The Matrix Exponential of K3

`make` a whirl using this: matrix[j, i] = project(exp(K(3) * θ) * p)
`make` a base map using this: matrix[i, j] = project(exp(f * θ * K(1) + f * -ϕ * K(2)) * q)
Get the source code: github.com/iam...
This animation begins with specifying a point in the three-dimensional sphere, which is a unit Quaternion number. As the video progresses in time, the point is pushed around the sphere along the K(3) axis by applying the exponential map repeatedly. It’s worth noting that the K(3) axis together with K(1), K(2) and the Identity axes form an abstract vector space. After exponentiation, the elements of the vector space represent the rotation of a four-dimensional orthogonal frame, a tetrad, also known as the symmetry group SO(4). The surprising fact about pushing points around the 3-sphere is that after exponentiating along K(3) for the equivalent of 360 degrees, the point becomes its own negative value (the antipodal point). You can see this change of sign directly at approximately second 22.5” of the animation. See it for yourself by comparing the shape of the 3-sphere between the first second and 22.5” of the video. This change will revert back to its initial sign by the end of the first half of the video though, since the rotation keeps going until the equivalent of 720 degrees is completed. So, the first second and the forty fifth second of the animation show the bundle with the same sign as a consequence of two complete turns. Nothing else is put in the code by hand, other than the natural result of continuous exponentiation for 720 degrees along the K(3) basis vector.
In 1931 Heinz Hopf discovered that the three-dimensional sphere is a bundle of circles over the skin of the globe. Furthermore, the tangent space of the sphere is defined as the space whose elements are orthogonal to radial vectors. The identification of Hopf splits the tangent space of the three-sphere into two different spaces: the horizontal and vertical subspaces. However, the 3-sphere is embedded in the four-dimensional Euclidean space, denoted by ℝ⁴. Therefore, the natural connection that puts the pair of subspaces back together smoothly, follows the basis vectors of the Lie algebra of SO(4). We take the exponentiation along the K(3) axis as motion in a vertical orbit, whereas the exponentiation along linear combinations of K(1) and K(2) bases, means a motion in horizontal subspaces. Here, the base maps represent a visualization of the horizontal subspaces while the whirls show the vertical orbits going through the base maps.
Also here, Starman tracks the parallel transport of a frame adapted to the three-sphere, such that every vector of the frame is tangent to the 3-sphere, while preserving mutual orthogonality. That means the track is made of all tangent frames. Through time, the animation shows a gradual change in both the direction and distance of the trajectory of the parallel transport of Starman’s frame. But, Starman switches its starting position on the 3-sphere every fifteen seconds. In order to know where Starman is located, the orbits of a few boundaries are drawn, including: Australia, Japan, United States of America, United Kingdom, Antarctica and Iran. Then, there’s a twist in the moving frame due to the curvature of the three-sphere. The natural connection is a fundamental tool for characterizing and measuring the twist of the trajectory. This is how Starman can do dead reckoning and find its way home among the stars.
See also:
Dead Reckoning With Connection One-Forms
• Dead Reckoning With Co...
The Push Along The K(2) Direction
• The Push Along The K(2...

[@TheMemesofDestruction]

Now that’s what I’m talking about! ^.^