Why Circle Squaring Is Impossible

I don't have a clue what "squaring a circle" is but I'm pretty sure that if you give me a big enough hammer I can pound a round peg into a square hole.

To everyone convinced that 'squaring a circle' is not possible...

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...Let me ask you a more simple question (so that I might better understand the problem)...

Is it possible to 'square a right triangle'?

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If you answer 'no', then I believe the original question doesn't mean much.

But if you answer 'yes', you might want to rethink 'squaring a circle'.

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Why?.... Because, 'squaring a circle' is easy if you can 'square a right triangle',...

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You just have to 'right triangle the circle' first.

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And 'right triangle(ing) a circle' is easy since the area of a circle is the same as the area of right triangle with one leg adjacent to the right angle equal in length to the circumference and the other leg adjacent to the right angle equal in length to the radius.

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Square away.

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What? are now not allowed to know the dimensions (notice 'dimensions' is plural, as in not just one) of the circle to be squared? Well that would just be silly, of course you can't make a square of the same area as a circle of unknown dimensions.

:)

Because, 'squaring a circle' is easy if you can 'square a right triangle',...
I've got homemade Vodka waiting for me....oh...and a girl...no math tonight...

That just shifts the problem slightly. You still need to construct a line segment of length pi, which is impossible, because pi has been proven to be an unconstructible number (using the rules of finite straightedge/compass construction).

I don't know what it is either..............I think I'm glad I don't.

I will admit that I like watching Ancient Aliens on the History channel. It's pretty fascinating how ancient civilizations were able to precisely cut, move and stack massive chunks of rock so tightly that they are almost water tight.

The different theories are interesting too. I guess what I'm saying is that it's kind of fun to listen to the different ideas on this stuff....................just don't fall in the soup.

I've always been a little bugged by pi. Since it seemingly has no end, wouldn't that imply that we can never know the exact true area of a circle?

But, since the inside of a circle is a finite space, an exact answer should be possible.

I understand that pi*r² works, but pi itself is frustrating...........at least to me.

Since I'm not a mathematical genius, the way I would square a circle would be to measure the circumference, divide by four, and make a square with four equal sides from that answer.

I'd write a book, but it would be very short.

That is a explanation

although...

"The set of irrational numbers is infinite - indeed there are "more" irrationals than rationals (when "more" is defined precisely)." --wikipedia

does not make sense when you really try to think about it...kind of like two kids arguing: No!!! You're wrong!!! Infinity times two...no tagbacks.

I guess there really is infinity times two...irrationally speaking

Would it look like this ?

Since it seemingly has no end

It has no end full stop!

Approximation is the closest we can get, just like in real life

USB, while theoretically impossible to achive absolute precision, it is certainly possible to get close enough to achieve a useful result. This is one of those problems that are extremely esoteric but not really all that useful.

Therefore, 'squaring the circle' is a geometric impossibility.

I think that it's things like this that have, (throughout history), driven some mathematicians insane.

We know that the area of a circle has to be a finite number, even though the best we can ever hope for, using pi*r², is an approximation.

Since we know that the area of a circle has to be a finite number, logically, creating a square with the exact same area should be a piece of cake.

I'm going out on a limb here, possibly to the point of deserving a flogging, (which I can handle), but it seems to me that squaring a circle has to be possible. We just haven't come up with an equation to do it.

This sounds like a silly question, but is the reverse also impossible?

Circling a square, or creating a circle with the same area as a square?

If you look at an area as an infitessimally small section of the volume for a shape of given restraints (circle, square) then the area could not be determined for any shape.

The area is determined using a plane, which has no vertical component (y-axis).

If the area was determined using a fixed section (y-axis) then one could determine the area through the measurement of volume, and squaring the circle would not be necessary.

I see it as a mis-application of differing terms, yet I have no better way.

The flaw in your thoughts is that the area of a circle is not approximately equal to pi*r^2 it is exactly equal to pi*r^2. There are many proofs of this relationship that go all the way back to Archimedes. You are making a common mistake of confusing the uncertainty of a measurement (finding the value of r in this case) and calculating with a real irrational number. My point here is that all uncertainty in calculating the area of a circle comes from knowing the value of r, the radius, not from the integer 2 or the irrational number pi.

The flaw in your thoughts is that the area of a circle is not approximately equal to pi*r^2 it is exactly equal to pi*r^2.

Then it should stand to reason that creating a square with the exact same area as a circle should not be impossible.

Now you're onto the second confusion. Clearly a square can have exactly the same area as a circle. However one cannot construct, or (to use your term) create a square with the exact same area as a circle.

I understand.

I should have looked at this earlier. No one mentioned the, 'finite number of steps using a compass and a straight edge' part.

If this is an accurate description of squaring a circle, it makes sense.

I'm not going to bother taking a compass and straight edge and attempting to do it.

Like most average people, I initially struggled with calculus until one day the light came on. Even then, however I always wondered if our math was all there was. Now I am not supporting our friend Dick, but is it actually possible that another (maybe higher) form of math exists that could compute more complex problems like squaring a circle. Evidently a lot of pretty good mathematicians tried for a long time before declaring it impossible (using current math tools).

You've also misunderstood the problem. The question at hand is not to calculate or compute a square that has the exact area as a circle, this is a trivial task. The question is to construct or fabricate a square with the exact same area as a given circle.

Yes, that's correct.

Our 'friend' Richard Dunn made the statement that he solved the problem of squaring the circle' by using a non-base 10 number system. He doesn't realize that the numbering system is irrelevant, it is the proportion of one length to another length that is the key to the problem -- in this case, a proportion between the two that cannot be constructed by the rules of geometric proof in a finite number of steps. (I thought I had explained that in my GA above, but I guess I was not explicit enough, as Kramarat didn't see my point.)

So measuring the circle's radius using inches or centimeters, with a decimal, binary, hexadecimal, trinary, or some other counting system -- it simply doesn't matter to the geometry of the problem. Algebraically, the problem can be 'solved' to whatever degree of precision you want. Just pick the number of decimal places you want to use for pi.

You're right. Once I read the wiki piece, I got it. Once I got it, I also got your explanation. A well deserved GA!

I wish we could stop talking about the person that started the other thread, neither he, nor the thread, are any longer relevant. I've learned more on this one.

It's interesting, and kind of sad, that so many people spent so many cumulative years, (I'm sure in some cases, lifetimes), trying to work this out.

I guess I need to include "Plain Facts for Circle-Squarers" in my reading assignment. Damnit RF.

"Construct" with compass and straightedge...got it now. I'm thick and slow but eventually get there.

For practical problems such as construction, accounting, surveying, etc., it doesn't matter. For the challenge of a puzzle, or logical/philosophical precision, or a kind of spiritual purity that mathematics has, it does matter.

Accepting certain rules and limitations can sharpen the enjoyment, such as in a piano sonata for the left hand, baking a cake from scratch, playing chess with a piece handicap, doing stonemasonry without mortar, etc.

You're a good guy Tornado.

Build me a square out of a circle......................please.

Trust me. I am working puzzles all the time.

Witnessing insanity in motion, (referring to the other thread), never makes me feel good.

We can do better.

If we find ourselves deriving pleasure from someone else's pain, then this site is useless.

In this, and an earlier point you raised, there are surely people who have become obsessed with problems of this kind. Not always bad: Even failed attempts have resulted in other mathematical advances. (The real action happens at the edge of the envelope.)

The "good" attempts occur when a person tries hard, but realizes that success hasn't happened yet. This can be frustrating, but it does matter that the attempter refrains from claiming more than actually accomplished. The "good" attempts also take into account the prior knowledge of the field. (As Einstein did, for example.)

The "bad" attempts occur when knowledgeable prior effort is ignored, the "rules" are changed, success is incorrectly asserted, and better informed persons are insulted. There is a difference between honest tries and mere bluster; the former can be respected, even if unsuccessful; the latter cannot.

This is all a little personal to me.

The first time I went crazy, everyone knew how to tie their shoelaces but me. It hurts.

I was about to say that, but you beat me to it. If the original post had made it clear that it means "using a ruler and compass to construct a square of same area as a given circle", it would have saved some confusion. Not everybody would understand that from "squaring the circle".

Codey

And that is exactly the hole I was in. Engineers however, are not always the best at expressing clearly what they mean. I had to take a technical writing class as part of my studies, and still I find I am misunderstood at times, through no fault of the reader.

The latest world record for the computation of "Pi" is outlined here:

http://www.super-computing.org/pi_current.html

It attests to 1,241,100,000,000 decimal digits.

And several hundred hours of computation!

The people concerned, the mathematical formulae are updated in this wiki article but I found them a few years ago in a book called Pi Unleashed by Arndt Haernel published in 2000 when the world record for Pi only contained 206.1 billion decimal places.

The book also conatins the many people that have been involved in the improvement in our understanding of Pi from the earliest times to the present day are outlined, their various calculations and formulae are presented.

When and if greater precision is found I am sure that the wiki will be updated!

By the way, if any one can show me how to insert a Pi character, and other characters including Tornados Number Theory sets I would be very grateful.

Tornado,, I am looking forward to expansion of your Number Theory. Any guidance on a good book as we move through?

Sleepy

It all settles down with the measurable significant workable digit, yes? No one has ever made point to point accuracy like say 3.14159....

It is limited by how accurate our technology could possibly measure.

Two 1 gram cotton weighed on the same weighing scale does not have the same weight agree? Yes, to grams perpective, say at nanograms-No.

Bring the workpiece to a machinist, he will make you As~=Ac. It may not be perfect but definitely approaching as dictated by how accurate his tools could measure.

'Usbport': Using your explanation, 'squaring a rectangle' might be equally impossible....

..for example a rectangle with one side length being an irrational number (lets say 2pi units) could not by your explanation be squared... So some rectangles could be squared, but others could not (depending on size and, oddly enough, units).

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...kind of hit or miss...not much of a rule.....

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....but then the same thing could be said of 'squaring a circle'...accepting your explanation, some circles cannot be squared.... but some can..

A circle with radius of 1/(pi^1/2) would happily participate in the 'squaring the circle' challenge and yield acceptable outputs as I understand it.

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...so as with 'squaring a rectangle' kind of hit or miss with 'squaring a circle' too... it depends on the circle, so it really isn't much of a rule.

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If you just have a longing for a rule related to this subject, I recommend the following,

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'There is no reasoning with the irrational'.

Squaring a rectangle is dead easy. The rectangle (sides a and b) area has area a×b. A square of side √(a×b) has the same area. An earlier post tells how to construct products and square roots. Squaring a triangle takes only another step or two.

It will only take a step or two, right? May be I try that today. lol

I was never really good at math and had to work very hard at it to emerge from college.

I remember fighting my way through calculus. I also remember asking my prof "How do you know you've got the right answer?". His response was "Well, Steve, you just do!" While he was able to stop himself halfway through the integration and decide he was going down the wrong path, I just sat there and asked myself "How did he know that?". He was so confident he had the right answer he never even bothered to test it. His was a brilliant mind and he was an engaging mathematician but I'm sure he was frustrated with me and probably felt like he was in a room full of idiots.

My makeup course at the university was even harder.

I use math everyday at my job. It's ironic how life can rear up and smack you sometimes. Very humbling. Nonetheless, I am undaunted. I like to poke things from different angles until I prevail.

This circle-square things sounds very interesting. Keep it coming!

Bad prof! Bad!

Just as an approximation to a solution 'feels' right, so does the 'method' of approach.

BUT while mathematics offers many ways to skin a given cat, the solution should always be provable.

My first Calc instructor taught at my local college at night, was a nuke weapons expert at the local military base, and got his start working for the US government out in a remote locale in New Mexico after escaping Germany.

We did proofs until we finally learned to love proofs. But the number of times from the back of the room I heard "Nein, nien, nein, verstehen nicht ..."* followed by things I never bothered to translate.

Crew cut, white short sleeve shirt, pocket protector - whole nine yards and the embodiment of old school German mathematicians. Rigorous.

Very daunting as I came to college as a night student with a full-time job and a weak background in math.

Obviously the crowd calls for more!

Press on Tornado!

USBPort's GA has explained some of this nicely.

I agree with kramarat that it can seem unsettling that some numbers cannot be expressed exactly as decimal fractions. Even 1/3 has the unending (but repeating) representation 0.33333.... No matter how many decimal places you go, you are only approximate, never exact; to be exact, one must use 1/3 (or 2/6 or other such equivalent expressions).

I was once flummoxed for a short time with (-1) x (-1) = +1. I have forgotten what explanation made that "click into place." One of the delights of mathematics is to have a counterintuitive idea become "unstuck". Sometimes it takes just the right explanation, which can vary from person to person. As this thread develops, I hope such explanations emerge.

(I have digressed, but I was expecting the thread to take some twists and turns. I will pick up the narrative this evening.)

One thing that I believe will help this discussion is a definition of terms used in number theory. Many people do not grasp the important subtle difference between a transcendental number and an irrational number. If this link still does not make sense, I do have my own explanation that people might like.

This certainly confirms my personal belief that taxation formulas are irrational.

Fractions of pennies do greatly annoy me. A beast plus half a beast = 1 exactly.

Then the beast is 2/3. Is its number 444?

_{even I am not so bad at math to think that 2/3 is equal to 444.}

It's all about the decimal place. Divvying up the poor little one in such a manner. Who cares about numbers that add to a nice even thou. I rather like those, actually.

If this link still does not make sense, I do have my own explanation that people might like.

The link made sense to me, but I'd still be interested in your explanation as well.

Ok, my simple explanation of number set theory progressing to a transcendental number. First an integer is a countable number. We happen to commonly use a ten digit number system to describe these numbers but three dozen and thirty six describe the exact same integer. A rational number is a number that can be defined by a ratio of integers. Many rational numbers cannot be precisely identified without including the mathematical operator of division. These are ratios of non divisible numbers. Trying to define these numbers using just the sum of decimal power integers leads to an infinite series of decimal digits. (1/3=0.333333333...) With a rational number these digits will eventually repeat. Irrational numbers are the numbers that cannot be defined by using only the operator of division. The most commonly defined irrational numbers are the algebraic that are the roots of polynomials. (Yeah I'm cheating a little here by rephrasing the Wikipedia article.) Trying to define these numbers in a decimal format will again produce an infinite series, but there will not be a repetition of digits. Then comes the transcendental irrational numbers that are defined by other mathematical operator functions than just the operators of algebra. What differentiates each of these sets of numbers is the mathematical operator used to construct this numbers.

What confuses many people with a discussion of numbers is that these numbers do not come from any measurement. Measurement always has a certain level of uncertainty. There is no uncertainty in any of the numbers defined by a mathematical function.

One critical concept that should be added for this discussion.

With the exception of identity cancellations (PI/PI=1, e-e=0), any algebraic function with a transcendental number in it produces a transcendental number. [sin(PI/2)=1 is a trigonometric not an algebraic function.]

Redfred,

I agree wholeheartedly, I look forward to an informed discusion, it's many and many a year ago since ths was a current topic with me.

If you had not proposed it I would have.

Sleepy

I've squared the circle!

Wouldn't that be " cubeing the sphere" ? :D

It has always amazed me how mathematicians can be so incredibly precise in their work yet still fall so incredibly far from reality.

Some years ago I went back to college at NDSU in Fargo ND to get a electrical engineering degree. In one of my calculus classes my professor was explaining how mathematical proofs can in fact prove that any one number can be equal to another number if enough BS got added to the equations. He even claimed that this mathematical proof system can be used in everyday daily life too!

I commented that it could not and anyone with the least bit of practical sense would know that. He said it could and he would happily prove it too!

I said well then what are you doing after class? He said nothing, why? I then said I want you to prove this can be used in practical daily life so would you mind helping me out for a little while.

He said sure and that he had a few hours before his next class so what was it I wanted help with?

I told him and the rest of the class I want you to use that proof to make the $30 I have left in my checking account equal to $30,000,000 and you can have half of it if it works and the bank believes it!

Well the class thought it was funny but at the end of the day I still only had the $30 though.

And that, is why he is a professor.

Are you saying that it is 'mathematically impossible' to square a circle, as opposed to actually taking a string with the two ends tied together, which can physically be laid out as a circle, and then using that same length of string to make a square, which has the same area as the circle? Am I right in thinking that you are saying that those dimensions cannot be reversed engineered to equal each other because PI is infinate?

Stretching a string is not the same thing as a straightedge/compass construction. (And would be imprecise from elasticity and thickness of the string.) The classic problem is a pure math question, requiring absolute exactness under specific "rules" (points have zero dimensions, line segments have length but zero width and height, plane segments have length and width but zero depth; the only "tools" allowed are compass and unmarked straightedge; and the steps of the construction must be finite, else it would never be completed.)

Pi is finite (being only a bit over 3.14), but its decimal (or fractional) expression never ends. (A few trillion digits are now known, though.) The infinite length of the expression does not of itself prevent its geometric construction; rather, it is the specific type of infinity according to the specific type of number.

Odd/even numbers have been mentioned. In time, we can discuss rational/irrational, algebraic/transcendental, and even real/imaginary numbers. (These terms have specific mathematical definitions that differ from ordinary meanings of the words. Some of the confusions that occur owe to this.)

"only "tools" allowed are compass and unmarked straightedge; and the steps of the construction must be finite"

I may be wrong but I want to give a try. Take a uniform thickness metal sheet (Al or Ag). decide a radius R on compass (1 unit), mark and cut or stamp a circle of of diameter 2R. Now cut the sheet to a rectangle of one side equal to R and other side as maximum as possible. Now we have 4 pieces left, length of 2 are bigger than R. Cut these two to small rectangles of bigger side R. Add these to previous rectangle of smaller side R in length. Cut smaller pieces in to rectangles of lengths R/2, R/4 etc. ang go on adding to original one just to finally arrive at a rectangle of smaller side R and bigger side to measure in terms of R. Calculate the area and find square root to get the side length of desired square.

I can't follow all that, but if "Cut smaller pieces in to rectangles of lengths R/2, R/4 etc. ang go on adding to original one ..." is to be continued indefinitey, it doesn't meet the requirement for a finite number of steps (post #34)

Codey

Yes, you are right. It will be continued to get more and more digit of accuracy. It seems finite steps is a long way for me (not a mathematician).

The perimeter of a circle having a radius R is 2*pi*R, and the circle has an area of pi*R².

Taking a string with the same length as the circle's perimeter (2*pi*R) and making a square gives you a square with the same perimeter (P) but with an area of [(1/4)*2*pi*R]², or 1/4*pi²*R². (Where each side of the square has a length of S = (1/4)*2*pi*R.)

Clearly, this square's area, 1/4*pi²*R² , is not the same as the circle's area, pi*R².

Letting R = 1 and using 3.1416 as an approximation for pi, the two areas are:

A_square = 2.4674

A_circle = 3.1416

Now you're just showing off.

Yippie!!! The big 200. I'd better shut up before they start running in reverse...................................._again.

Shall I tell you how man times 405 has come and gone?????????????

And while I'm here, assuming someone solves this riddle, what good will it do? What are the practical implications?

Uh. I think that one thing has been clearly demonstrated.

Attempts to solve this problem can lead to mental anguish and delusional behavior.

I think I'll leave it alone. I'm going to stick with temporary bouts of liquid insanity.

A tweak here and a little fraud there and it all looks different. I posted this in the PM avalanche and think it was overlooked as a fraud. It could have gone the other way.

Is this illegal?

Some references to the other thread may be nearly inevitable, but I'm hoping to concentrate mostly on explanations of this topic that can be understood by lay people.

By the grace of Martin Gardner, I was diverted from the obsessions that have captivated so many. Actually, I think it tends to be otherwise bright persons who are more susceptible. For weaker students, even medium-hard problems seem impossible, so they can accept the idea of genuine impossibility.

Well put.

The people that refuse to accept the idea of genuine impossibility usually end up on one of two paths. They go insane trying to prove otherwise, or they prove otherwise.

Either way, it's not a game for the weak to be playing.

then it would seem that the correct number of turns of string around the circle would in fact give the square. (ie, 1.6 turns.) and would do so for any radius?

chris

I stand corrected.

(√(∏R²))²

So whats so hard about that?

If you really want to nit pick its practical application take ∏ to around 30 - 40 decimal places. Regardless of how big your circle is in reality its still going to place you into the area of near sub molecular resolution levels and as far as I know once you hit the molecular level you cant machine a solid object any more precise!

& for our next quest

We'll be trisecting an angle with only a compass & a straightedge

Like the old joke about engineers and mathematicians and the bag of gold across the room:

The engineers on the forum are going to insist that 'close enough' is good enough, and the mathematicians are going to keep pointing out inconsistencies.

Two different exercises with two different groups using two different sets of rules.

Two different exercises with two different groups using two different sets of rules.

Sounds just Like Washington and out present Politics.

OK, it can't exactly be done, but as a engineer, if your boss demanded it, I bet you could come up with an answer. Am I right? I think you could have found a better way to roast our friend Dick. Hopefully, some day, the letters will stop.

Getting back to the idea of how this topic depends on certain classes of numbers--

If you are given line segments of unit length and two other lengths a and b, it is possible by simple geometric constructions to perform these operations (assuming one already knows how to construct parallel and perpendicular lines and bisect a line segment):

a+b: On a line, mark a segment of length a; away from one end mark length b; the two segments add to a+b.

a-b: Mark a as before, but this time turn backward to mark b; this gives a-b.

a×b: Construct x and y axes; on the x-axis, mark the unit length at (1,0) and length a at (a,0); on the y-axis, mark length b at (0,b); draw line through (1,0) and (0,b); draw a parallel line through (a,0). Where this line intersects the y-axis, by similar triangles, y = a×b.

1/a: Again using xy axes, mark points (a,0) and (0,1); draw line segment between; draw perpendicular line through (0,1); this line intersects the x-axis at (-1/a,0), again by similar triangles.

b/a: = b×(1/a) by combining the previous two constructions.

√a: Mark (a,0) and (-1,0) on the x-axis; bisect the resulting line segment of length a+1; draw a semicircle with the this segment as the diameter. Where the semicircle intersects the y-axis at (0,y), by similar triangles y = √a.

Thus, by geometric construction, one can obtain sums, differences, products, quotients, and square roots of given quantities.

(to be continued)

GA and welcome to the mad house.

(Wow 16 digits accuracy, thats impressive.)

good to hear. welcome to CR4.

VGA.

fishingguy raises a point that I also had in mind for later. GA.

Even though the circle squaring problem is more of a pure math thing rather than applied, the study of it can spin off useful results.

As noted the circle may not be squared due to the Transcendental nature of Pi.

The image above of the 16 decimal solution to this problem is the most accurate solution to this problem in the world. This solution was achieved using the GeoGebra geometric software package which is available for download from the web as freeware. To go beyond this point with regards to achieving a higher degree of accuracy would involve a significant effort and may not be achievable.

As noted by another poster and myself work with the Circle Squaring Problem can have spin offs that are applicable to other areas. In this case from our work my team developed this geometric method of calculating Pi.

In 1666 Sir Issac Newton used a geometric construct to derive a formula for the calculation of Pi. Wolfram Mathworld see also (Wells 1986, p. 50; Borwein et al. 1989; Borwein and Bailey 2003, pp. 105-106)

The use of a geometric construct from which to derive a formula for the calculation of Pi is an accepted practice as documented above.

An earlier post described how geometric constructions could be used to produce sums, differences, products, reciprocals, quotients, and square roots.

And that is all that can be produced. The proof in this direction is more complicated, but basically one can use analytic geometry to show that the point of intersection of any pair of lines or circles previously constructed by the rules has coordinates that are the sum, product, etc., of what has has been done before.

These numbers are called the constructible numbers. They include all the rational numbers, and their successive square roots (i.e., 4th roots, 8th roots, etc.). But not other roots, such as cube roots of numbers (that are not perfect rational cubes to start with).