Category: math

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

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

Frégier’s Theorem and Frégier Points

Due to the properties of the Frégier point, Frégier’s theorem provides a practical means of constructing with straightedge and square the tangent to a conic at any point on the respective conic. Constructing a Frégier point is very easy and the procedure is the same for any type of conic section (parabola, hyperbola, ellipse or circle). Right… Continue reading Frégier’s Theorem and Frégier Points

A Very Easy Method to Inscribe a Regular Pentagon in a Circle

The classical way of inscribing a regular pentagon in a circle is discussed in Book IV, Proposition 11 of Euclid’s Elements. The method of Euclid is a bit complicated since it requires first to draw a golden triangle (a 72-72-36 triangle) and then to inscribe the triangle in the given circle. In this post I… Continue reading A Very Easy Method to Inscribe a Regular Pentagon in a Circle

Lill’s Method and Geometric Solutions to Quadratic Equations With Complex Roots

I already wrote a paper called “Lill’s Method and Graphical Solutions to Quadratic Equations” that shows how to solve quadratic equations using Lill’s method. In this post I want to show an alternative way of solving quadratic equations that have complex or imaginary roots. This method was briefly mentioned in the paper “Geometric Solution of… Continue reading Lill’s Method and Geometric Solutions to Quadratic Equations With Complex Roots

The Hexagram/Six Pointed Star/Star of David and the Golden Ratio

Many people that are familiar with the golden ratio ( Φ = 0.5(1+ 50.5) which is about 1.61803), know that it appears very often in the proportions of the pentagon and the pentagram. In this post I want to present a diagram that connects the golden ratio with the regular hexagram or the six pointed… Continue reading The Hexagram/Six Pointed Star/Star of David and the Golden Ratio

Symbolism: Allegory of Arithmetic by Laurent de La Hyre

In this post I want to discuss the symbolism found in the painting “Allegory of Arithmetic” by the baroque painter Laurent de La Hyre. This painting belongs to La Hyre’s series of paintings on the liberal arts. Since Martianus Capella ( 5th century AD), the seven liberal arts were divided in trivium (grammar, logic/dialectics and… Continue reading Symbolism: Allegory of Arithmetic by Laurent de La Hyre

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