README.md

Hopf Fibration Visualizer

Welcome to the Hopf Fibration Visualizer! This website allows you to explore and visualize the Hopf Fibration, a fundamental concept in topology and differential geometry.

About the Hopf Fibration

The Hopf Fibration, discovered as the first example of a map from a higher-dimensional sphere to a lower-dimensional sphere that is not null-homotopic, was a groundbreaking revelation in mathematics. Contrary to the prevailing belief at the time, the Hopf Fibration demonstrated that such mappings exist, challenging the analogy with homology groups.

The Hopf map $f: S^3 \rightarrow S^2$ arises in various contexts and can be generalized to a map $S^7 \rightarrow S^4$. For any point $p$ in the sphere $S^2$, its preimage $f^{-1}(p)$ is a circle $S^1$ in $S^3$. Descriptions of the Hopf map, also known as the Hopf fibration, abound.

Mathematical Representation

The 3-sphere, $S^3$, defined as a submanifold of $\mathbb{R}^4$, and the 2-sphere, $S^2$, a submanifold of $\mathbb{R}^3$, are mathematically represented as:

$S^3 = {(X_1, X_2, X_3, X_4) : X_1^2 + X_2^2 + X_3^2 + X_4^2 = 1}$

$S^2 = {(x_1, x_2, x_3) : x_1^2 + x_2^2 + x_3^2 = 1}$

The Hopf map transforms points $(X_1, X_2, X_3, X_4)$ on $S^3$ to points $(x_1, x_2, x_3)$ on $S^2$ according to the equations:

$x_1 = 2(X_1X_2 + X_3X_4)$ $x_2 = 2(X_1X_4 - X_2X_3)$ $x_3 = X_1^2 + X_3^2 - X_2^2 - X_4^2$

Usage

To use the Hopf Fibration Visualizer, simply visit the following URL:

https://hopf-fibrations-4vwvchlou-ulizesr.vercel.app

Feedback and Contributions

Feedback and contributions are highly appreciated! If you encounter any issues, have suggestions for improvements, or want to contribute to the project, feel free to open an issue or submit a pull request.