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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Banach-Tarski Paradox

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) ┬ácan be split into finitely many pieces and, using only rigid motions, can be rearranged […]

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The Fibonacci Sequence and the Golden Ratio

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 […]

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Summing the Natural Numbers

Infinite sums are weird. Some converge and some do not. One might think it easy to conclude that if we sum the natural numbers, we would get . But is this correct? Believe it or not, we can rearrange the sum of the natural numbers to be equal to . We will give the proof […]

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