3-sphere

In mathematics, a hypersphere or 3-sphere is a special shape that exists in four dimensions, much like how a regular sphere lives in three dimensions. It is the set of all points that are the same distance from a central point in four-dimensional Euclidean space. The space inside a 3-sphere is called a 4-ball.

Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect ⟨0,0,0,1⟩ have infinite radius (= straight line). In this picture, the whole 3D space maps the surface of the hypersphere, whereas in the next picture the 3D space contained the shadow of the bulk hypersphere.

Even though it lives in four dimensions, the surface of a 3-sphere is three-dimensional. This means that if you were traveling on its surface, you could move in three different directions, similar to how we can move north, south, east, and west on the surface of our planet, but with an extra direction possible.

The 3-sphere is an important example of a 3-manifold, which is a space that locally looks like regular three-dimensional space, even though it might be curved or stretched in larger shapes. It helps mathematicians understand the properties of higher-dimensional spaces and has applications in various areas of geometry and physics.

Definition

A 3-sphere is a special shape in four-dimensional space. Imagine a regular sphere, which is the set of all points that are the same distance from the center in three-dimensional space. In the same way, a 3-sphere is the set of all points that are the same distance from a center point in four-dimensional space.

The unit 3-sphere, which has a radius of 1 and is centered at the origin, is very important in mathematics. It can also be described using complex numbers or quaternions, which are special number systems that help us understand four-dimensional space better.

Properties

The 3-sphere is a special shape in four-dimensional space. Imagine a regular sphere, which you can think of as a ball in three dimensions. The 3-sphere is like that ball but stretched into four dimensions. It is the set of all points that are the same distance from a central point in four-dimensional space.

When you cut through a 3-sphere with a flat three-dimensional slice, you get a regular two-dimensional sphere. This slice starts as a single point, grows into a sphere, gets bigger until it passes through the middle of the 3-sphere, and then shrinks back to a point. The 3-sphere has special properties that make it interesting to mathematicians, such as being able to shrink any loop back to a point without leaving the surface.

Homotopy groups of S³
k	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
π_k(S³)	0	0	0	Z	Z₂	Z₂	Z₁₂	Z₂	Z₂	Z₃	Z₁₅	Z₂	Z₂⊕Z₂	Z₁₂⊕Z₂	Z₈₄⊕Z₂⊕Z₂	Z₂⊕Z₂	Z₆

Topological construction

A 3-sphere can be made by joining two 3-dimensional balls at their edges. Imagine two balls that are the same size. Their outer edges are like 2-dimensional spheres. When you join these edges together, matching every point on one edge to the matching point on the other edge, you create a 3-sphere. The place where they are joined is called the equatorial sphere.

Another way to think about a 3-sphere is by removing just one point from it. What remains is like normal 3-dimensional space. This idea is similar to how removing a point from a regular sphere leaves something that looks like a flat plane. This way of seeing a 3-sphere helps mathematicians study its properties.

Main article: Topologically
Main articles: "Glueing" together, 3-balls, real-valued function, stereographic projection, conformal, exponential map, unit disk, Lie group

Coordinate systems on the 3-sphere

The four coordinates for a 3-sphere add up to one, meaning they are not all independent. To describe the 3-sphere more simply, we can use three coordinates, similar to how we use two coordinates (like latitude and longitude) to describe a regular sphere. However, because of the special shape of the 3-sphere, we need at least two different sets of coordinates to cover the whole space.

The Hopf fibration can be visualized using a stereographic projection of S3 to R3 and then compressing R3 to a ball. This image shows points on S2 and their corresponding fibers with the same color.

One common way to describe the 3-sphere uses angles similar to those on a regular sphere, called hyperspherical coordinates. These coordinates include angles ψ, θ, and φ, which help us picture points on the 3-sphere. Another useful set of coordinates is called Hopf coordinates, which help us understand the 3-sphere in terms of complex numbers. Finally, stereographic coordinates are another way to map the 3-sphere onto a flat space, similar to how we can flatten a globe onto a map.

latitude longitude coordinate charts hyperspherical coordinates round metric volume form versor unit imaginary quaternion Euler's formula quaternions and spatial rotations Hopf bundle torus circle Hopf fibration stereographic projection hyperplane atlas

Group structure

When we think of the 3-sphere as a group of special numbers called unit quaternions, it gains an important structure through multiplication. This makes the 3-sphere a special type of group called a Lie group, which is smooth and has three dimensions.

Only certain spheres, like the 1-sphere and the 3-sphere, can be Lie groups. The 3-sphere is connected to a special group of matrices called SU(2), meaning they share the same structure in a particular way. This connection helps mathematicians understand the properties of both the 3-sphere and SU(2).

In literature

In the book Flatland, written by Edwin Abbott Abbott in 1884, and its sequel Sphereland by Dionys Burger from 1965, a 3-sphere is called an oversphere.

Some writers and thinkers, like those in The Divine Comedy, have imagined the universe in ways that match the idea of a 3-sphere. This idea is also explored in art and architecture.