Hopf Fibration

About this Program

This is a 3D visualization of the Hopf Fibration, a mapping of points on $S^3 \to S^2$ visualized with "fibers". This program is written in C++ and is rendered in OpenGL.

Theory

The usual three dimensional sphere, $S^2$, is an object said to live in $\mathbb{R}^3$ described by

$$S^2 = {(x, y, z) \in \mathbb{R}^3 \vert x^2 + y^2 + z^2 = r^2}$$

For some $r\in \mathbb{R}$. Similarly, the four dimensional sphere, $S^3$, is defined as

$$S^3 = {(x, y, z, w) \in \mathbb{R}^4 \vert x^2 + y^2 + z^2 + w^2 = r^2}$$

Notice that coordinates in $\mathbb{R}^4$ may be written as coordinates in $\mathbb{C}^2$ and coordinates in $\mathbb{R}^3$ may be written as coordinates in $\mathbb{C}\times \mathbb{R}$.

$$(x_1, x_2, x_3, x_4) = (z_0, z_1)$$

Similarly,

$$(x_1, x_2, x_3) = (z, x)$$

Then, the Hopf Map satisfies

$$p(z_0, z_1) = (2z_0z_1^*, |z_0|^2 - |z_1|^2)$$

It can be verified that $p(z_0, z_1) \in S^2$ for all $(z_0, z_1)\in \mathbb{C}^2$.

A possible inverse Hopf Map is then given by the following. Let $(a, b, c) \in \mathbb{R}^3 \backslash {(0, 0, -1)}$. Then, the inverse Hopf Map $f$ of $(a, b, c)$ is given by

$$f(a, b, c) = \frac{1}{\sqrt{2(1 + c)}}((1 + c)\cos(\theta), a \sin(\theta)\cdots$$

$$\cdots- b \cos(\theta), a\cos(\theta) + b\sin(\theta), (1 + c)\sin(\theta))$$

As $\theta$ changes, we sweep out a great circle of $S^3$. However, we cannot visualize this and thus we must stereographically project the set of points from $\mathbb{R}^4 \to \mathbb{R}^3$.

In summary, this project samples various points in $S^2$ and via the inverse Hopf Map and stereographic projection, returns fibers which we can render in 3D space.

Usage

1. Clone this repository

2. From the root of the repository,

   xcopy res build\Release\res /E /I
   mkdir build
   cd build
   cmake -G "Visual Studio 17 2022" -A Win32 ..
   

3. Navigate to /build and build the project using release build either in Visual Studio or in terminal via MSBuild.

4. Run build/release/main.exe

Future Work

• Clean up code
• Figure out optimizations for calculating new points
• Create more initial point distributions on $S^2$

Acknowledgements

This project uses OpenGL abstraction classes from "The Cherno"'s tutorial series. Also, the Wikipedia Article on the subject was a very helpful resource.