4 ancient mathematical problems that demonstrate that the impossible was possible – 12/24/2023 – Science

There is a set of classic problems from ancient mathematics that seem charmingly simple. But in fact, it’s not just difficult to solve them — it’s impossible.

It took millennia to prove this impossibility. Meanwhile, geniuses such as Euclid, Archimedes, René Descartes, Isaac Newton and Carl Friedrich Gauss, as well as artists and intellectuals, tried to find a solution to these problems, without success.

But his attempts were not in vain. They were inspiring and boosted the development of mathematics.

It is not known exactly how these problems arose, but the most famous of them — looking for the square of the circle — already appears in the Rhind papyrus, an Egyptian document from around 4,000 years ago.

What is known is that it was the ancient Greeks who presented these problems precisely, in mathematical terms.

Briefly, the objectives of these problems were to find:

• squaring the circle
• the trisection of the angle
• doubling the cube
• the inscription of all regular polygons in a circle

Expressed this way, it may seem confusing, but in fact, what is being asked is:

• draw a square whose area is the same as a given circle
• divide an angle into three equal angles
• draw a cube that is twice the size of another
• divide a circle into equal parts

It’s clearer that way, isn’t it?

But, as American writer Donald Westlake (1933-2008) said, “whenever something seems easy, it’s because there’s a part you haven’t heard.” Or in this case, that we didn’t say.

You can only solve these problems in the style used in ancient Greece. In other words, in addition to something to draw a drawing on, something to draw on and your mind, you can only use a compass and a ruler without markings.

Why?

“This is a good question. And there are several answers”, mathematician David Richeson, author of the book “Tales of Impossibility”, told BBC News Mundo (the BBC’s Spanish service). (“Tales of impossibility”, in free translation).

“One answer is that the compass and straightedge are recorded very clearly in the postulates of the fundamental mathematics book ‘Euclid’s Elements’ [cerca de 300 a.C.]”, he explains.

“Another is that they represent the most basic tools that have always been used. With a rope you can draw a straight line, and if you attach one end to the ground, with the other you can draw a circle.”

“But also for its simplicity and elegance”, says the mathematician. “For me, the surprising thing is not so much what you can’t do, but everything you can do with these tools.”

You can, for example, bisect an angle (divide it into two equal angles) easily.

(1) Place the compass on the apex of the angle and draw an arc. (2) Place the compass at one of the intersection points of the arc and the lines and draw an arc. (3) Do the same at the other intersection point. (4) Draw a line between the vertex of the angle and the point of intersection of the two arcs.

“The bisection of an angle is something we learned in geometry class at school. It’s very simple”, highlights Richeson. “But the question that interested the Greeks is: if you have an angle, could you divide it into three equal parts?”

“The answer is: sometimes, yes, but there is no general rule for that.”

The mathematician continues: “This doesn’t mean that these problems are unsolvable, regardless of the tools you use. But, with classical Euclidean tools, it is impossible to solve them.”

Archimedes, one of the greatest mathematicians in history, demonstrated that, if the ruler has only two marks, it is possible to measure exactly one distance, which would be enough to proceed with the trisection of any angle, according to Richeson. “In other words, if his tools were a little more sophisticated, these problems could be solved.”

But it’s not worth it. The challenge is to solve problems while respecting the rules of the game, which is irresistible for brilliant minds…

…very bright

The first mathematician known to attempt to square the circle was Anaxagoras, famous for being the first to introduce philosophy to Athens, Greece, in the 5th century BC

Anaxagoras was arrested for stating that the Sun is not a god, but a rock that burns bright red, and that the Moon reflects its light, according to the historian Plutarch (46-120 AD).

He spent his time in prison trying to construct, with just a ruler and compass, a square with the same area as a circle. But his efforts were in vain.

His contemporary Hippocrates of Chius, one of the mathematicians whose work was synthesized in Euclidean geometry, achieved an encouraging partial solution: the Hippocratic lunula, the first quadrature of a curvilinear figure in history.

It would take 23 centuries for the great Swiss mathematician and physicist Leonhard Euler (1707-1783) to find two new types of lunulas that could be transformed into squares, in 1771. But his discovery would not contribute to squaring the circle, as was previously thought. think.

This is just the beginning of a long list of mathematicians, amateurs or not, who have tried to achieve this goal, armed with just the two tools.

“Leonardo da Vinci [1452-1519] He spent a period of time really fascinated by mathematics and geometry and tried to solve these problems, but he also incorporated his artistic talent to create drawings with them”, highlights Richeson.

And da Vinci was not the only Renaissance man trying to solve classical problems. The most famous artist of the German Renaissance, Albrecht Dürer (1471-1528), was another of the most important mathematicians of that era.

In the second volume of his work “The Four Books of Measurement”, Dürer provided approximate methods for squaring the circle, using ruler and compass constructions. And he also provided a method to obtain, in a very approximate way, the trisection of the angle with Euclidean tools.

For Richeson, one of the most fascinating stories talks about the construction of regular polygons – that is, the division of the circle into equal parts.

“This has always been a notoriously complicated problem,” he says. “Several of them were known to be made, but not all of them. Some, like polygons with 7, 9 and 17 sides, were unknown, and for many years people wondered if they were impossible.”

From the time of classical Greece until the end of the 18th century, there was no significant progress using Euclidean tools alone. Until the German mathematical prodigy Carl Friedrich Gauss (1777-1855) appeared.

“In 1796, Gauss was just a teenager, but he turned out to be one of the most famous mathematicians in history. He demonstrated that it is possible to construct a regular polygon with 17 sides.”

“It was one of his first discoveries — something that was impossible for generations of mathematicians,” says Richeson.

It is also necessary to keep in mind that, as these problems are theoretical and not practical, the evidence of their resolution is more important than the resolution itself. And the in-depth analysis carried out by Gauss to prove his discovery opened the doors to later ideas about the so-called Galois theory.

Therefore, if you were wondering what the benefit of so many brilliant minds having worked so hard, trying to achieve something that, in many cases, could be achieved with other tools, this is an example of a feedback process that generated a lot of other knowledge.

“Trying to solve these problems really pushed mathematics forward, but also, as mathematics developed, people went back to old problems and saw if new discoveries helped solve them,” explains the expert. “It was kind of back and forth over the centuries.”

But not everything is possible

Trying to solve these problems contributed to the progress of mathematics, but demonstrating its impossibility depended on these advances.

“We had to wait for the invention of analytic geometry, algebra, calculus, complex numbers, a deep understanding of the number π, and even a little number theory,” says Richeson, “and that was part of the reason it took so long time.”

In the case of squaring the circle, for example, “the death knell occurred when it was discovered that π is a transcendental number.”

After centuries of an obsession that even received a name in ancient Greece —tetragonidzein, or squaring the circle—, the search came to an end.

Squaring the circle was not just an ambition of more or less famous luminaries, who brought advances to knowledge with their efforts. Thousands of people over the years have suffered from what British mathematician Augustus De Morgan (1806-1871) called morbus cyclometricus — the squaring-the-circle disease that he said affected misinformed enthusiasts.

One of these people was the Argentine accountant and amateur mathematician Elías O’Donnell. In 1870, he published a book with “the most intimate awareness that, in this treatise, the desired exact resolution of the square of the circle is demonstrated, in the most convincing and rigorous way”, as declared by the author, on the first page of the work .

“And, however serious this statement may seem, it will be true for all centuries of posterity.”

But, since 1801, it was already known, thanks to Gauss, that π (the area of ​​the circle with radius 1) is transcendent and, therefore, squaring the circle is impossible.

In 1882, another German mathematician, Ferdinand Von Lindemann (1852-1939), demonstrated that, in fact, π is a transcendental number.

And, 45 years earlier, the French mathematician Pierre Wantzel (1814-1848) had proven, in one of the seven pages of an article he authored, that the other three problems are also insoluble.

All of this is amazing, because proving that something is impossible is immensely difficult… and important.

“Usually, when we think something is impossible, we believe it is very difficult, that it might take a long time or something like that,” explains Richeson. “But when a mathematician demonstrates that something is impossible, it means that, from a logical point of view, it cannot happen: there is no way to trisect a general angle. There is no way to square the circle.”

“It’s not just ‘we’re not smart enough’, ‘we’re not trying hard enough’ or ‘we need more time. It’s ‘let’s stop here: it’s impossible’.”

“There are several famous impossibility theorems in mathematics and they are all highly revered because negation was demonstrated: that something cannot happen”, continues the mathematician. “And this is an incredible success.”

But that doesn’t mean people give up.

In 1897, for example, the Indiana Senate in the United States discussed a bill to legalize a method of squaring the circle discovered by the physician and amateur mathematician Edwin L. Goodwin.

The law sought to “introduce a new mathematical truth”. It was initially accepted by a committee, until it was finally rejected.

It is said that there is no mathematician who has not received solutions on squaring the circle, doubling cubes or trisecting angles by email, from people convinced of having found the solution.

“They insist because they don’t understand the meaning of ‘impossible'”, explains Richeson. And also because the supposed solutions “are easy to describe and play with.” Therefore, they try, believe they have solved it “and send the solutions to mathematicians at universities”.

“There will definitely be an error somewhere, whether it’s mathematical or with the rules. So maybe they found a way to solve some of these problems, but not using the classical rules.”

Euclid built an entire framework of wisdom and enabled the creation of new ideas, as his contemporaries and subsequent generations continued trying to advance knowledge, using only a ruler and compass.

In the case of these four problems, perhaps it was suspected since ancient Greece that their solution would be impossible. But trying to solve them was very enriching.

This text was originally published here.