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"Psssst... Try this at home. Instructions appear below!"

An ancient secret? 

Rope stretchers of ancient Egypt used knotted ropes to form right angles for the construction of pyramids. Surveyors also used knotted ropes and pegs for other measurements. Later, the Greeks introduced a 'rule' that constructions had to be drawn with compass and straight edge only.

Today, we know squaring the circle with compass and straightedge is impossible. Yet what if older methods were allowed.

Could the circle have been squared with this peg and rope method?

Squaring a circle with peg and rope!
(Tips  to square a circle drawn from a music CD appear further below.)

Step 1. A rope of Unit length (1) is pegged at an end and rotated around O to draw a pink circle.

Step 2. A red rope is laid on half the circumference and cut to present a red (lined) semi-circle.

Step 3. The red rope the length of the semi-circle is straightened and extended left from O to S.

Step 4. Half the rope length of SA then draws the Green dashed circle around C.

Step 5. A perpendicular (black dashed) line is drawn from O, meeting the Green circle at P.
Step 6. The line from O to P is a side our our desired square.

Step 7. The line from O to P draws the required circles to form the blue square OPMN which has the same area as the initial pink unit circle.

(Image simplified following a suggestion from Geoffrey Cadman.)

HOLIDAY FUN!

Try this at Home... 

When I try this by hand, I'm going to square a circle drawn around a music CD. Wrapping the string 360° around the CD will be relatively easy and sit above the pencil line around the CD.

Then I will cut the string in half (carefully) and extend this from the center of the circle, which will be found from the intersection of perpendicular bisectors of a triangle inscribed in the the circle.

The end of this string will be point S in the diagram. Then another string will be used to bisect the distance from the edge of the circle through the center to S. This midpoint C will let me create the green dashed circle.

Drawing the perpendicular dashed black line from O will give me the first square side OP and the rest should follow.

The final result may be a little 'wonky' yet no more 'wonky' than the many published diagrams drawn from precise instructions. The math is correct, yet hand created constructions inevitably compound errors along the way, as the following examples show. Wonky diagram 1 ~ Wonky diagram 2

 Let me know how you go!
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People tried to square the circle for thousands of years using straight edge and compass alone. From the time of Archimedes, (3rd Century BCE) the above construction should have emerged, yet appears not to have. Perhaps the world became hypnotised by Euclid's straightedge and compass constructions and abandoned the older methods that might have led to the above construction.

Sometimes I feel like I know what people were thinking about thousands of years ago. Others often seem to project modern ideas back through time. 

We think we know it all, yet here is something special the world appears to have never seen! 

https://tube.geogebra.org/student/m702787

Thank you for reading my posts in 2015. I wish you and your family a Happy and Healthy Holiday Season and New Year.

Jonathan Crabtree
P.S. A single more detailed image of the above, with the notes below, is at www.jonathancrabtree.com/mathematics/wp-content/uploads/2014/11/SquareCircleHiRes.png

NOTES:

1. Given we accept limits and the reals, the concept behind the construction is exact. Archimedes converted a circle into a triangle and gave a proof. See my previous post.

If cutting an onion makes you cry, at least its area is as easy as pi...

2. The method of constructing the first side OP of the square OPMN is in Descartes' 1637 La Géométrie, via Euclid's Elements, (300 BCE) Book VI Prop. 13, “To two given lines to find a mean proportional.” Thus SO is to OP as OP is to 1 and we write this as SO : OP :: OP : 1 . (The yellow triangles SOP and POA are similar, so their side lengths are proportional.) In the proportion the product of the extreme (outer) terms = the product of the mean (inner) terms, SO × 1 = OP × OP. Thus SO = OP2 and √SO = OP. Thus OP is the side of the square with the same area as the area of the given circle.
3. Squaring the circle cannot be done with compass and straight edge. Yet before the compass, in both India and Egypt, circles were drawn via 'peg and rope/cord' methods. As the construction shown does NOT appear in the Indian ?ulbasūtras (Rules of the cord, 600 BCE) which circled the square, it may be assumed the general method of finding a mean proportional had not emerged, either in India, (which made use of the Pythagorean theorem before Pythagoras) or in Greece. Reversing the Indian method of circling the square is entirely different to this construction. Similarly, the Indian method of converting a rectangle into a square is different to Euclid's.
4. Irrational numbers can be constructed with compass and straight edge, such as √2, which is the hypotenuse of a square with unit sides. As π is unable to be constructed with compass and straight edge alone, (being transcendental), we use rope to straighten the semi-circle, and create a line length of π from which we create the required square via proportion.

ENDNOTE
Did the ancient Egyptians know this method of squaring a circle? Egyptian rope stretchers created right angles for the corners of their pyramids via the Pythagorean theorem. Had they drawn a few more circles, they might have stumbled across this method. Perhaps they did, yet it appears to not have been recorded in extant historical records.

Also at:
www.jonathancrabtree.com/mathematics/an-ancient-secret-how-to-square-a-circle-with-peg-and-rope/