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Ineedinformation

Registered Member
This is a hypothetical question am not looking for and answer I just need to understand the method ok here goes
If pi was exactly 3.14 how can I create a line segment using the ratio 1:square root of pi please describe in detail my math is very rusty.

Then how would I use this information to perform th e impossible task of squaring the circle this is of a comic script am working on but I want it to seem convincing to the audience lol thanks for the help if anyone knows??
 
Ineedinformation said:
This is a hypothetical question am not looking for and answer I just need to understand the method ok here goes
If pi was exactly 3.14 how can I create a line segment using the ratio 1:square root of pi please describe in detail my math is very rusty.

Then how would I use this information to perform th e impossible task of squaring the circle this is of a comic script am working on but I want it to seem convincing to the audience lol thanks for the help if anyone knows??
Click to expand...

I didn't understand. A line segment has length. What does it mean to you to create a line segment using a ratio? Suppose I ask you to create a line segment of ratio 1:1. Are you going to draw a line 1 inch long? Not sure what you mean by this.
 
Aqueous Id said:
I didn't understand. A line segment has length. What does it mean to you to create a line segment using a ratio? Suppose I ask you to create a line segment of ratio 1:1. Are you going to draw a line 1 inch long? Not sure what you mean by this.
Click to expand...

Use any unit of measurement just pretend I was squaring the circle and I was actually solving one of the questions of antiquity the exact quantity for pi is not known so if it was hypothetically 3.14 what is the method I must follow to carry out the task correctly outlined with the rules allow by the people that posed the question originally? How would I use 3.14 to square the circle with the straight edge and compass? I need step by step instructions on how the method would be used if that was the correct quantity for pi?
 
Given a length of 1, you extend it by 1:3.14 (this 314/100 = 157/50 which is an annoying ratio, but legitimately constructible). Then construct a semicircle on the length of 4.14 and at the junction where the length of 1 meets the length of 3.14 construct a right angle line that meets the semicircle. This new segment is in the ratio to $$1:\sqrt{\frac{157}{50}}$$ as requested.

However, the ratio $$19 : 9 \sqrt{14}$$ is closer to $$\sqrt{\pi}$$ than $$\sqrt{3.14}$$. [Edit: Like the difference between a regular 127-gon and a regular 57-gon at approximating a circle.]
The ratio $$1 : \left( \sqrt{ \frac{3}{10} } +\sqrt{ \frac{3}{2} } \right)$$ is closer still. [Edit: Very close -- better than approximating a circle with a regular 463-gon.]
And the ratios $$167:296$$ or $$ 2034 : 1087 \sqrt{11}$$ are both much closer to $$\sqrt{\pi}$$ than $$\sqrt{3.14}$$, but are awkward to construct. [Edit: Corresponding, respectively to about a 1534-gon and a 20311-gon.]

So if you aren't married to the particular approximation $$\sqrt{3.14}$$ here is a way to get a closer approximation with I think a minimum of effort:
Layout straight line ABCDEFG such that AB=13, BC=7, CD=5, DE=1, EF = 14, FG = 10. From center B and radius = AB, construct a semicircle above the line from A to E. From center D and radius = AD construct a semicircle below the line from A to G. Construct the perpendicular bisector of AF, which passes through the upper semicircle at new point H, through point C and through the lower semicircle at new point I. Then it follows from the Pythagorean theorem that HI = √120 + √600 and thus that AC:HI are in the ratio 20:(√120 + √600) or 1:(√(3/10) + √(3/2) ). A circle with radius AC has area = 400 π ≈ 1256.637... , a square with side HI has area (√120 + √600)² = 720 + 240√5 ≈ 1256.656... while the approximation (400 × 3.14 ) = 1256 exactly, which is not as good an approximation as the square with side HI.​
 
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Squared circle using straight edge and compass, no?

squaredcircle_zpsddc49c0b.png
 
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While Motor Daddy has a lot of superfluous lines in this alleged construction, it appears that this is not a good approximation of squaring the circle.

Let the red circle be the original, centered at point A. Let the rightmost point of the circle be labeled B. Then AB is the radius of this circle. To the best of my ability to determine, the right circle is centered at B with radius AB. These two circles intersect each other in points C (upper) and D (lower) which appears to be equal to the side of the blue square. The ratio of lengths AB:CD = 1:√3 and so this cannot be a good approximation of 1:√π.


1:√3 is an approximation to 1:√π that you get by squaring the regular hexagon, not a circle.


If A = (0,0), B = (1,0) then analytic geometric describes the four circles as:
  • $$x^2 + y^2 - 1 = 0$$
  • $$(x-1)^2 + y^2 - 1 = 0$$
  • $$x^2 + (y+1)^2 - 1 = 0$$
  • $$x^2 + (y-1)^2 - 1 = 0$$
and the diagonal lines as:
  • $$x + y + 1 = 0 $$
  • $$x + y - 1 = 0 $$
  • $$x - y - 1 = 0$$
  • $$x - y + 1 = 0$$
and the vertical lines as:
  • $$x = -1$$
  • $$x = \pm \frac{\sqrt{3}}{2}$$
  • $$x = \pm \frac{1}{2}$$
  • $$x = \pm \frac{2 - \sqrt{2}}{2}$$
and the horizontal lines as:
  • $$y = 1$$
  • $$y = \pm \frac{\sqrt{3}}{2}$$
  • $$y = \pm \frac{\sqrt{2}}{2}$$
  • $$y = 0$$
 
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rpenner said:
While Motor Daddy has a lot of superfluous lines in this alleged construction, it appears that this is not a good approximation of squaring the circle.

Let the red circle be the original, centered at point A. Let the rightmost point of the circle be labeled B. Then AB is the radius of this circle. To the best of my ability to determine, the right circle is centered at B with radius AB. These two circles intersect each other in points C (upper) and D (lower) which appears to be equal to the side of the blue square. The ratio of lengths AB:CD = 1:√3 and so this cannot be a good approximation of 1:√π.
Click to expand...

Radius = 1
Diameter = 2
Circumference = 6.28
3.14*1=3.14 square units of area for the radius =1 circle.

A square of 3.14 units of square area that is equal to the circle area has sides of length $$sqrt{3.14}=1.772004514666935$$ units.


A 1 square unit square has sides of 1 unit of length, making the center point .5 units from center to side along the same axis.

Now what??

The diagram should be correct.
 
ABC is an equilateral triangle with sides of 1. Bisect that for a hypotenuse of 1 and sides of .5 and .8660. Each side of the square is 1.7320.
 
el es said:
ABC is an equilateral triangle with sides of 1.
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Sides of 1 means 2 of those triangles equals 1 square unit of area, or each triangle has an area of .5 square units. The square is one square unit, with sides of 1 unit of length. The hypotenuse of the square is a side of the square that is 4 trinagles in area, or, the hypotenuse is the square root of 2, or 1.414213562373095 units.
 
Motor Daddy said:
A 1 square unit square has sides of 1 unit. The hypotenuse is longer than 1 unit, agree?
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Where's the hypotenuse on an equilateral triangle?
If the sides are 1 then the area HAS to be less than 0.5 (since the height is, by definition, less than 1).
Drop a line from the vertex to split it into two and you form two right-angled triangles: the base of which is 0.5 and the hypotenuse is 1.
Area of a triangle with all sides of length 1 is ~0.43.
 
Back to approximations of pi. The rational approximation of π, 355/113 is so good, it is better than approximating a circle with a regular 6224-gon. Thus the constructible ratio √113:√355 is the second-best constructible approximation to √π on this page.

The schoolboy rational approximation of π, 22/7, is better than approximating a circle with a regular 90-gon. So even constructing √7:√22 is much easier and somewhat more accurate than attempt to construct √50:√157 as you requested in the OP.
 
Motor Daddy said:
Dare you attempt a refutation of my squared circle?
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It was immediately contested in [post=3222271]post #8[/post] which began with
rpenner said:
While Motor Daddy has a lot of superfluous lines in this alleged construction, it appears that this is not a good approximation of squaring the circle.
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and the the next paragraph describes the main points of the construction with circle centers A and B and intersection points between the circles as C and D. In [post=3222328]post #10[/post] another poster calculated the height of the equilateral triangle ABC:
el es said:
ABC is an equilateral triangle with sides of 1. Bisect that for a hypotenuse of 1 and sides of .5 and .8660. Each side of the square is 1.7320.
Click to expand...
which is in full agreement with [post=3222271]post #8[/post] which describes the ratio AB:CD of to be 1:√3 not 1:√π as required to square the circle.

The onus was on you to refute my interpretation of your unlabeled diagram in [post=3222223]post #7[/post] by actually explaining what all the lines were in a way consistent with the diagram but inconsistent with my interpretation. Clearly if the diagram was properly labelled, it would be easier to describe in words. But it appears that all posters that understand geometry agree with my interpretation in [post=3222271]post #8[/post].

That you would only now suspect your claim of squaring the circle was hotly contested and a reasonable alternative interpretation discussed does not make you look good.
 
rpenner said:
It was immediately contested in [post=3222271]post #8[/post] which began with
​
and the the next paragraph describes the main points of the construction with circle centers A and B and intersection points between the circles as C and D. In [post=3222328]post #10[/post] another poster calculated the height of the equilateral triangle ABC:
​
which is in full agreement with [post=3222271]post #8[/post] which describes the ratio AB:CD of to be 1:√3 not 1:√π as required to square the circle.

The onus was on you to refute my interpretation of your unlabeled diagram in [post=3222223]post #7[/post] by actually explaining what all the lines were in a way consistent with the diagram but inconsistent with my interpretation. Clearly if the diagram was properly labelled, it would be easier to describe in words. But it appears that all posters that understand geometry agree with my interpretation in [post=3222271]post #8[/post].

That you would only now suspect your claim of squaring the circle was hotly contested and a reasonable alternative interpretation discussed does not make you look good.
Click to expand...

I responded to your post with facts. Do you dispute those facts?
 
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