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"
In this post, we will explore a few ways to derive the volume of the unit dimensional sphere in . Let’s begin with an important question: What is the value of the following integral: . This is known as a Gaussian integral, and is related to one of the most important concepts seen in basic […]Read more "The Gaussian and Spherical Volume"
“Eureka! Eureka!” shouts could be heard from around town as a brazen Archimedes streaked through the streets of Syracuse bare. “Eureka!” he shouted all the way home. He felt nothing but pure excitement at his latest discovery. He had just come from the bath where he stuck his foot in and saw the water rise. […]Read more "Eureka! The Legend"
According to a timeline of the history of mathematics by William Richardson at Wichita State University, the earliest known evidence of mathematics is the evidence of counting around 50,000 B.C. Since then mathematics has advanced tremendously. The advancement of mathematics is experienced by school children, yet very often it goes unnoticed and unappreciated. Students begin by learning […]Read more "Abstraction"
Mathematicians create entire worlds from a small set of rules. This can be seen with the advent of real numbers. The world around the real numbers is still growing even today. However, there are always roadblocks that stop mathematicians dead in their tracks. For example, in the real numbers, there is no solution to because […]Read more "Quaternion"
The Möbius strip is a curious object.The strip was discovered simultaneously and independently by German mathematicians August Ferdinand Möbius and Johann Benedict Listing. Möbius took the concept farther than Listing by exploring the properties of this strip. Let us discuss some of these properties. If you happen to have a ribbon or something […]Read more "Topological Twists: The Möbius Strip"
Data is a large part of our everyday lives. We personally take in all kinds of data everyday. Things such as weather, political polls, meal costs, and much much more make up most of our daily lives. Companies collect data as well. Medical, consumer, weather, and other kinds of data have been collected for years. […]Read more "Topological Data Analysis part 1"