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

## Adding and Substracting 1 and the First n Primes to Get the Next Prime (Part 2)

In part 1 I dealt with even-indexed primes. In this post I want to show the first few examples of how to obtain the odd-indexed prime numbers using the addition or subtraction of 1 and all the other preceding primes. The odd-indexed primes are covered by the OEIS sequence A031368. The summation for each prime… Continue reading Adding and Substracting 1 and the First n Primes to Get the Next Prime (Part 2)

## Adding and Substracting 1 and the First n Primes to Get the Next Prime (Part 1)

In this post I want to show a few examples of how a prime number can be obtained from all the preceding prime numbers and 1 using addition and subtraction. In Part 1 I’ll deal with only even-indexed primes: a(n)=prime(2n). 3 is the first even-indexed prime, 7 is the second, 13 is the third, 19… Continue reading Adding and Substracting 1 and the First n Primes to Get the Next Prime (Part 1)

## A Prime Counting Sequence and Andrica’s Conjecture

In this post I want to discuss a prime counting sequence similar to OEIS sequence A066888. The sequence A066888 counts the number of primes between 2 consecutive triangular numbers. On the OEIS page of the sequence there is a conjecture that says that there is at least one prime number between 2 consecutive triangular numbers.… Continue reading A Prime Counting Sequence and Andrica’s Conjecture

## 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

## 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

## Symbolic Literacy: Justice and the Ostrich

Egyptians were the first to symbolically associate the ostrich with justice. Maat, the goddess of justice, is usually depicted wearing an ostrich feather. The image above shows a scene from the Book of the Dead that depicts the “Weighting of the heart” ritual. In the “Weighting of the heart’ ritual, the heart of of the… Continue reading Symbolic Literacy: Justice and the Ostrich