Newton's Cube 3 Circular Projective Geometry

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Quadrature has eluded mathematicians until because they have been using Euclidean Geometry. You can only ever get a close approximation unless you use my geometry of complex space!

Circular Projective Geometry
Projective geometry is an interesting but rather abstract area of mathematics that I came across and considered its suitability for helping to provide a (or near) solution for quadrature (the conversion of a circle into a square). I actually got much better results than I expected as I was approaching quadrature from the non-mathematical problem of solving Newton’s Cube.
Many mathematical genii have tried their hand at solving quadrature but it was the equivalent of turning lead into gold.
Although existing methods of projective geometry were fascinating I did not expect them to help me with quadrature. I believed I had to start with a circle whereas the normal approach in projective geometry may have been to start with a square which seems, wrongly, easier.

Establishing a New Projective Geometry.
To outline my new projective geometry a simple approach is taken by taking the projection of an ellipse upon a circle. This produces a projection curve which is close to being a straight line.

This paper then identifies this curve by what I call the “elastic curve” and shows that the possibility now exists to apply this new mathematics to form a square from a circle and so solve Newton’s Cube.
The discovery of elastic curves is fundamental in solving quadrature (at least by projective geometry) and uncovers the theoretical and numerical advantage of using 'elastic curves' rather than exponential curves.

Traditional Projective Geometry.
Traditionally projective geometry has used lines projected from a set figure to generate a new curve.
I saw that inclining a circular in a plane can create an ellipse and so the possibility of doing this is very good with projective geometry. Also if we find the focal point of an ellipse and tie a string to them then by allowing a pen to map the behaviour of that string creates an ellipse (properties of an ellipse). This gets me half way to a square easily if the ellipse can truly be considered in this manner. I investigated using these strings outside the ellipse to form a projective geometry:

See figure "Projection Lines 1"

Continued on page 4

I considered the behaviour of arcs and tangents to an ellipse to see if they was any advantage to using them:

See figure “Projection Lines for an ellipse”

 We can consider the arc AB fixed and revolve point P around the ellipse but the arc length will increase (for example AC). If we keep the arc AB fixed instead and rotate P we can see that a this is better but still the locus is not a square.

An example of this is given in the link below:

LINK HERE

In figure one we use tangents taken from the ellipse to generate a new set of points P which comprise the new curve. In this conventional form the set of chords QQ' are used to determine the points Q and Q'.