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2 fixed transposition of superscripts, reordered list

One of the most beautiful structures, in my mind, is the classical Hopf fibration, which allows you to visualize the $3$-sphere $S^3$ as a smooth circle bundle over the $2$-sphere. When you view $S^3$ minus a point as $\mathbb R^3$, one can actually draw very nice pictures of this fibration. It's doubly interesting to me because it involves the isomorphism of $S^2$ with $\mathbb C\mathbb P^1$ from complex analysis.

There are actually 4 such Hopf fibrations (spheres which are total spaces of fibre bundles whose base and fibre are also both spheres):

1) $S^3$ S^1$is an$S^1$S^0$ bundle over $\mathbb C R \mathbb P^1 \cong S^2$S^1$. 2)$S^7$S^3$ is an $S^3$ S^1$bundle over$\mathbb H C \mathbb P^1 \cong S^4$, the quaternionic projective lineS^2$.

3) $S^{15}$ S^7$is an$S^7$S^3$ bundle over $\mathbb O H \mathbb P^1 \cong S^8$S^4$, the octonionic quaternionic projective line. 4)$S^0$S^{15}$ is an $S^1$ S^7$bundle over$\mathbb R O \mathbb P^1 \cong S^1$S^8$, the octonionic projective line.

1 [made Community Wiki]

One of the most beautiful structures, in my mind, is the classical Hopf fibration, which allows you to visualize the $3$-sphere $S^3$ as a smooth circle bundle over the $2$-sphere. When you view $S^3$ minus a point as $\mathbb R^3$, one can actually draw very nice pictures of this fibration. It's doubly interesting to me because it involves the isomorphism of $S^2$ with $\mathbb C\mathbb P^1$ from complex analysis.

There are actually 4 such Hopf fibrations (spheres which are total spaces of fibre bundles whose base and fibre are also both spheres):

1) $S^3$ is an $S^1$ bundle over $\mathbb C \mathbb P^1 \cong S^2$.

2) $S^7$ is an $S^3$ bundle over $\mathbb H \mathbb P^1 \cong S^4$, the quaternionic projective line.

3) $S^{15}$ is an $S^7$ bundle over $\mathbb O \mathbb P^1 \cong S^8$, the octonionic projective line.

4) $S^0$ is an $S^1$ bundle over $\mathbb R \mathbb P^1 \cong S^1$.