typo.

docs/src/hopffibration.md

@@ -6,7 +6,7 @@ Description = "How the Hopf fibration works."
 
 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 computing its points via a bundle atlas 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). The basic idea is that since the Hopf map takes the three-dimensional sphere into the two-dimensional sphere, we can pull the skin of the globe back to the three-sphere and visualize it.
 
-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 a monolithic visualization. It also makes sense to visualize differential operators in the Minkowski space-time as vectors in a cross-section of the Hopf bundle and then study the properties of spin-transformations. The choice of a gauge transformation (or trivialization) along with Lorentz transformations of Minkowski spacetime should not have any effect on physical laws. It is therefore a great model to understand these transformations and walk the road to reality. 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. The book Mark J.D. Hamilton (2018) explains the Standard Model to students of both mathematics and physics, covers both the specific gauge theory of the Standard Model and generalizations, and is highly accessible and self-contained. Then, the definitions are going to be used to explain the source code in terms of computational methods and types.
+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 a monolithic visualization. It also makes sense to visualize differential operators in the Minkowski space-time as vectors in a cross-section of the Hopf bundle and then study the properties of spin-transformations. The choice of a gauge transformation (or trivialization) along with Lorentz transformations of Minkowski spacetime should not have any effect on physical laws. It is therefore a great model to understand these transformations and walk the road to reality. 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. The book *Mathematical Gauge Theory* explains the Standard Model to students of both mathematics and physics, covers both the specific gauge theory of the Standard Model and generalizations, and is highly accessible and self-contained. Then, the definitions are 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``.
 
@@ -242,7 +242,7 @@ However, step seven makes sure that the coordinates in the linear 2-space are in
 
 ## Compute a Four-Screw
 
-We are going to execute a motion around a closed loop in the Lie group ``SL(2, \mathbb{C})``, and then multiply every point in the Hop bundle by an element of the loop. A four-screw is a subset of the Lie group ``SL(n, \mathbb{K}) = \{A \in Mat(n \times n, \mathbb{K}) | det(A) = 1]\}``, square matrices of `Complex` numbers whose volume form (determinant) equals 1. Here, the number ``n = 2`` and the field ``\mathbb{K} = \mathbb{C}``. A four-screw is a kind of restricted Lorentz transformation where a z-boost and a proper rotation of the celestial sphere are applied. The transformation lives in a four-complex dimensional space and it has six degrees of freedom (the same number of dimensions as ``SO(4)``). By parameterizing a four-screw one can control how much boost and rotation a transformation shuld have. Here, `w` as a positive scalar controls the amount of boost, whereas angle `ψ` controls the rotation component of the transform. But, the parameterization accepts *rapidity* as input for the boost. So we take the natural logarithm of `w` (``log(w) = \phi``) in order to supply the transformer with the required rapidity argument. First, we set `w` equal to one in order to preserve the scale of the Argand plane and animate the angle `ψ` through zero to 2π for rotation. The name `progress` denotes a scalar from zero to one for instantiating a different transformation at each frame of the animation.
+We are going to execute a motion around a closed loop in the Lie group ``SL(2, \mathbb{C})``, and then multiply every point in the Hopf bundle by an element of the loop. A four-screw is a subset of the Lie group ``SL(n, \mathbb{K}) = \{A \in Mat(n \times n, \mathbb{K}) | det(A) = 1]\}``, square matrices of `Complex` numbers whose volume form (determinant) equals 1. Here, the number ``n = 2`` and the field ``\mathbb{K} = \mathbb{C}``. A four-screw is a kind of restricted Lorentz transformation where a z-boost and a proper rotation of the celestial sphere are applied. The transformation lives in a four-complex dimensional space and it has six degrees of freedom (the same number of dimensions as ``SO(4)``). By parameterizing a four-screw one can control how much boost and rotation a transformation shuld have. Here, `w` as a positive scalar controls the amount of boost, whereas angle `ψ` controls the rotation component of the transform. But, the parameterization accepts *rapidity* as input for the boost. So we take the natural logarithm of `w` (``log(w) = \phi``) in order to supply the transformer with the required rapidity argument. First, we set `w` equal to one in order to preserve the scale of the Argand plane and animate the angle `ψ` through zero to 2π for rotation. The name `progress` denotes a scalar from zero to one for instantiating a different transformation at each frame of the animation.
 
     if status == 1 # roation
         w = 1.0