What does it mean to "square the circle"?

[RandFan]

I just visited a site called timecube that to tell you the truth is beyond my comprehension skills. I'm guessing this guy shouldn't be left alone with small children or house pets.

But hey, general relativity was difficult for me so who knows.

What is squaring the circle?

I am wiser than any god or scientist, for I have squared the circle and cubed Earth's sphere

Is this a mathmatical concept or is he claiming to have changed the earth?

Hey stupid - are you too dumb to know there are 4 different simultaneous 24 hour days within a single rotation of Earth?

I simply don't have a clue what this means. I guess I am too dumb to know.

 

[Zep]

Gene Ray is the Internet's favourite looney. Consider him the pillory of the Internet, only less coherent.

Therefore don't bother your head about what he speaks of. It's commonly assumed Gene has no idea of what he is speaking about either.

 

[epepke]

I just visited a site called timecube that to tell you the truth is beyond my comprehension skills. I'm guessing this guy shouldn't be left alone with small children or house pets.

To "square the circle" is to construct a square with the same area as a circle of a given radius. It was a big math problem for a long time.

Of course, now we know that it can't be done except without using transcendental numbers (in this case, &pi. (or pi for the funny-browsered). And construction is mathematically equivalent to arithmetic over natural numbers. It's been a fair chunk of time since we've known and can prove that.

"Squaring the circle" thus comes across as rather archaic, so it seems like a parody to me.

 

[Beleth]

Re: Re: What does it mean to "square the circle"?

To "square the circle" is to construct a square with the same area as a circle of a given radius. It was a big math problem for a long time.

I think part of the challenge was that you were limited in the tools you could use, i.e. just a straightedge and a compass (the circle-drawing kind, not the north-pointing kind).

You can't get pi by just using those two devices.

The other classic straightedge-and-compass construction impossibilities are "trisecting the angle" (drawing an angle that is exactly 1/3 the size of a given angle) and "duplicating the cube" (making a cube that is exactly twice the volume of a given cube) because, just like pi, 1/3 and the cube root of 2 are not possible to make with just a straightedge and a compass.

 

[Ladewig]

The initial problem involved using only a straightedge and compass. It is quite impossible to succeed with that limitation.

 

[epepke]

Re: Re: Re: What does it mean to "square the circle"?

I think part of the challenge was that you were limited in the tools you could use, i.e. just a straightedge and a compass (the circle-drawing kind, not the north-pointing kind).

Yes. But these tools were the ones allowed in the Euclid game that was the big deal for thousands of years. Also, construction with compass and straightedge was for a long time considered the same as the use of rational numbers, which the Greek tradition liked very much.

Now, of course, we know that construction is a bit more powerful, as it can construct numbers of the form n = a + b * c^0.5, where a, b, and c are either rational or some n from another construction.

 

[Cabbage]

The above posts explained what it means to "square the circle", but there are a couple of inaccuracies.

With a compass and straightedge (and some designated unit of length), we can add lengths, subtract, multiply, and divide, along with taking square roots (and, of course, any finite combination of all five). So, for example, all rational numbers are constructible, along with things like sqrt(2) or sqrt(9 - sqrt(5 + 7/6)).

It's been mentioned that angle trisection in general is not constructible, however, this is not because 1/3 is not constuctible (1/3 is, in fact, constructible). Here's an explanation of why trisection is not constructible: Certainly a 60^o angle is constructible. If we can trisect any given angle, then a 20^o angle must also be constructible, and hence b=cos(20^o) is also constructible. However, b is a root of 4x³ - 3x - 1/2 = 0. This polynomial can't be factored over the rationals, so all of its roots will involve taking cube roots, which are not constructible. Therefore, angle trisection is not possible in general.

To finish, I should mention that it is certainly possible to trisect an arbitrary angle using a compass and ruler (a straightedge with one unit length marked off). Occasionally some poor soul discovers such a method, thinking he/she has actually solved an ancient and unsolvable problem, when in fact they're merely ignorant of the distinction between ruler and straightedge.

 

[RandFan]

This is so cool. The Internet is so amazing. Anytime a question pops into my head and I either google for an instant answer or ask a group who are likely to have the answer for me in a short period of time.

I thought it was some impossible problem. I understand why it was difficult for a period and now solvable.

Thank you.

P.S. I guess I should post my querie regarding the meaning of life in Philosophy and Religion.

 

[Krazy Miller]

I learn something everyday!

 

[Lendri]

I'd like to recommend the book "The Trisectors" by Underwood Dudley to everyone. It gives a fascinating insight into the minds of cranks, in this case people who think they have solved the trisection problem. Some are complete loons, while others merely ignorant. All seem to waste years or even decades on an entirely futile task, something to remember when casually debating some of the wilder thinkers out there.