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. 

• Mathematics