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

The Vertex, Axis of Symmetry and Corresponding Fregier Points for Parabola Points

In my blog post “Frégier’s Theorem and Frégier Points” I introduced Fregier’s theorem. In this post I want to show an alternative way of constructing or finding the corresponding Fregier points for points on a Parabola. This alternative method makes use of the Vertex point and the parabola’s axis of symmetry. I am not sure… Continue reading The Vertex, Axis of Symmetry and Corresponding Fregier Points for Parabola Points

Whittaker’s Root Series: Going Transcendental

Whittaker’s Root Series formula is an interesting method that can be used to calculate the root with the smallest absolute value of a polynomial equation. The formula creates a geometrically convergent infinite series using the determinants of a special class of Toeplitz matrices. These Toeplitz matrices are generated using the coefficients of the polynomial equation.… Continue reading Whittaker’s Root Series: Going Transcendental

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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

My archive.org Collection

On archive.org I have a small collection of scanned books or brochures. Right now I only have 7 uploads and they are in Romanian or English. For convenience I will embed the scanned materials on this page and maybe I will give a short description. The Dutch and Flemish Cabinet Galleries (National Gallery of Art… Continue reading My archive.org Collection

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