Close enough??
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 like the pink, this one.
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.
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.
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?
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. 
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. 
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
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.
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!
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.
Come to think of it - I know (\sqrt\pi) can be drawn.
The question is how to do it with straight edge and compass. 
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??
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.
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. 
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.