Geometric interpretation of quartic equation (with graph)

Draw a similar shape, where the radius R, the distance α and the angle ω with ω>0° have random values.

 

The consequences are

 

1. x1 + x2 + x3 + x4 = 0

 

Of course, the values of the points x are the roots of a reduced quartic equation. Then

 

2. The radius R is calculated by the coefficients of this equation, while it is the only radius for which the rectangularity of the lines of the figure applies.

 

This is part of the geometric interpretation of the quartic equation that leads to multiple analytical solutions.

 

Can this work for higher degree equations?

 

The following result shows whether it would be possible for such an equation (with degree n) to have a similar model with parameters R, α, ω (where the angles formed by the lines at point 0,0 are equal to 360°/n each).

 

If R,α and n have constant values with R>|α|>=0 and n>2, then for every angle ω will be valid

 

x1+x2+x3+...+xn=δcos(nω)

 

where δ is a constant. Therefore δ will be equal to the sum of the roots for ω=0°. When n is an odd number it will be δ=0 only if α=0, while when n is an even number it will always be δ=0 due to the symetry that these systems have with respect to the y-axis for ω=0°.

 

So the indications are encouraging for equations that have an even degree, but the problem remains open.

 

"The single biggest problem we face is that of visualisation", Richard Neiman.

Done