Squared Circle

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Squared Circle

Postby Motor Daddy » Sat Aug 28, 2021 4:25 pm

Close enough??

Squared Circle.png
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Re: Squared Circle

Postby obsrvr524 » Sat Aug 28, 2021 8:10 pm

-
Appears awfully close but two questions -
  • Have you checked the math?
  • How did you place the 2 vertical lines that are closest to center?
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Re: Squared Circle

Postby WendyDarling » Sat Aug 28, 2021 8:20 pm

D2729D21-53F8-4172-A97B-7FA527CC441D.jpeg
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I like the pink, this one.
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Re: Squared Circle

Postby Motor Daddy » Sat Aug 28, 2021 9:33 pm

obsrvr524 wrote:-
Appears awfully close but two questions -
  • Have you checked the math?
  • How did you place the 2 vertical lines that are closest to center?


I created the pic in Paint. The maximum accuracy of paint was reached and I can not verify any closer.

I posted this on a Math forum a while ago and one of the Genius members claimed that using Trig the side length of the square was 1.77 units. It should be the square root of pi side length, which is 1.7725 units.

I can't verify his measurements or Paints measurements, so I don't know which, if either, is correct. It is damn close, though, using only straight edge and compass.

The vertical lines closest to the center were arrived at by a series of straight lines until I had enough lines to run them vertically through the centers of the circles they vertically run though the centers of.
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Re: Squared Circle

Postby obsrvr524 » Sat Aug 28, 2021 9:39 pm

Motor Daddy wrote:The vertical lines closest to the center were arrived at by a series of straight lines until I had enough lines to run them vertically through the centers of the circles they vertically run though the centers of.

I assumed that you drew the center lines of those circles before the circles. If the lines were drawn after the circles, how did you know where to draw the circles?
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Re: Squared Circle

Postby Motor Daddy » Sat Aug 28, 2021 9:59 pm

obsrvr524 wrote:
Motor Daddy wrote:The vertical lines closest to the center were arrived at by a series of straight lines until I had enough lines to run them vertically through the centers of the circles they vertically run though the centers of.

I assumed that you drew the center lines of those circles before the circles. If the lines were drawn after the circles, how did you know where to draw the circles?


I made the drawing in Paint as if I were drawing it with only a straight edge and compass, which is the premise of the squared circle, that the area is the same, and drawn only with straight edge and compass.

I don't remember the EXACT order, but I started with drawing the center circle. I then drew a straight line horizontally through the center of the circle. From the intersection of the line and circle edge on both sides I created the circle on the left and right of the center circle, so the intersection was the center of each circle. That gave me more intersections to draw the lines horizontally on top and bottom of the 3 circles, which gave me more intersections to draw more straight lines.

The only lines and circles that were made was when there was enough points to make those intersections, and therefore new straight lines or circle centers.

The actual outermost square was formed almost last, it was not filled in, or drawn as a boundary from the beginning. It took many many lines to get those outer corners of the outer square.
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Re: Squared Circle

Postby obsrvr524 » Sat Aug 28, 2021 10:13 pm

Motor Daddy wrote:I then drew a straight line horizontally through the center of the circle. From the intersection of the line and circle edge on both sides I created the circle on the left and right of the center circle, so the intersection was the center of each circle. That gave me more intersections to draw the lines horizontally on top and bottom of the 3 circles, which gave me more intersections to draw more straight lines.

The only lines and circles that were made was when there was enough points to make those intersections, and therefore new straight lines or circle centers.

I followed you that far. But I still don't see how the upper 2 and lower 2 circles could be horizontally positioned other than just arbitrary guess.

And if they are not mathematically situated, how did any maths geek calculate the square?
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Re: Squared Circle

Postby Motor Daddy » Sat Aug 28, 2021 10:16 pm

I take that back, I didn't start with drawing the center circle. I started by drawing that horizontal line that runs through the center circle, and then picked a point on the line and that was the point the center of the circle was drawn from. From there there was a horizontal line running through the center of the circle. The points where the line exits the circle on both sides is the center where each side circle was drawn.

It's been a couple years, now. I have it framed above my computer desk! :) It's beautiful artwork, at the very least. ;)
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Re: Squared Circle

Postby obsrvr524 » Sat Aug 28, 2021 10:20 pm

-
Ok I see how you could horizontally position those upper circles - half of the radius from center vertical.

It is an interesting feat. I wonder if it can be made exactly right. :D
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just the same Satanism as always -
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    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: Squared Circle

Postby Motor Daddy » Sat Aug 28, 2021 10:23 pm

obsrvr524 wrote:I followed you that far. But I still don't see how the upper 2 and lower 2 circles could be horizontally positioned other than just arbitrary guess.

And if they are not mathematically situated, how did any maths geek calculate the square?


The circles were created by using intersections of lines. I honestly don't remember the exact order.

He claimed that trig could work out the angles and length of lines. I don't know Trig, so beats the heck out of me. :) He made an excellent diagram and showed all angles and used Trig to calculate the side lengths.

The circles have a radius of 1 unit, so the outer square is 4 units x 4 units
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Re: Squared Circle

Postby obsrvr524 » Sat Aug 28, 2021 10:28 pm

Motor Daddy wrote:He claimed that trig could work out the angles and length of lines. I don't know Trig, so beats the heck out of me. :) He made an excellent diagram and showed all angles and used Trig to calculate the side lengths.

The circles have a radius of 1 unit, so the outer square is 4 units x 4 units

I imagine I could work out the math now.

Did you come up with that technique yourself?
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just the same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: Squared Circle

Postby Motor Daddy » Sat Aug 28, 2021 10:33 pm

obsrvr524 wrote:
Motor Daddy wrote:He claimed that trig could work out the angles and length of lines. I don't know Trig, so beats the heck out of me. :) He made an excellent diagram and showed all angles and used Trig to calculate the side lengths.

The circles have a radius of 1 unit, so the outer square is 4 units x 4 units

I imagine I could work out the math now.

Did you come up with that technique yourself?


Yep, just opened Paint and had at it! ;)

I was on another forum many moons ago, and had a previous attempt that I thought was right at the time. But as I look back now, rpenner was right! The side length on that one was 1.73 units, not 1.77 as I claimed.
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Re: Squared Circle

Postby obsrvr524 » Sat Aug 28, 2021 10:43 pm

-
They say it is impossible to make it precisely accurate - but I don't know if that is a proven fact or just a highly educated guess.

Either way - it's a brilliant accomplishment. =D>
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just the same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
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Re: Squared Circle

Postby Motor Daddy » Sat Aug 28, 2021 10:48 pm

obsrvr524 wrote:-
They say it is impossible to make it precisely accurate - but I don't know if that is a proven fact or just a highly educated guess.

Either way - it's a brilliant accomplishment. =D>


Thanks. Like I said, I enjoy looking at it as art. It is very interesting to look at the different lines and triangles and circles. I had it printed on Photo paper and bought a nice frame for it. I love looking at it. It never gets old.

The reason they say it is impossible is because there is no finite number for the square root of Pi, so how can you make the side length of the square a finite size when there isn't a finite square root of Pi?

We went round and round for pages and pages of back and forth about Pi and infinity and all that noise! :)

In the end I like my painting! :)
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Re: Squared Circle

Postby obsrvr524 » Sat Aug 28, 2021 10:58 pm

Motor Daddy wrote:The reason they say it is impossible is because there is no finite number for the square root of Pi, so how can you make the side length of the square a finite size when there isn't a finite square root of Pi?

That doesn't - on the surface - sound like a legitimate argument. Pi itself is an irrational quantity but can still be drawn. I don't see why the sqrt couldn't be also.
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just the same Satanism as always -
    • separate the bottom from the top,
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    • blame each for the sins of the other
    • - until they beg you to take charge.
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Re: Squared Circle

Postby obsrvr524 » Sat Aug 28, 2021 11:06 pm

-
Come to think of it - I know \(\sqrt\pi\) can be drawn.

The question is how to do it with straight edge and compass. :-k
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just the same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
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Re: Squared Circle

Postby Motor Daddy » Sat Aug 28, 2021 11:27 pm

obsrvr524 wrote:-
Come to think of it - I know \(\sqrt\pi\) can be drawn.

The question is how to do it with straight edge and compass. :-k


It has to have the same area as the circle, which has a radius of 1 unit.

Area for a circle is Pi(r^2), so the area of the circle is 3.14159.. square units

Therefore, the area of the square is 3.14159.. square units, and therefore the side length of the square is the square root of Pi.

My argument is that, how can the circle have a finite area if Pi is infinite, for all practical purposes??
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Re: Squared Circle

Postby obsrvr524 » Sat Aug 28, 2021 11:44 pm

Motor Daddy wrote:My argument is that, how can the circle have a finite area if Pi is infinite, for all practical purposes??

This gets back to that base-unit issue. It isn't that pi is infinite - but that trying to express pi in a rational base will yield an infinite string of digits.

But what if you use base-pi.

You know that you can -
    draw a \(2\pi\) length line merely by rolling a radius 1 disc across a paper exactly once.
    Divide that in half with a compass and you have exactly \(\pi\)
    Duplicate that length vertically from the end of the \(\pi\) length (making a corner or "L").
    Draw a line from the tip of each \(\pi\) segment and you have a line exactly \(\pi\sqrt2 \)

But they don't let you use a disc - so the trick is how to do it using only the compass instead of a disc. It doesn't seem to be a math problem as much as a mechanical problem.
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just the same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: Squared Circle

Postby Motor Daddy » Sun Aug 29, 2021 12:09 am

obsrvr524 wrote:
Motor Daddy wrote:My argument is that, how can the circle have a finite area if Pi is infinite, for all practical purposes??

This gets back to that base-unit issue. It isn't that pi is infinite - but that trying to express pi in a rational base will yield an infinite string of digits.

But what if you use base-pi.

You know that you can -
    draw a \(2\pi\) length line merely by rolling a radius 1 disc across a paper exactly once.
    Divide that in half with a compass and you have exactly \(\pi\)
    Duplicate that length vertically from the end of the \(\pi\) length (making a corner or "L").
    Draw a line from the tip of each \(\pi\) segment and you have a line exactly \(\pi\sqrt2 \)

But they don't let you use a disc - so the trick is how to do it using only the compass instead of a disc. It doesn't seem to be a math problem as much as a mechanical problem.


You can use a compass and draw ANY SIZE circle and claim that has a radius of 1 unit, since by definition, the radius can be ANY unit. The circle therefore has an area of 3.14159.. square units OF THOSE UNITS.

The trick is to make the square THE SAME AREA as the circle, which automatically means the sides have to have a length of square root of Pi OF THOSE UNITS.

How do you make a side length of the square root of an infinite string of digits?

At some point, the measurements break down. There is no way you can measure an infinite string of units. You have to truncate or round off, at some point. Which point is that, 3.14? 3.14159?? Where do you stop the nonsense and calculate to that amount of decimal places, as accurately as your ruler can measure? If so then I did it! I can not measure any more accurately than what I did. I simply don't have the tools, and nobody does!

It is not possible to measure to any degree of accuracy past, say, a dozen decimal places, which is NOWHERE close to the square root of Pi.
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Re: Squared Circle

Postby obsrvr524 » Sun Aug 29, 2021 12:37 am

-
I think you missed my point.

My point was that a length being an irrational number with an infinite string of decimals (such as \(\pi\) or \(\sqrt2\) doesn't stop you from drawing a line of that length - you can even draw a line of the multiplication of two irrational numbers - \(\pi\sqrt2\).

I do have question as to how to draw a line that is certain to be the sqrt of an arbitrary length - such as \(\pi\). I can get the sqrt of some irrational numbers but I haven't figured out how to get the \(\sqrt\pi\) - yet. O:)

And that is why I asked if it had been proven to be impossible or just assumed to be impossible by those who have tried and failed.
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just the same Satanism as always -
    • separate the bottom from the top,
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    • the light from the dark, and
    • blame each for the sins of the other
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Re: Squared Circle

Postby Motor Daddy » Sun Aug 29, 2021 12:48 am

obsrvr524 wrote:-
I think you missed my point.

My point was that a length being an irrational number with an infinite string of decimals (such as \(\pi\) or \(\sqrt2\) doesn't stop you from drawing a line of that length - you can even draw a line of the multiplication of two irrational numbers - \(\pi\sqrt2\).


You can draw a line of that length, but how would you differentiate between 3" and 3.14159", when you only have capability to measure to the closest .1".

If all you have is a ruler marked for Hundredths of an inch, how would you know if the line is 3.14159"? The best you could do is say the line is 3.14", and you don't know if it's 3.14159" or 3.14172".

See what I'm saying? You may be able to calculate to 27 decimal places, but with a ruler that only has Hundredths of an inch, the rest is nonsense to you. You can measure to 2 decimal places, and the remaining 25 decimal places may as well be truncated, you have no purpose for them!
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Re: Squared Circle

Postby obsrvr524 » Sun Aug 29, 2021 12:56 am

Motor Daddy wrote:
obsrvr524 wrote:-
I think you missed my point.

My point was that a length being an irrational number with an infinite string of decimals (such as \(\pi\) or \(\sqrt2\) doesn't stop you from drawing a line of that length - you can even draw a line of the multiplication of two irrational numbers - \(\pi\sqrt2\).


You can draw a line of that length, but how would you differentiate between 3" and 3.14159", when you only have capability to measure to the closest .1".

If all you have is a ruler marked for Hundredths of an inch, how would you know if the line is 3.14159"? The best you could do is say the line is 3.14", and you don't know if it's 3.14159" or 3.14172".

See what I'm saying? You may be able to calculate to 27 decimal places, but with a ruler that only has Hundredths of an inch, the rest is nonsense to you. You can measure to 2 decimal places, and the remaining 25 decimal places may as well be truncated, you have no purpose for them!

Who said that you have to measure it?

The only issue is drawing it and knowing that it is the right length. And you can do that in a variety of ways.

The first proof that an irrational number can be drawn was by some bloke back in the Pythagorean days when the establishment said that it couldn't be done - but he simply drew a 1 unit square with a diagonal and proved that the diagonal had to be an irrational number - \(\sqrt2\) (because it had to be both an even and also an odd number at the same time).

To get \(\pi\) - just roll a 1 unit disc across a paper exactly once and you have exactly \(\pi\) length.
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just the same Satanism as always -
    • separate the bottom from the top,
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    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: Squared Circle

Postby Motor Daddy » Sun Aug 29, 2021 1:07 am

obsrvr524 wrote:
To get \(\pi\) - just roll a 1 unit disc across a paper exactly once and you have exactly \(\pi\) length.


How long is that line, 6.28 units or 6.28318 units?
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Re: Squared Circle

Postby obsrvr524 » Sun Aug 29, 2021 1:09 am

Motor Daddy wrote:
obsrvr524 wrote:
To get \(\pi\) - just roll a 1 unit disc across a paper exactly once and you have exactly \(\pi\) length.


How long is that line, 6.28 units or 6.28318 units?

Neither one.

It is EXACTLY \(\pi\) in length (forgiving the errors of pencil and paper).
Member of The Coalition of Truth - member #1

              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just the same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: Squared Circle

Postby Motor Daddy » Sun Aug 29, 2021 1:11 am

obsrvr524 wrote:
Motor Daddy wrote:
obsrvr524 wrote:
To get \(\pi\) - just roll a 1 unit disc across a paper exactly once and you have exactly \(\pi\) length.


How long is that line, 6.28 units or 6.28318 units?

Neither one.

It is EXACTLY \(\pi\) in length (forgiving the errors of pencil and paper).


Pi is not a length, it is a ratio of the length of the diameter to the circumference.

So if you say the radius is 1, then the diameter is 2. Now, in order to figure out the length of the line, you have to multiply Pi x 2. What number are you going to use for Pi to multiply by 2?
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