What happens to the volume of a sphere in higher dimensions? To answer this question, we will focus our attention on the unit -sphere in Euclidean space. That is the sphere in (-space) with radius centered at the origin. For example, in , the unit sphere is the collection , which satisfies the equation […]Read more "Vanishing Volume: The Curious Case of The Sphere"
This is the third and final post on the volume of a sphere. The other two can be accessed by the following links, “Coordinates in 3-Space” and “The Volume of a Sphere with Calculus” As the title suggests, this will be a derivation without the use of Calculus. This proof is Greek in origin, in […]Read more "The Volume of a Sphere (without Calculus)"
This post was inspired by a calculus student and will be in three parts: This on the development of different coordinate systems, one with calculus and one without. We are taught in school that the volume of a sphere with radius is . In this post we shall look at a development in calculus that will not […]Read more "Coordinates in 3-Space"
1, 1, 2, 3, 5, 8, 13, 21, 34, … These numbers form one of the most recognizable sequences in the world. It is known as the Fibonacci sequence and it’s named after Leonardo Pisano, who was also known as the 13th century mathematician Fibonacci. The story goes, Fibonacci was working out the growth rate […]Read more "The Fibonacci Sequence and the Golden Ratio"
We are going to go through a (light) development of some trig identities by using geometry. We will not get into specifics or prove these claims in this post. There might be a post later to prove these statements. So without further ado let’s start with a circle. Choose two points on the circle that […]Read more "An Interesting Trig Circle"