Squaring the Circle - Old School

[Acorn]

Using just a straight edge and compasses, it has been proven that it is impossible to construct a square with the same area as a given circle.

What confounds me, even now, is that Archimedes, introduced his eponymous spiral into the mix and came up with a solution.

I mentioned it in passing in https://www.talkgraphics.com/showthr...555#post619555 that I was looking a bit more deeply into what he got up to with his spiral: one turn inscribes one-third the area of its enclosing circle. My Area of Archimedean Spiral.xar tried to depict this in a XAR file to see how accurate Xara's implementation is. I got 99.843% accuracy.

My final diagram tried to follow Archimedes' construction logic, using XDPX, and I got Area of Circle.xar, where the green circle, the yellow triangle and the blue square should all have the same area (three times the red sleeping bird).
The method involves drawing the tangent to the spiral at the end of the first turn. This bit is a bit ropey.

In the end I managed 99.852%.

So Xara, you could do better!

Acorn

[gwpriester]

Dunno if this counts.

Create a circle with the Ellipse Tool.

Select the circle then select the Quick Shape Tool, set the number of sides to 4 and press the Create Polygons button.

The resulting square fits exactly inside of the circle.

Increase the circle by 141.37% and the Square and the Circle are the same size?

[Acorn]

Dunno if this counts.
Create a circle with the Ellipse Tool.
Select the circle then select the Quick Shape Tool, set the number of sides to 4 and press the Create Polygons button.
The resulting square fits exactly inside of the circle.
Increase the circle by 141.37% and the Square and the Circle are the same size?

Cheers Gary, the point of "Old School" is no one at the time knew a very accurate value for pi.
It is the mindset of what these people could do without tools we currently have with a seeming deeper insight.

I was chasing the insights, like the volume of a cone is one-third that of the enclosing cylinder, all done using the same geometric approach.

Your scaling simply puts the circle inside the square, the diameter and the side of the square are now the same length.
This is not the same as the areas being the same - sorry.

For your approach, you didn't need to scale up just clone and rotate 45 degrees and then Shift+Drag the circle of box to fit Yellow square).
The scaling factor ought to have been Root 2 = 1.4142 (141.42%).

To get equal areas, the scaling factor would actually be 125.32% and, saldly, this does not lead to a simple construction.



The white square has the same area as the green circle.

Acorn

[gwpriester]