E8 and the 20-Group Twist: The Fundamental Representative Geometry of the Quasicrystalline Spin Network

Klee Irwin with Quantum Gravity Research discusses E8 and the 20-group
The fundamental representative geometry of the Quasicrystalline Spin Network is a group of 20 tetrahedra. 

What is so interesting about the 20-Group for unification physics? 

It relates to higher dimensions, just as the mathematics of modern gravitational and particle physics does.

In this video I discuss geometric details about the 20-Group, a structure that is a fundamental part of the quasicrystalline spin network, our conjectured point space on which the team at Quantum Gravity Research aim to model our first-principles unified quantum gravity theory which we refer to as emergence theory.



We start with the E8 crystal. The E8 crystal can be thought of as the maximum density packing of eight dimensional spheres, in the same way we would stack oranges in the supermarket is the maximum density that we can pack those spheres.

We take this very special lattice in eight dimensions, and we take a slice of it, and we project it to 4 dimensions, where we get a quasicrystal that is made entirely of regular 3D tetrahedra. But each tetrahedron lives in a different three-dimensional space next to its neighbor, in the same way that the different faces of a cube live in different two dimensional spaces from the other faces of the cube.

Around any one point in this four-dimensional quasicrystal that comes from E8, there are twenty of regular 3D tetrahedra. And, their convex hull that would encapsulate them, is a regular three -dimensional icosahedron, so it’s an icosahedron segmented by twenty regular tetrahedra. But in three dimensions, we have the Quasicrystalline Spin Network. 

What’s interesting is that in three dimensions, these regular tetrahedra cannot close up. In 4D we can take twenty of them and they all kiss faces, and they share vertices and they share edges. Around every edge in this four-dimensional space, there are five regular tetrahedra and around every point, or vertex, there are twenty regular tetrahedra.

And in 3D, this is not the case. When you keep all twenty in 3D, and each of the twenty tetrahedra have their center vertex coincident, or shared, with the nineteen others, there’s gaps that open up. And if you evenly distribute those gaps, so that you space these twenty tetrahedra around a shared point at the center, you’ll have these thirty gaps between the faces of the tetrahedra.

Each tetrahedron has four different planes, four different equilateral triangle faces. So when you take twenty tetrahedra, then you have a total of eighty of these two dimensional faces. Now, normally when you have them evenly spaced, in this twenty group with the gaps, you would have forty parallel classes of these triangular faces, and they do not touch. If you want them to touch, you rotate the faces of the tetrahedra, which rotates each entire tetrahedron, by a special golden ratio angle, which comes ultimately from the four-dimensional space of this 4D quasicrystal.

And when you do that rotation, you have to make a choice of right or left, so you have to rotate them all right or all left, or they’ll crash. 

So let’s say you rotate all of them right by this golden ratio angle. What happens is all the faces then kiss, and they splay or separate, but they kiss. And what happens is these forty plane classes collapse to only ten plane classes. 

This is important for the physics that we use because we follow Minkowski’s theorem where what is important is the number of ratios in a network of objects, and the number of parallel line classes. 

So here we collapse to the minimum possible in this configuration, and where the edges cross on this group of twenty tetrahedra, it divides the edge of a tetrahedron into the golden ratio, and if you were to rotate it the opposite direction, it would divide each edge into the golden ratio on the other side of the edge.

So the edge crossing points of this twenty group, where the edge is divided at the golden ratio, form the thirty vertices of a special Archimedian solid called the icosadodecahedron, which has icosahedral symmetry. 

The representative polytope of the Quasicrystalline Spin Network is the icosadodecahedron, in this sense.

Comments

  1. Could a simple geometrical process forms the potential for the E8 lattice and quasicrystals? The geometry for this process is in the form of spherical symmetry forming and breaking. The electron is the most spherical object in the Universe and this process is relative to the surface of the sphere. The electron charge is squared representing the radius of the sphere being squared and multiplied by 4pi to form the surface area. In this theory the surface area of the sphere represents a dynamic two dimensional boundary condition with the information encoded by the movement of positive and negative charge. The inner concaved surface representing negative charge and the outer convexed surface representing positive charge. As a process of continuous energy exchange or continuous creation the holographic principle is in play here with the information of our three spatial dimensions of our everyday life written on a two-dimensional surface, like a hologram. The light photon forms the movement of charge with the flow of EM fields in three dimensional space. The momentum of light in the form of the spontaneous absorption and emission of photon energy forms the driving force for this process with charge being an innate part of all matter. When objects touch it is charge that makes contact and when the atoms bond and break there is an exchange of photon energy with the movement of charge. In this theory we have an emergent future unfolding with each photon electron interaction with the wave particle duality of light and matter in the form of electrons forming a blank canvas that we can interact with. This potential is in geometrical form with an infinite number of line symmetries within a sphere that in this theory represents an infinite number of potential time lines. With each one being at right-angles to surface of the sphere representing EM fields always being at right-angles to each other as a process of continuous change, continuous creation. This geometrical process gives us the potential to turn the possible into the actual with the spontaneous absorption and emission of photon energy continuously forming the ever changing world of our everyday life.

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