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):
$S^1$ is an $S^0$ bundle over $\mathbb R \mathbb P^1 \cong S^1$.
$S^3$ is an $S^1$ bundle over $\mathbb C \mathbb P^1 \cong S^2$.
$S^7$ is an $S^3$ bundle over $\mathbb H \mathbb P^1 \cong S^4$, the quaternionic projective line.
$S^{15}$ is an $S^7$ bundle over $\mathbb O \mathbb P^1 \cong S^8$, the octonionic projective line.
