3-sphere

In mathematics, a hypersphere or 3-sphere is a special shape that exists in four dimensions. It is like a regular sphere, but in a space with one more direction.

The 3-sphere is made 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 you could move in three different directions on its surface, just like we can move in different directions on the surface of our planet.

The 3-sphere is an important example of a 3-manifold. It helps mathematicians understand higher-dimensional spaces and has uses in geometry and physics.

Definition

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

The unit 3-sphere, with a radius of 1 and centered at the start point, 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. Think of a regular sphere, which looks like 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.

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, 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 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 always add up to one, so they are not all independent. We can make it easier to describe the 3-sphere by using just three coordinates. This is like using two coordinates, such as latitude and longitude, to describe a regular sphere.

Because of the special shape of the 3-sphere, we need more than one set 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, called hyperspherical coordinates. These angles, ψ, θ, and φ, help us picture points on the 3-sphere. Another useful set of coordinates is called Hopf coordinates, which help us understand the 3-sphere using complex numbers. Stereographic coordinates are another way to map the 3-sphere onto a flat space, like flattening 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 gets an important structure by multiplying them. This makes the 3-sphere a special type of group called a Lie group. It is smooth and has three dimensions.

Only some spheres, like the 1-sphere and the 3-sphere, can be Lie groups. The 3-sphere is linked to a special group of matrices called SU(2). They share the same structure in a special way. This helps mathematicians learn more about 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 seen in art and architecture.