# Emergence of an Aperiodic Dirichlet Space from the Tetrahedral Units of an Icosahedral Internal Space

^{*}

## Abstract

**:**

## 1. Introduction

#### Organization of the Paper

## 2. Aperiodic Internal Space and Clifford Motors

- (1)
- (2)
- (3)
- The bridge between the Dirichlet quantized lattice and the physical Minkowski space is illustrated through the language of Cartan sub-algebra and Dynkin–Coxeter graphs [11]. The comprehension of quasicrystalline forms through non-crystallographic Dynkin–Coxeter graphs and Lie algebras has been discussed in detail in the work of Koca et al. [12]

#### 2.1. Clifford Spinors in Dirichlet Coordinates

#### 2.1.1. Spinors of $C{l}_{3}$

#### 2.1.2. Dirichlet Coordinates

#### 2.1.3. Spinors in Dirichlet Coordinates

#### 2.2. Emergence of the 20G by the Action of Clifford Motors

#### 2.2.1. The Primal Five-Group

- (1)
- The primal 5G is the simplest unit of the 20G icosahedral cluster that fixes the chirality of the 20G as has been mentioned earlier.
- (2)
- The pentagonal gap associated with the primal 5G (as well as the other 5Gs), as seen in the rightmost graphic in Figure 3, is a direct consequence of the fact that the 20G built from it has the minimal number of plane classes (see discussion in [1]). This makes the 5Gs canonical building blocks of the 20G.

#### 2.2.2. Generation of the Auxiliary 5Gs

#### 2.3. Emergence of the Dirichlet Quantized Quasicrystal

## 3. Emergence of the 6D Dirichlet Quantized Host, ${\mathsf{\Lambda}}_{{D}_{3}}\u2a01\varphi {\mathsf{\Lambda}}_{{D}_{3}}$

#### 3.1. Vertices of 20G as Embeddings in ${\mathsf{\Lambda}}_{{D}_{3}}\u2a01\varphi {\mathsf{\Lambda}}_{{D}_{3}}$ Lattice

#### 3.2. Representation of ${\mathsf{\Lambda}}_{{D}_{3}}\u2a01\varphi {\mathsf{\Lambda}}_{{D}_{3}}$ and Its Root System

## 4. Emergence of $SU\left(5\right),{E}_{6}$ and ${E}_{8}$ from ${\mathsf{\Lambda}}_{{D}_{3}}\u2a01\varphi {\mathsf{\Lambda}}_{{D}_{3}}$

#### 4.1. Transformations that Encode Emergence of Spacetime Fabric

#### 4.2. Spectral Norm of Transformation Matrices

#### 4.2.1. Physical Interpretation of the Spectral Norms

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Table A1.**Vertex Coordinates of the 20G, $(x+\varphi {x}^{\prime},y+\varphi {y}^{\prime},z+\varphi {z}^{\prime})$.

x | y | z | $x+y+z$ | ${x}^{\prime}$ | ${y}^{\prime}$ | ${z}^{\prime}$ | ${x}^{\prime}$$+{y}^{\prime}$$+{z}^{\prime}$ |
---|---|---|---|---|---|---|---|

0 | −2 | 2 | 0 | 0 | 0 | 0 | 0 |

−2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |

−2 | −2 | 0 | −4 | 0 | 0 | 0 | 0 |

−2 | −1 | 1 | −2 | 1 | −1 | 0 | 0 |

−1 | 1 | 0 | 0 | 1 | −2 | −1 | −2 |

−1 | 0 | −1 | −2 | 2 | −1 | 1 | 2 |

2 | −1 | −1 | 0 | −1 | −1 | 0 | −2 |

1 | 1 | 0 | 2 | −1 | −2 | 1 | −2 |

1 | 0 | 1 | 2 | −2 | −1 | −1 | −4 |

2 | −1 | 1 | 2 | −1 | −1 | 0 | −2 |

1 | −1 | 2 | 2 | 1 | 0 | −1 | 0 |

1 | −2 | 1 | 0 | 0 | 1 | 1 | 2 |

−1 | 1 | 0 | 0 | 1 | −2 | 1 | 0 |

1 | 2 | 1 | 4 | 0 | −1 | 1 | 0 |

0 | 1 | −1 | 0 | −1 | −1 | 2 | 0 |

0 | 2 | −2 | 0 | 0 | 0 | 0 | 0 |

−2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |

−2 | 0 | −2 | −4 | 0 | 0 | 0 | 0 |

2 | 1 | −1 | 2 | −1 | 1 | 0 | 0 |

1 | 2 | −1 | 2 | 0 | −1 | −1 | −2 |

1 | 1 | −2 | 0 | 1 | 0 | 1 | 2 |

−2 | 1 | 1 | 0 | 1 | 1 | 0 | 2 |

−1 | 2 | 1 | 2 | 0 | −1 | 1 | 0 |

−1 | 1 | 2 | 2 | −1 | 0 | −1 | −2 |

−2 | 1 | −1 | −2 | 1 | 1 | 0 | 2 |

−1 | 0 | 1 | 0 | 2 | 1 | −1 | 2 |

−1 | −1 | 0 | −2 | 1 | 2 | 1 | 4 |

1 | −1 | 0 | 0 | −1 | 2 | −1 | 0 |

2 | 1 | 1 | 4 | −1 | 1 | 0 | 0 |

1 | 0 | −1 | 0 | −2 | 1 | 1 | 0 |

2 | −2 | 0 | 0 | 0 | 0 | 0 | 0 |

2 | 0 | −2 | 0 | 0 | 0 | 0 | 0 |

0 | −2 | −2 | −4 | 0 | 0 | 0 | 0 |

−1 | −2 | 1 | −2 | 0 | 1 | 1 | 2 |

1 | −1 | 0 | 0 | −1 | 2 | 1 | 2 |

0 | −1 | −1 | −2 | 1 | 1 | 2 | 4 |

1 | −2 | −1 | −2 | 0 | 1 | −1 | 0 |

−1 | −1 | 0 | −2 | 1 | 2 | −1 | 2 |

0 | −1 | 1 | 0 | −1 | 1 | −2 | −2 |

1 | 0 | −1 | 0 | −2 | −1 | 1 | −2 |

−1 | 1 | −2 | −2 | −1 | 0 | 1 | 0 |

0 | −1 | −1 | −2 | −1 | 1 | 2 | 2 |

−2 | −1 | −1 | −4 | 1 | −1 | 0 | 0 |

−1 | −2 | −1 | −4 | 0 | 1 | −1 | 0 |

−1 | −1 | −2 | −4 | −1 | 0 | 1 | 0 |

2 | 0 | 2 | 4 | 0 | 0 | 0 | 0 |

0 | 2 | 2 | 4 | 0 | 0 | 0 | 0 |

2 | 2 | 0 | 4 | 0 | 0 | 0 | 0 |

−1 | −1 | 2 | 0 | −1 | 0 | −1 | −2 |

0 | 1 | 1 | 2 | −1 | −1 | −2 | −4 |

1 | 0 | 1 | 2 | −2 | 1 | −1 | −2 |

1 | −1 | −2 | −2 | 1 | 0 | 1 | 2 |

0 | 1 | −1 | 0 | 1 | −1 | 2 | 2 |

−1 | 0 | −1 | −2 | 2 | 1 | 1 | 4 |

1 | 1 | 0 | 2 | −1 | −2 | −1 | −4 |

0 | 1 | 1 | 2 | 1 | −1 | −2 | −2 |

−1 | 2 | −1 | 0 | 0 | −1 | −1 | −2 |

−1 | 0 | 1 | 0 | 2 | −1 | −1 | 0 |

1 | 1 | 2 | 4 | 1 | 0 | −1 | 0 |

0 | −1 | 1 | 0 | 1 | 1 | −2 | 0 |

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**Figure 1.**The action of a spinor s, with the axis given by $\mathbf{s}$, on the vector $\mathbf{v}$ rotates the latter by an angle $\theta $ resulting in the vector ${\mathbf{v}}^{\prime}$. The length of $\mathbf{s}$ is proportional to the angle of rotation $\theta $. In this figure, the planes containing vectors ($\mathbf{v}$, $\mathbf{s}$) and (${\mathbf{v}}^{\prime}$, $\mathbf{s}$) are orthogonal to the plane of rotation. Moreover, the spinor-axis denoted by the vector $\mathbf{s}$ is also normal to the plane of rotation.

**Figure 2.**The tetrahedron hosted by a cubic section of a Face Centered Cubic (FCC) lattice is the simplest platonic solid that forms the base of our construction of the icosahedral 20-group (20G). The spherical nodes are some of the lattice points of an FCC lattice. The node at the center of the cubic section forms the central vertex (center) of the 20G. All tetrahedra that constitute the 20G share one of their vertices at this center.

**Figure 3.**Construction of the first five-group (5G), known as the primal 5G, by the action of the spinor ${s}^{\left(5\right)}$ on successively generated tetrahedra. The ${s}^{\left(5\right)}$ spinor is denoted here by the red arrow.

**Figure 4.**Construction of the auxiliary 5Gs by the action of the family of ${s}^{\left(3\right)}$ spinors on the primal 5G is shown here beginning with top left and moving row wise. The last graphic at the bottom right shows a gap diametrically opposite to the primal 5G. The family of ${s}^{\left(3\right)}$ spinors is designed by successive rotation of the ${s}^{\left(3\right)}$ spinors by the ${s}^{\left(5\right)}$ spinors starting with the first ${s}^{\left(3\right)}$ spinor defined by Equation (9). The ${s}^{\left(3\right)}$ family of spinors is shown here by the non-red arrows.

**Figure 5.**The graphic on the left shows the closure of the gap in the last graphic of Figure 4 by action of the Clifford shifter on the primal 5G. The middle and right graphics show the correspondence of the vertices of the pentagonal gap of the primal 5G and the newly created 5G. These color-matched vertex pairs form the direction axis for the shift operations on the corresponding tetrahedra of the primal 5G.

**Figure 6.**The Dirichlet quantized quasicrystal referred to by Fang et al. [1] as theQuasicrystalline Spin Network (QSN).

**Figure 7.**The graphic depicts the doubling of dimensions by the action of the Dirichlet quantized spinor, where successively generated tetrahedra have vertices with Dirichlet coordinates with non-golden and golden parts, each being hosted by the three-dimensional crystallographic root system ${D}_{3}$. In this figure, the newly-generated tetrahedron has a golden (colored brown) and non-golden component (colored cyan). Each of their vertices are at a distance $\frac{1}{\varphi}$ and one units from the corresponding vertices of tetrahedra hosted by the original FCC lattice (middle), respectively. For the sake of visual clarity, the lattice hosting these golden and non-golden components are translated in space and shown on the sides of the original lattice.

**Figure 8.**The six-dimensional (6D) Dirichlet quantized lattice hosting the 20G. In other words, the geometric fabric of the icosahedral object (20G) is unraveled and presented as a 6D host ${\mathsf{\Lambda}}_{{D}_{3}}\u2a01\varphi {\mathsf{\Lambda}}_{{D}_{3}}$.

**Figure 9.**Superposition of the golden and non-golden components of the Dirichlet coordinate frames revealing the orthogonal relationship between them.

**Figure 10.**Dynkin graph of ${\mathsf{\Lambda}}_{{D}_{3}}\u2a01\varphi {\mathsf{\Lambda}}_{{D}_{3}}$.

**Figure 11.**A graphical representation of the emergence of lattices that constitute the unified world space beginning with the Dirichlet quantized host lattice, ${\mathsf{\Lambda}}_{{D}_{6}^{3\u2a01\varphi 3}}{\displaystyle \underset{m=1}{\overset{6}{\u2a01}}}I{I}^{m,0}$. The solid dotted (dashed) lines denote the addition of edges between (removal of) nodes (edges and/or nodes). Note: In Dynkin graphs, nodes represent root vectors. Each hollow dotted line denotes the merging of two nodes. The solid green circles represent the six physical dimensions that are hidden in the internal space; subsequently, two of these hidden dimensions bridge the internal space and the physical Minkowski spacetime. The hollow circles represent nodes in the Euclidean (non-curved) space. The transformations ${T}_{i},{\stackrel{\u02d8}{T}}_{2},\phantom{\rule{3.33333pt}{0ex}}i=\{1,2,3\}$ act on the corresponding Cartan matrices and regulate the emergence of higher dimensional lattices.

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**MDPI and ACS Style**

Sen, A.; Aschheim, R.; Irwin, K.
Emergence of an Aperiodic Dirichlet Space from the Tetrahedral Units of an Icosahedral Internal Space. *Mathematics* **2017**, *5*, 29.
https://doi.org/10.3390/math5020029

**AMA Style**

Sen A, Aschheim R, Irwin K.
Emergence of an Aperiodic Dirichlet Space from the Tetrahedral Units of an Icosahedral Internal Space. *Mathematics*. 2017; 5(2):29.
https://doi.org/10.3390/math5020029

**Chicago/Turabian Style**

Sen, Amrik, Raymond Aschheim, and Klee Irwin.
2017. "Emergence of an Aperiodic Dirichlet Space from the Tetrahedral Units of an Icosahedral Internal Space" *Mathematics* 5, no. 2: 29.
https://doi.org/10.3390/math5020029