Why can't a circle be squared, but a lune can?

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In summary, the conversation discusses the concept of squaring a circle, which has been a challenge for mathematicians for centuries. The author mentions reading about Hippocrates' proof that the lune is squarable, but that the circle itself cannot be squared due to its dependence on the transcendental number pi. The author also questions why other curved shapes, such as the lune, can be squared. The expert summarizer explains that it is not always possible to square a lune, but a specific type of lune, known as the lune of Hippocrates, is squarable due to its rational area.
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The Anomaly
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I'm new to this forum, and I am very excited to be a member of this esteemed community. Anyway, I've been reading the Journey Through Genius, by William Dunham, and I am truly loving it. However, I had a question about a statement in the text, and I thought I'd ask if you guys could help me out with it.

I was reading about Hippocrates' proof that the lune is squarable, and the author said that despite hundreds of years of effort, the circle could not be squared. In other words, that there could not be made a square that has the same area as a circle. This was then proved in the nineteenth century by Ferdinand Lindemann.

Anyway, I understand both Hippocrates proof, and Lindemann's proof, and I believe I fully understand why it'd be impossible to make a square that is the same area as a circle. It makes sense--I mean, pi is transcendental and all, and as such it can not be drawn. However, what I don't understand is why we can still square a curved shape such as a lune. Would not a lune simply be a section of a circle? I mean, doesn't it have that same curve that a circle has--just that it is not a perfect circle?

Basically, what makes a circle so intrinsically special that it can not be squared, while other curved ones can be? Is it simply because their areas are not linked to a transcendental number like the circle is?
 
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What is special about a circle is just what you said above: its area depends on the constant [itex]\pi[/tex] which is transcendental. It is NOT true that a lune can alwats be squared. What is true is that a certain lune, the lune of Hipocrates, which is formed by intersecting a circle of Radius R with a circle of radius [itex]\sqrt{R/2}[/itex] passing through points on the first circle a quarter circle apart (see http://en.wikipedia.org/wiki/Lune_of_Hippocrates), has area (1/2)R2, a rational number, and so is "squarable".
 
  • #3
HallsofIvy said:
What is special about a circle is just what you said above: its area depends on the constant [itex]\pi[/tex] which is transcendental. It is NOT true that a lune can alwats be squared. What is true is that a certain lune, the lune of Hipocrates, which is formed by intersecting a circle of Radius R with a circle of radius [itex]\sqrt{R/2}[/itex] passing through points on the first circle a quarter circle apart (see http://en.wikipedia.org/wiki/Lune_of_Hippocrates), has area (1/2)R2, a rational number, and so is "squarable".

Alright, thanks for the answer. That's just what I wanted to know.
 

What is the concept of quadrability of a lune?

The quadrability of a lune refers to the ability to construct a square with the same area as a given lune, which is a shape formed by two circular arcs intersecting at their endpoints.

Who first studied the quadrability of a lune?

The concept was first studied by ancient Greek mathematician Archimedes in the 3rd century BC.

What is the significance of quadrability of a lune?

The quadrability of a lune is significant because it demonstrates the relationship between geometry and algebra, and it was one of the earliest examples of a mathematical problem that could not be solved with traditional geometric methods.

How is the quadrability of a lune solved?

The problem can be solved using the method of exhaustion, which involves dividing the lune into an infinite number of smaller parts and finding the sum of their areas. This sum can then be compared to the area of a square to determine if they are equal.

Are there any real-life applications of the quadrability of a lune?

While the concept itself may not have many direct applications, the method of exhaustion used to solve it has been applied to other mathematical problems and has had a significant impact on the development of calculus and other branches of mathematics.

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