Fixed typos and wrote an introduction.

docs/src/hopffibration.md

@@ -4,11 +4,11 @@ Description = "How the Hopf fibration works."
 
 # The Hopf Fibration
 
-The Hopf fibration is a fiber bundle with a two-dimensional sphere as the base space and circles as the fiber space. It is the geometrical shape that relates Einstein's spacetime to quantum fields. In this model, we visualize the Hopf fibration by first calculating its points via a bundle chart and then rendering the points in 3D space via stereographic projection. The projection step is necessary because the Hopf fibration is embedded in a four-space, yet it has only three degrees of freedom as a three-dimensional shape. The idea that makes this model more special and interesting than a typical visualization is the idea of [Planet Hopf](http://drorbn.net/AcademicPensieve/Projects/PlanetHopf/), due to Dror Bar-Natan (2010). So, if the base space is a two-dimensional sphere much like the skin of the globe, then we can model the Earth as a sphere and skin the horizontal sections of the bundle. Into the bargain, the Earth rotates about its axis every 24 hours. That spinning transformation of the Earth, together with the non-trivial product space of the Hopf bundle, can be encoded naturally into the visualization. It makes a lot of sense no matter how ridiculous, especially when we try to visualize differential operators in the Minkowski space-time and investigate the properties of spin-transformations. The following explains how the source code for generating animations of the Hopf fibration works (alternative views of Planet Hopf).
+The Hopf fibration is a fiber bundle with a two-dimensional sphere as the base space and circles as the fiber space. It is the geometrical shape that relates Einstein's spacetime to quantum fields. In this model, we visualize the Hopf fibration by first calculating its points via a bundle chart and then rendering the points in 3D space via stereographic projection. The projection step is necessary because the Hopf fibration is embedded in a four-space, yet it has only three degrees of freedom as a three-dimensional shape. The idea that makes this model more special and interesting than a typical visualization is the idea of [Planet Hopf](http://drorbn.net/AcademicPensieve/Projects/PlanetHopf/), due to Dror Bar-Natan (2010). So, if the base space is a two-dimensional sphere much like the skin of the globe, then we can model the Earth as a sphere and skin the horizontal sections of the bundle. Into the bargain, the Earth rotates about its axis every 24 hours. That spinning transformation of the Earth, together with the non-trivial product space of the Hopf bundle, can be encoded naturally into the visualization. It makes a lot of sense no matter how ridiculous, especially when we try to visualize differential operators in the Minkowski space-time and investigate the properties of spin-transformations. The following explains how the source code for generating animations of the Hopf fibration works (alternative views of Planet Hopf). We follow the beginning of chapter 4 of [Mark J.D. Hamilton (2018)](https://doi.org/10.1007/978-3-319-68439-0) for a formal definition of the Hopf fibration as a fiber bundle. Then, the definition is going to be used to explain the source code in terms of computational methods and types.
 
 First, let ``E`` and ``M`` be smooth manifolds. Then, ``\pi: E \to M`` is a surjective and differentiable map between smooth manifolds. Meaning, every element in ``M`` has some corresponding element in ``E`` via the map ``\pi``. Now, let ``x \in M`` be a point. A *fiber* of ``\pi`` over point ``x`` is called ``E_x`` and defined as a non-empty subset of ``E`` as follows: ``E_x = \pi^{-1}(x) = \pi^{-1}(\{x\}) \subset E``. The singleton of ``x`` is taken to the manifold ``E`` by the inverse of the map ``\pi``. However, to have a set of more than one point let ``U`` be a subset of ``M``, ``U \subset M``. Then, we have ``E_U = \pi^{-1}(U) \subset E``. In this case, ``E_U`` is the part of ``E`` above the subset ``U``.
 
-Next, define a global section of the map ``\pi`` like this: ``s: M \to E``. Considering the definition of ``pi: E \to M``, the definition of the global section implies that the composition of ``\pi`` and ``s`` is the identity map ``\pi \ o \ s = Id_M`` over ``M``. A section such as ``s`` can be a local one if we take a subset of ``M`` in the domain, ``U \subset M``. Then, a local section is defined as ``s: U \to E``. In a similar way the definition of the local section implies that the composition of ``\pi`` with it is the idenity map over the subset: ``\pi \ o \ s = Id_U``. For all points ``x`` in subset ``U``, the section ``s(x)`` is in the fiber ``E_x`` of ``\pi`` above ``x``, if and only if ``s`` is a local section of ``\pi``. In this pointwise case, the map ``\pi`` is restricted to subset ``U``. In other words ``\pi: E \to U``, where ``U \subset M``.
+Next, define a global section of the map ``\pi`` like this: ``s: M \to E``. Considering the definition of ``\pi: E \to M``, the definition of the global section implies that the composition of ``\pi`` and ``s`` is the identity map ``\pi \ o \ s = Id_M`` over ``M``. A section such as ``s`` can be a local one if we take a subset of ``M`` in the domain, ``U \subset M``. Then, a local section is defined as ``s: U \to E``. In a similar way the definition of the local section implies that the composition of ``\pi`` with it is the idenity map over the subset: ``\pi \ o \ s = Id_U``. For all points ``x`` in subset ``U``, the section ``s(x)`` is in the fiber ``E_x`` of ``\pi`` above ``x``, if and only if ``s`` is a local section of ``\pi``. In this pointwise case, the map ``\pi`` is restricted to subset ``U``. In other words ``\pi: E \to U``, where ``U \subset M``.
 
 In general, for two points ``x \not = y \in M`` that are not equal, the fibers ``E_x`` and ``E_y`` of ``\pi`` over ``x`` and ``y`` may not be embedded submanifolds of ``E``, or even be diffeomorphic. That means, there may not be a differentiable and invertible map that takes fiber ``E_x`` into fiber ``E_y``, and the tangent spaces of ``E_x`` and `` E_y`` over points ``x`` and ``y`` may not be naturally linear subspaces of the tangent space of ``E``. But, it is different in the special instance where manifold ``E = M \times F`` is the product of ``M`` and the general fiber ``F`` and ``\pi`` as a map is the projection onto the first factor ``\pi: M \times F \to M``. If that is the case, then fibers ``E_x, E_y \in F`` of ``\pi`` over the two distinct points ``x \not = y \in M`` are embedded submanifolds of ``E`` and diffeomorphic. To explain it more clearly, given that condition, there exists an invertible and smooth map taking one fiber to the other, and the tangent spaces of the fibers are directly summed with their respective dual subspaces at points in the fibers to span the whole tangent space of manifold ``E`` at points of ``\pi`` over ``x`` and ``y``.  Fiber bundles are the generalization of products ``E = M \times F`` as twisted products.