Geometric Problems of Classical Greece: Angular Trisection – 02/02/2021 – Marcelo Viana

The three classic problems of geometry – multiplying the cube, dividing the angle into three, and squaring the circle – caught the attention of the great mathematicians of ancient Greece and challenged generations for more than two millennia before they were solved in the 19th century.

We do not know when they were first formulated. Writings of the philosopher Proclus from the 5th century BC. Chr. Already suggest solutions, but the problems are older. Perhaps the main novelty in the Greek formulation is the rule that it must be solved with a ruler (not graduated) and a compass, that is, only with lines and circles.

At any angle, it’s easy to cut it in half (split in half): draw a circle centered on the vertex of the angle. Look at the points where this circle intersects the sides of the angle and draw two circles centered on those points. As long as the compass is open enough, these circles intersect at two points. The line that goes through these two points divides the angle in half.

The problem with angle trisection is to look for such a construction that divides the angle into three equal parts. Some special cases are simple: for example, it is not difficult to cross the right angle (90 degrees).

The ancient Greeks knew methods of breaking every angle with other curves such as cones or spirals, and many others have been discovered over the centuries. There are even curves called trisectizes that are designed for this purpose.

However, the search for a general method of dividing the angle into three with just a ruler and a compass withstood all efforts until the Frenchman Pierre Wantzel (1814–1848) proved in 1837 that such a method cannot exist.

Wantzel has shown that a measurement angle A can be divided if and only if the polynomial 4×3-3x-cos (A) has a property called reducibility, which is not valid for most values ​​of A.

Wantzel’s work is a forerunner and contemporary of the theory of his young and talented (tragically ending, dueling) colleague Évariste Galois (1811-1832), which was only published posthumously in 1846.

One final note: a friend noticed that January was a palindromic month: the month and year numbers (12021) form a sequence of numbers that stay the same when read in reverse order. It was the start of a series that will end in September 2029 (92029). But then it will take some time.

Can you figure out what’s going to happen next?

Answers are welcome via email to viana.folhasp@gmail.com.

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