The geometry of impossibility

"Squaring the circle" is back. A colleague reports having heard the phrase several different times in close succession recently, in the kind of frequency that is the sign of a new - or returning - vogue phrase.

She was hearing it in connection with Hamas, the Islamic Resistance Movement, whose surprise victory in the Palestinian elections of Jan. 25 has forced observers of Middle Eastern politics to do some recalculating. Many Palestinians are delighted to see the Fatah party, long regarded as inept and corrupt, getting the boot in favor of Hamas, seen as honest, competent, and disciplined.

But for many in Israel, which has suffered hundreds killed in Hamas suicide bombings, the prospect of negotiating the future with a Hamas government representing the Palestinians is nothing less than horrifying. Yes, Israelis have wanted a real negotiating partner with a solid mandate from the Palestinian people. But how, they ask, can they be expected to negotiate with a party whose charter calls for the destruction of Israel?

"We are going to have to find a creative way of squaring the circle," an Israeli official told Richard Beeston of The Times of London.

"To square the circle" is, metaphorically speaking, to perform an inherently impossible task. The phrase is rooted deeply in classical Greek geometry. It meant drawing a square with the same area as a given circle. If one confines oneself to the traditional tools of classical geometry - a straightedge and a compass - the task is impossible.

NATO seems to have a knack for trying to square the circle. Ten years ago, the Atlantic Alliance, which had arguably lost its raison d'être with the collapse of the Soviet Union and the end of the cold war, was moving into a new role as the democratic club that former Soviet satellites were eager to get into.

The challenge was how to afford Eastern Europe the protection of the NATO umbrella without riling the Russians to the point of precipitating the very instability the alliance was intended to prevent. "Squaring the circle," they called it then. In fact, it wasn't just a clever phrase that people kept using; it hardened into a set phrase, a term of art used with a very specific meaning.

But one of the most interesting discussions of "squaring the circle" that I've run across is a Dartmouth College course, "Geometry in Art and Architecture." It goes back not just to the Greeks but to the Egyptians, who, faced with the Nile rearranging their real estate every year, had to develop a system of "earth measure" (the literal meaning of the Greek elements that have come into English as "geometry") to keep track of what was where.

The online course notes read in part:

"If flooding of the Nile symbolized the annual return of watery chaos, then geometry, used to reestablish the boundaries, was perhaps seen as restoring law and order on earth....

"The square, with its four corners like the corners of a house, represents earthly things, while the circle, perfect, endless, infinite, has often been taken to represent the divine or godly. So squaring the circle is a universal symbol of bringing the earthly and mundane into a proper relationship with the divine."

No wonder it seems an impossible task.

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