**Hypersphere**

**11/12/16**

*“What does a four-dimensional sphere look like?”*

Let’s think about something. A **circle** can be classified as an object in 2-dimensional space whose boundary is composed of all of the points equidistant from a center point, where the radius is composed of **two cartesian coordinates** ( r=sqrt(x^2+y^2)) and the **area is proportional to the square of the radius** (A=pi*r^2). A **sphere** in turn is the 3-dimensional version of this, where all of the points equidistant from a center makes up the object, with the radius being composed **three cartesian coordinates **and the **volume is proportional to the cube of the cube of the radius** (V=4/3 * pi * r^3 ), and if were to take a two-dimensional cross-section, we would obtain a circle. .But what if we were to take this concept into higher dimensions? Well, let’s explore the concept. A four-dimensional sphere (termed a **hypersphere **by mathematicians) would have to be **described with four cartesian coordinates**, and the **“hyper-volume” would be proportional to the fourth power of the radius **(V= ½ * pi^2 * r^4),and if were to take a three dimensional cross section of a hypersphere, we would find a regular three-dimensional sphere.

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