1. Introduction In plane geometry, the idea of an “angle” has long been built into our intuition. The image of two rays meeting at a point sparks a vague notion of how “far apart” the two rays are: this notion is made rigorous with the definition of a , the angle required to intercept a circular arc with equal length and radius. But what happens when we leave our sheet of paper and enter the real world? How can we…
1. Introduction The first installment of this post investigated the question of computing the volume of a unit ball (and by extension any ball) of dimensions. The main result we obtained was that the volume can be computed recursively as , where In this post, we will use this result to prove an explicit formula for . 2. The Formula A quick search online yields the following explicit formula for : where the function extends the…
1. Introduction This post investigates an interesting question I had been thinking about: we all know that the area of a circle of radius equals , and the volume of a sphere of radius equals . But what if we enter the territory of even higher dimensions? We should be able to say something about the “volume” of an n-dimensional sphere, where “volume” is interpreted as the equivalent measure of space in that many dimensions (e.g. area for 2D, regular…