# Vanishing Volume: The Curious Case of The Sphere

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 […]

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# The Gaussian and Spherical Volume

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 […]

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It is a common occurrence in mathematics that when something does go wrong, it goes terribly wrong. This exact phenomenon occurs with the Banach-Tarski Paradox. Informally, it says that one can take a sphere (in 3 or more dimensional space)  can be split into finitely many pieces and, using only rigid motions, can be rearranged […]