Lill’s Method and the Derivative of tan(θ) and the Powers of tan(θ)

Lill’s method can be used to obtain the derivatives of polynomial equations in a graphical manner. The method to obtain the derivatives is described in the paper “NOTE ON LILL’S METHOD OF SOLUTION OF NUMERICAL EQUATIONS” by B. MEULENBELD. The paper complicates the matter by having Lill representations of polynomials that don’t have perpendicular segments.… Continue reading Lill’s Method and the Derivative of tan(θ) and the Powers of tan(θ)

Quadratics with Complex Roots, Lill’s Circle and the Hyperbola

I already have a few papers that show how Lill’s method can be used to solve quadratic equations. In my paper “Lill’s Method and Graphical Solutions to Quadratic Equations” I showed 2 methods that can solve quadratics with real roots and 1 method for solving quadratics with complex roots. That paper also mentions that the… Continue reading Quadratics with Complex Roots, Lill’s Circle and the Hyperbola

Littlewood Polynomials of Degree n with Closed Lill Paths

In this post I attempt to answer this question: How many Littlewood polynomials of degree n have a closed Lill path? A Littlewood polynomial is a polynomial that has all its coefficients equal to 1 or -1. Littlewood polynomials seem to be studied for their properties related to autocorrelation. A polynomial has a closed Lill… Continue reading Littlewood Polynomials of Degree n with Closed Lill Paths

The Tridecagon, Hyperbola and Lill’s Method

The regular tridecagon is another regular polygon that cannot be constructed using a compass and straightedge. In this post I want to show how the tridecagon can be constructed using the intersection of a circle and a hyperbola. In my previous posts “The Heptagon, Hyperbola and Lill’s Circle” and “The Nonagon, Hyperbola and Lill’s Method”… Continue reading The Tridecagon, Hyperbola and Lill’s Method

Lill’s Method, Prime Numbers and Tangent of Sum of Angles

In this post I want to explore again the property discussed in my paper “Lill’s Method and the Sum of Arctangents”. I’ll apply the property to this question: If tan(θ1)=2, tan(θ2)=3,tan(θ3)=5,…,tan(θn)=n-th prime number, then what is tan(θ1 + θ2 +… θn)? The question can be easily solved with a calculator. We’ll see that the answer… Continue reading Lill’s Method, Prime Numbers and Tangent of Sum of Angles

The Nonagon, Hyperbola and Lill’s Method

The nonagon is another polygon that cannot be constructed with ruler and the compass (see OEIS sequence A004169). However, the nonagon can be constructed using conics (see OEIS sequence A051913). In this post I want to show how we can use the intersection of the Lill circle of the polynomial x3 – 0.75x + 0.125… Continue reading The Nonagon, Hyperbola and Lill’s Method

The Heptagon, Hyperbola and Lill’s Circle

The heptagon cannot be constructed with just a ruler and a compass. However, in this post I’ll show how you can construct the heptagon using a hyperbola and the Lill’s circle of a third degree polynomial. The heptagon construction is very similar in nature to my trisection construction that I presented in my “Trisection Hyperbolas… Continue reading The Heptagon, Hyperbola and Lill’s Circle

Trisection Hyperbolas and Lill’s Circle

In my previous blog post about trisection “Angle Trisection: a Neusis Construction using Lill’s Method and Lill’s Circle”, I showed a “mechanical” method for trisecting an angle smaller than 90 degrees. The neusis construction involved the Lill’s method representation of cubic equations of the form x3  -3tan(θ)x2 -3x + tan(θ). These cubic equations have 3 real… Continue reading Trisection Hyperbolas and Lill’s Circle

Angle Trisection: a Neusis Construction using Lill’s Method and Lill’s Circle

The problem of trisecting an angle using just a compass and a straightedge is an impossible problem. However, the problem of angle trisection can be solved if we are allowed the use of a marked ruler. Geometric constructions that use a marked ruler are called neusis constructions. You can learn more about the topic of… Continue reading Angle Trisection: a Neusis Construction using Lill’s Method and Lill’s Circle

Lill’s method and the Philo Line for Right Angles

(Note: This is an article from 2016 that I posted on Hubpages. In this article I used Riaz’s way of illustrating Lill’s method. To become more familiar with Lill’s method I recommend going through the links provided in my Lill’s Method page. Also see my Papers page and my Lill’s method articles/posts. ) In this… Continue reading Lill’s method and the Philo Line for Right Angles