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Squaring the circle has become synonymous with something unattainable.
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- Author, Dalia Ventura
- Scroll, BBC News World
2 hours ago
There are a number of classic problems from ancient mathematics that seem charmingly simple. In fact, solving them is not only difficult, but even impossible.
It took thousands of years 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 unsuccessfully to find a solution to these problems.
But his attempts were not in vain. They were inspiring and encouraged the development of mathematics.
It is not known exactly how these problems arose, but the most famous of them the quest to square the circle appears as early as the Rhind Papyrus, an Egyptian document from about four thousand years ago.
What is known is that it was the ancient Greeks who presented these problems precisely and in mathematical terms.
In short, the goal of these problems was to find:
• Squaring the circle
• the trisection of the angle
• Doubling the cube
• the inscription of all regular polygons in a circle
Put like this it may seem confusing, but what is actually being asked is:
• Draw a square whose area is equal to that of 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, right?
But as the American writer Donald Westlake (19332008) said: “If something seems simple, it is because there is a part that you have not heard.” Or in this case, that we have not said it.
These problems can only be solved in the style of ancient Greece. In other words, in addition to something to draw, something to draw with, and your mind, all you can use is a compass and a ruler without markings.
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Ruler (unmarked) and compass are the only tools that can be used to solve the classic challenges.
Why?
“That's a good question. And there are multiple answers,” mathematician David Richeson, author of Tales of Impossibility, told BBC News Mundo (the BBC’s Spanish service).
“One answer is that the compass and ruler are very clearly recorded in the postulates of the basic mathematics book Euclid's Elements [cerca de 300 a.C.]”, he explains.
“Another reason 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 you can draw a circle with the other end.”
“But also because of 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.”
For example, you can easily bisect an angle (divide it into two equal angles).
(1) Place the compass on the top of the square and draw an arc. (2) Place the compass at one of the intersections of the arc and lines and draw an arc. (3) Do the same at the other intersection. (4) Draw a line between the vertex of the angle and the intersection of the two arcs.
“We learned how to bisect an angle in geometry class at school. It’s very simple,” emphasizes Richeson. “But the question that interested the Greeks was: If you had an angle, could you divide it into three equal parts?”
“The answer is: sometimes yes, but there is no general rule.”
The mathematician continues: “That doesn’t mean that these problems are unsolvable, no matter what tools you use. But it is impossible to solve them with classical Euclidean tools.”
Archimedes, one of the greatest mathematicians in history, showed that it is possible to accurately measure a distance if the ruler has only two marks, which, according to Richeson, would be enough to proceed to trisecting any angle. “In other words, if your 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 to brilliant minds…
…very bright
The first known mathematician to attempt to square the circle was Anaxagoras, known for doing so in the 5th century BC. was the first to introduce philosophy to Athens, Greece
According to the historian Plutarch (46120 AD), Anaxagoras was arrested because he claimed that the sun was not a god but a stone burning bright red and that the moon reflected its light.
He spent his time in prison constructing a square with the same area as a circle using only a ruler and compass. But their 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 until the great Swiss mathematician and physicist Leonhard Euler (17071783) found two new types of lunulas that could be converted into squares in 1771. But his discovery would not contribute to squaring the circle of thinking thoughts as before.
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Hippocrates' lunula was the first of only five lunulas that can be converted into squares using a ruler and compass.
This is just the beginning of a long list of mathematicians, amateur or not, who have tried to achieve this goal using only the two tools.
“Leonardo da Vinci [14521519] “For a while he was fascinated by mathematics and geometry and tried to solve these problems, but also used his artistic talent to create drawings,” emphasizes Richeson.
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Leonardo da Vinci's Vitruvian Man conjured up the problem of squaring the circle in the 15th century, but did not attempt to solve it.
And da Vinci wasn't the only Renaissance man trying to solve classical problems. The most famous artist of the German Renaissance, Albrecht Dürer (14711528), was one of the most important mathematicians of that time.
In the second volume of his work “The Four Missals,” Dürer provided approximate methods for squaring the circle using ruler and compass constructions. And it also provided a method to get the trisection of the angle in a very approximate way using Euclidean tools.
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The famous German Renaissance artist Albrecht Dürer tried unsuccessfully to solve the problem of squaring the circle.
One of Richeson's most fascinating stories concerns 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. “Some of them were known to have been made, but not all. 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 math prodigy, Carl Friedrich Gauss (17771855), appeared.
“In 1796, Gauss was still a teenager, but he emerged as one of the most famous mathematicians in history. He showed that it was 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.
Since these problems are theoretical and not practical, it must also be taken into account that the proof of their solution is more important than the solution itself. And the detailed analysis that Gauss carried out to prove his discovery opened the doors for later ideas about the socalled Galois Theory.
So if you're wondering what the benefit is of having so many brilliant minds working so hard and trying to achieve something that in many cases could be achieved with other tools, then this is an example of a feedback process , which gave rise to many other things knowledge.
“Trying to solve these problems really advanced mathematics, but as mathematics developed, people also went back to old problems and saw whether new discoveries helped solve them,” the expert explains. “It’s been a bit of back and forth over the centuries.”
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“The Ancient of Days” by William Blake (17571827) depicts Urizen (the incarnation of conventional wisdom and law in his mythological universe) with a compass (for him, the symbol of reason that limits the imagination).
But not everything is possible
Attempting to solve these problems contributed to the progress of mathematics, but proving their impossibility depended on these advances.
“We had to wait for the invention of analytic geometry, algebra, analysis, complex numbers, a deep understanding of the number π, and even a little number theory,” says Richeson, “and that was one of the reasons it took so long .” Time.”
For example, in the case of squaring the circle, “the death knell occurred when it was discovered that π was a transcendent number.”
After centuries of 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 the ambition of more or less famous luminaries whose efforts advanced knowledge. Thousands of people over the years suffered from what the British mathematician Augustus De Morgan (18061871) called cyclometricus disease the squaringthecircle disease, which he said afflicted misinformed enthusiasts.
One of these people was the Argentine accountant and amateur mathematician Elías O'Donnell. In 1870 he published a book with “the deepest awareness that in this treatise the desired exact resolution of the square of the circle is demonstrated in the most convincing and rigorous manner,” as the author explained in the first page of the work.
“And no matter how serious this statement may seem, it will remain true for all centuries of posterity.”
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Detail of the cover of Elias O'Donnell's book, which aimed to solve the problem of squaring the circle.
However, thanks to Gauss, it was known since 1801 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 (18521939), proved that π is indeed a transcendent number.
And 45 years earlier, the French mathematician Pierre Wantzel (18141848) had proven on one of the seven pages of an article he wrote that the other three problems were also insoluble.
This is all amazing because proving something is impossible is immensely difficult… and important.
“When we think something is impossible, we usually think that it is very difficult, that it might take a long time, or something like that,” Richeson explains. “But when a mathematician proves that something is impossible, that means that from a logical point of view it cannot happen: there is no way to trisect a general angle. There’s no way to square the circle.”
“It's not just about 'We're not smart enough', 'We're not trying hard enough' or 'We need more time'. It's about 'Let's stop here: it's impossible'.”
“There are several famous impossibility theorems in mathematics and they are all highly revered because the negation has been shown: that something cannot happen,” the mathematician continues. “And that is an incredible success.”
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Promising attempts to solve the squaring of the circle have transferred the problem from geometry to graph theory, but using computers instead of rulers and compasses.
But that doesn't mean people give up.
For example, in 1897, the Indiana Senate in the United States discussed a bill to legalize a method of squaring the circle discovered by physician and amateur mathematician Edwin L. Goodwin.
The law aimed to “introduce a new mathematical truth.” It was initially approved by a committee until it was ultimately rejected.
It is said that there is not a mathematician who has not received solutions to squaring the circle, doubling cubes, or trisecting angles via email from people who are convinced they have found the solution.
“They insist because they don't understand the meaning of 'impossible,'” Richeson explains. And also because the supposed solutions are “easy to describe and play with”. So they try to believe they have solved it “and send the solutions to mathematicians at universities.”
“There will definitely be an error somewhere, be it mathematical or with the rules. So maybe they found a way to solve some of these problems, but not with the classic rules.”
Euclid built a comprehensive framework of wisdom and enabled the creation of new ideas, while his contemporaries and subsequent generations continued to seek to expand their knowledge using only a ruler and a compass.
These four problems were perhaps suspected even in ancient Greece to be impossible to solve. But trying to solve them was very rewarding.