has a surprising property that it does not share with its cousin. the ellipsoid of revolution. Which one?

Carlo Frabetti He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among them Damn physics, Damn mathematics or The big game. He was a screenwriter The Cristal ball.


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