mathematics
Article
Emergence of an Aperiodic Dirichlet Space from
the Tetrahedral Units of an Icosahedral Internal Space
Amrik Sen *, Raymond Aschheim and Klee Irwin
Quantum Gravity Research, Los Angeles, CA 90290, USA; raymond@quantumgravityresearch.org (R.A.);
Klee@quantumgravityresearch.org (K.I.)
* Correspondence: amrik@quantumgravityresearch.org; Tel.: +1-310-574-6917
Academic Editor: Lokenath Debnath
Received: 23 February 2017; Accepted: 18 May 2017; Published: 26 May 2017
Abstract: We present the emergence of a root system in six dimensions from the tetrahedra of
an icosahedral core known as the 20-group (20G) within the framework of Clifford’s geometric
algebra. Consequently, we establish a connection between a three-dimensional icosahedral seed,
a six-dimensional (6D) Dirichlet quantized host and a higher dimensional lattice structure. The 20G,
owing to its icosahedral symmetry, bears the signature of a 6D lattice that manifests in the Dirichlet
integer representation. We present an interpretation whereby the three-dimensional 20G can be
regarded as the core substratum from which the higher dimensional lattices emerge. This emergent
geometry is based on an induction principle supported by the Clifford multi-vector formalism
of three-dimensional (3D) Euclidean space. This lays a geometric framework for understanding
several physics theories related to SU(5), E6, E8 Lie algebras and their composition with the algebra
associated with the even unimodular lattice in R3,1. The construction presented here is inspired by
Penrose’s three world model.
Keywords: aperiodic Dirichlet lattice; icosahedral symmetry; Clifford spinors and Lie algebras
1. Introduction
This paper investigates the emergence of higher dimensional E8 lattice geometry from
the spatially-physical three-dimensional (3D) world of icosahedral space. This mathematical structure
is encoded here within the language of Lie algebra.
We begin by presenting here the induction of a six-dimensional (6D) embedding of lattice
geometry, Λ
D3
⊕
φ3
6
:= ΛD3
⊕
φΛD3 , from the icosahedral symmetry group of a 20-tetrahedra—20-group
(20G)—cluster that is at the core of a newly-discovered 3D quasi-crystalline structure called the
quasicrystalline spin network [1]. This 6D composite lattice forms the core of an eight-dimensional
(8D) internal space. This internal space manifests physically in a four-dimensional (4D) Minkowski
spacetime, represented here by the unimodular lattice I I3,1, and together constitutes an aperiodic
Dirichlet space. The subscripts in this notation denote the dimension of the root system, and the
superscripts denote the base dimensions from which the higher root system is deduced. The presence
of the golden mean, φ, is a result of Dirichlet decomposition of 3D space and is elucidated in subsequent
sections. Moreover, the notation ΛD3 denotes the lattice associated with the Lie algebra D3.
The purpose of the current paper is to demonstrate an inductive framework for the creation
of a higher dimensional lattice from the simple components of 3D quasicrystalline forms.
This approach is distinct from the one by Dechant [2] where the birth of E8 is deduced through
the non-crystallographic intermediary root systems, H3 and H4, but is inspired by Dechant’s use of
Clifford spinors as an induction tool for creating higher dimensional forms from 3D icosahedra. In this
paper, the induction is presented through an intermediary host structure. The precise representation
Mathematics 2017, 5, 29; doi:10.3390/math5020029
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Mathematics 2017, 5, 29
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of this intermediary host is ΛD3
⊕
φΛD3
⊕
Π6,0, but will often be represented in a concise manner
as ΛD3
⊕
φΛD3 , where the representation of the physical dimensions is suppressed, and only the 6D
embedding of the internal space is referred to as the intermediary host.
Organization of the Paper
Following the introductory section, we define spinors and shifters in the language of Clifford’s
geometric algebra in Section 2. Dirichlet coordinates are also introduced in Section 2. The construction
of the 20G is explained in detail in Section 2. In Section 3, we show the emergence of the Dirichlet
quantized host in 6D as a consequence of the spawning of dimensions within the Dirichlet integer
representation. In Section 4, we present a pathway to connect the Dirichlet host with a higher
dimensional lattice through a series of transformations of Cartan matrices within the framework of Lie
theory. We also comment on the properties of these transformations that reveal the emergent geometry.
Finally, in Section 5, we provide concluding remarks and discuss the direction of future research.
2. Aperiodic Internal Space and Clifford Motors
In this section, we demonstrate that an icosahedral core comprising a group of twenty tetrahedra
is constructed from the vertices of a tetrahedron hosted by a Face Centered Cubic (FCC) lattice by using
Clifford motors—motors are the composition of rotors/spinors and translators/shifters— (cf. p. 132
in [3]) with Dirichlet integer coefficients. This is followed by dilation of the inter-tetrahedral gaps
based on a metric prescribed by an infinite Fibonacci word. The resulting structure has icosahedral
symmetry and a quasicrystalline diffraction pattern with a finite number of vertex-types [1,4–6]. The
crux of the mathematics presented in this paper relies on three key machinery.
(1)
The use of spinors as an induction mechanism [2]. The complexification process, in general,
provides a pathway for generating higher dimensional geometric algebra from lower dimensions.
This is explicitly illustrated through examples in the work of Muralidhar [7]
(2)
Expressing coefficients of the spinor as an ordered pair in Dirichlet space [8–10] allows
one to mathematically grasp the imagery of higher dimensional space in the language of
geometric algebra
(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]
Surely, a strong hypothesis is made in the constructive presentation of this paper, viz.,
about the a priori existence of an internal space [13] and a physical Minkowski spacetime [14].
This is rather a matter of philosophical ontology and is beyond the scope of this paper.
2.1. Clifford Spinors in Dirichlet Coordinates
Simply speaking, spinors are rotors. The action of a spinor on a vector results in a rotation of
the vector in Euclidean space. In this section, we formally introduce Clifford spinors as our primary
rotor mechanism for the geometric construction discussed subsequently.
2.1.1. Spinors of Cl3
The icosahedral core is generated from the simplest Platonic solid [15] by the action of spinors.
Spinors are pure rotors, i.e., the action of a spinor on a vector results in the rotation of the vector by
a desired angle around the axis defined by the spinor. As mentioned, the starting point of our inductive
framework is the 3D Euclidean space defined by R3. In the language of Clifford’s geometric algebra,
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we work within the algebra of Cl3 of R3. Therefore, we use the 3D spinor, s, defined by the elements of
the even sub-algebra of Cl+
3 over the subspace R
⊕∧2 R3 [16]. Thus:
s = s0 + s1e23 + s2e31 + s3e12 ≡ s0 + se123,
(1)
where the axis of rotation (of the spinor) is defined by the axis-vector s = s1e1 + s2e2 + s3e3 and
the bivector se123 := s1e23 + s2e31 + s3e12. We note that (se123)2 = −|s|2. The magnitude (length) of
the bivector (hence that of the spinor) is proportional to the angle by which it rotates the object-vector,
v. This is evident by re-writing the spinor in Equation (1) as follows:
s = e
1
2 se123 = cos
θ
2
+ e123ŝ sin
θ
2
,
(2)
whereŝ =
s
|s| is the unit vector that denotes the direction of the spinor-axis and the angle of
rotation, θ = |s| (Compare the role of the bivector se123 of Cl3 in R3 and iθ in C in the exponential
form demonstrating a rotation. Here, e123ŝ = −ŝe123.).
Finally, the rotation of the vector,
v = v1e1 + v2e2 + v3e3, is given by:
v s−→ svs−1 = svs̄
ss̄
= svs̄,
(3)
because s ∈ Spin(3) =⇒ ss̄ = 1. Moreover, s is a two-fold covering, which means that the action of
s and −s on v by the way of svs̄ and (−s)v(−s̄) is identical. The action of the spinor s on a vector v
is graphically illustrated in Figure 1.
Figure 1. The action of a spinor s, with the axis given by s, on the vector v rotates the latter by an
angle θ resulting in the vector v′. The length of s is proportional to the angle of rotation θ. In this
figure, the planes containing vectors (v, s) and (v′, s) are orthogonal to the plane of rotation. Moreover,
the spinor-axis denoted by the vector s is also normal to the plane of rotation.
2.1.2. Dirichlet Coordinates
Recall that our objective is to generate an icosahedral core that has a characteristic five-fold
rotational symmetry. This entails a representation of the coordinate frame by quadratic integers in
the quadratic field Q(
√
D), D = 5. D = 5 defines the quadratic integer ring with five-fold symmetry
(Note that the three-fold symmetry corresponding to the 2π3 rotations has the axes of rotation passing
through the vertices of a dodecahedron that have coordinates in the Dirichlet frame. Hence, the
Dirichlet integer decomposition used in this paper follows entirely from pentagonal symmetry and the
implications of the three-fold symmetry in the coordinate representation is accounted for automatically).
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The units of this ring corresponding to the solution of Pell’s equation (x2 − Dy2 = 1 with D = 5)
are of the form ±φn, n ∈ Z, and φ = 1+
√
D
2
is the golden ratio when D = 5. Thus, the quadratic
integer ring corresponding to the desired pentagonal rotational symmetry has algebraic integers of
the form Z[φ] = a + φb, a, b ∈ Z, which are known as Dirichlet integers. The Dirichlet coordinate
frame is defined as a set of coordinates that span the ring of Dirichlet integers [8–10] (this ring has
also been studied in the context of other objects with five fold symmetry like the Penrose tilings
(pp. 60–64 in [17])). This decomposition of the coordinates into two orthogonal golden and non-golden
parts lies at the core of the doubling of dimensions representative of an emergent geometry, as
explained in subsequent sections. We use the notations a + φb and (a, b) interchangeably, both
implying the presence of two orthogonal coordinate frames, viz. non-golden (first component) and
golden (second component) coordinate frames. Therefore, the symmetry group of the icosahedron
ensures, a priori, that the Dirichlet coordinates are sufficient to host all of the vertices of the 20G and
the quasicrystalline spin network (QSN). Consequently, the spinor axes and the spinors, defined in the
next section, are Dirichlet quantized (normalized).
2.1.3. Spinors in Dirichlet Coordinates
For the construction of the icosahedrally symmetric 20G, we use a family of two successive spinors
with θ = 2π
q ,
2π
p with axes of q−fold and p−fold rotations (p. 47 in [15]). Here, q = 5 and p = 3,
as evident by the Schläfli notation for the icosahedron, i.e., {p, q} = {3, 5}. The origin is located at
the center of a cubic section of an FCC lattice that hosts tetrahedrons in eight corners in such a way
that all eight tetrahedra share a common vertex at this origin. The starting point of our geometric
construction using spinors is one of these eight tetrahedra, as will be discussed in the next section.
The spinor-axis corresponding to the 2π5 rotations is given in Dirichlet coordinates as follows:
s(5) = (0 + φ0)e1 + (−1− φ3)e2 + (2 + φ1)e3
= (0, 0)e1 + (−1,−3)e2 + (2, 1)e3.
(4)
It is important to note that the axis defined above is Dirichlet normalized in the sense that
the coefficientss̊(5)
i =s̊
(5)
i1 + φs̊
(5)
i2 = (s̊
(5)
i1 ,s̊
(5)
i2 ), i = 1, 2, 3;s̊
(5)
i1 ,s̊
(5)
i2 ∈ Z are Dirichlet numbers (denoted
by˚here). Likewise, the spinor-axis corresponding to the first set of 2π3 rotations is given in Dirichlet
coordinates as follows:
s(3) = (−1 + φ0)e1 + (−1 + φ0)e2 + (1 + φ0)e3
= (−1, 0)e1 + (−1, 0)e2 + (1, 0)e3.
(5)
We also note that:
|s(5)| =
√
(0 + φ0)2 + (−1− φ3)2 + (2 + φ1)2
and:
|s(3)| =
√
(−1 + φ0)2 + (−1 + φ0)2 + (1 + φ0)2.
The spinors defined in Equations (1) and (2) have Dirichlet coefficients, i.e.,
s(5) = (s̊(5)
01 + φs̊
(5)
02 ) + (s̊
(5)
11 + φs̊
(5)
12 )e23
+ (s̊(5)
21 + φs̊
(5)
22 )e31 + (s̊
(5)
31 + φs̊
(5)
32 )e12
≡ (s̊(5)
01 ,s̊
(5)
02 ) + (s̊
(5)
11 ,s̊
(5)
12 )e23 + (s̊
(5)
21 ,s̊
(5)
22 )e31
+ (s̊(5)
31 ,s̊
(5)
32 )e12
≡ s(5)e123,
(6)
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To demonstrate this natural occurrence of the Dirichlet coordinates in the expression of the spinor
defined by Equation (2), we note that θ = 2π5 , cos θ/2 = φ/2 where φ =
1+
√
5
2
is the golden ratio. Next,
we examine the bivector components of the spinor, i.e., the terms e123ŝ(5) sin θ2 = − sin
θ
2ŝ
(5)e123,
e123ŝ(5) sin
θ
2
= − sin 2π
10
1
|s(5)|
{
(s̊(5)
11 + φs̊
(5)
12 )e23
+ (s̊(5)
21 + φs̊
(5)
22 )e31 + (s̊
(5)
31 + φs̊
(5)
32 )e12
}
= − sin 2π
10
1
|s(5)|
{
(s̊(5)
11 ,s̊
(5)
12 )e23 + (s̊
(5)
21 ,s̊
(5)
22 )e31
+ (s̊(5)
31 ,s̊
(5)
32 )e12
}
= (0, 0)e23 +
(
−1
2
, 0
)
e31 +
(
−1
2
,
1
2
)
e12.
(7)
Thus, the Dirichlet normalized spinor corresponding to the 2π5 rotations is defined as follows:
s(5) = 2
(
cos
θ
2
+ e123ŝ(5) sin
θ
2
)
= (0, 1) + (0, 0)e23 + (−1, 0)e31 + (−1, 1)e12,
(8)
with the spinor axis defined by Equation (4). It is noteworthy to compare Equations (6) and (8), which
define this spinor in the Dirichlet coordinates (the multiplying factor of 2 in Equation (8) ensures that
the spinor is Dirichlet normalized, i.e., the coefficients in the definition of the spinor are Dirichlet
integers (and not made of fractional numbers) of the minimum unit value permissible). Likewise,
the Dirichlet normalized spinor corresponding to the first set of 2π3 rotations is defined below:
s(3) = (1, 0) + (−1, 0)e23 + (−1, 0)e31 + (1, 0)e12,
(9)
with the spinor axis defined by Equation (5). The Dirichlet normalized spinors for the subsequent
set of 2π3 rotations are obtained by successive rotations of s
(3) by s(5). Summarizing, we use one s(5)
spinor and a family of s(3) spinors obtained by successive rotations by s(5) starting with the first s(3)
spinor defined by Equation (9).
2.2. Emergence of the 20G by the Action of Clifford Motors
In this section, we demonstrate the action of the spinors s(5) and s(3) on the vertices of successive
tetrahedra starting with a tetrahedron hosted by a cubic section of an FCC lattice, which bears
the crystallographic root system, D3. A cubic section of an FCC lattice hosts eight different tetrahedra
in each of its eight corners. These eight tetrahedra belong to two different families of either right-
or left-handed chiral groups. However, once we pick one of these tetrahedra as the origin of our
construction (see Figure 2 below), the chirality of the ensuing collection of tetrahedra remains invariant
under the action of the Clifford spinors.
2.2.1. The Primal Five-Group
In Figure 3, we show the construction of the primal five-group (5G) (the primal 5G is of
fundamental importance, as we show subsequently that this 5G is the generator of the remaining
tetrahedral clusters and, hence, can be regarded as the generator of the 20G by the action of the Clifford
motors) by the action of the spinor s(5) on the successive tetrahedra (specifically, the vertices of the
Mathematics 2017, 5, 29
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tetrahedra that are position vectors with reference to the central vertex, called the center) starting with
the originating tetrahedron shown in Figure 2. The primal 5G (along with the four other 5Gs) are the
canonical units of the 20G for two reasons.
(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.
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(5) on successively generated tetrahedra. The s(5) spinor is denoted here by the red arrow.
2.2.2. Generation of the Auxiliary 5Gs
The generation of the remaining 5Gs is described below by the combined action of a family of s(3)
spinors followed by the action of a family of Clifford shift vectors upon the primal 5G. This sequence
of rotations followed by a set of translations (shifts) together constitute the action of Clifford motors
(spinors followed by shifters).
Action of s(3) spinors on the primal 5G: the primal 5G is rotated by the s(3) spinor given by
Equation (9). This means each vertex (and hence, their position vectors with respect to the center)
Mathematics 2017, 5, 29
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of each tetrahedron of the primal 5G is rotated by s(3). The resulting tetrahedral cluster, along with
the newly generated 5G shaded in green, is shown in the top left graphic of Figure 4. Consequently,
other new sets of 5Gs are generated by the successive action of s(3) spinors, which are themselves
generated by the rotation of their predecessors by the s(5) spinor as shown in the sequence of
graphics in Figure 4. It must be noted that each generation of 5Gs shares some tetrahedral units
with its predecessor.
This process of producing auxiliary 5G clusters may be continued in this manner until production
of the 20G. Clearly, something more efficient can be done to accomplish this. In the last graphic of
Figure 4, we observe that upon the completion of the sixth generation of 5G clusters, a gap befitting
a 5G is created that is diametrically opposite to the primal 5G. This means that a reflection of the primal
5G, by a suitable plane passing through the center and parallel to the plane containing the primal
pentagonal gap, followed by an appropriate chirality correction, can be used to fill the aforementioned
gap and complete the generation of the 20G.
Figure 4. Construction of the auxiliary 5Gs by the action of the family of s(3) 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(3) spinors is designed by successive
rotation of the s(3) spinors by the s(5) spinors starting with the first s(3) spinor defined by Equation (9).
The s(3) family of spinors is shown here by the non-red arrows.
Action of Clifford shifters on the primal 5G: the strategy described in the previous paragraph
to fill the gap and complete the generation of the 20G is analogous to a simple shift operation of
the primal 5G by a family of Clifford shifters along an appropriate direction. We begin with the first
tetrahedron (the one shown in Figure 2 above) of the primal 5G and shift it along the shift vector:
n→ = (0e1 + 0e2 + 0e3)− (0e1 − 2e2 + 2e3)
= 0e1 + 2e2 − 2e3,
(10)
by a magnitude |n→| =
√
02 + 22 + (−2)2 = 2
√
2. The direction of the shift vector n→ is determined
by the edge of the first tetrahedron of the primal 5G that is closest to the s(5) spinor or, equivalently,
by the edge containing the center and the vertex that forms the pentagonal gap of the primal 5G.
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The magnitude of the shift is determined by the distance between the shaded face of the starting
tetrahedron shown in Figure 2 and the plane containing the center with a normal vector given by n→.
This choice of the shift operation guarantees the symmetry of the newly-generated final 5G, as well as its
chirality. We repeat this shift operation for each of the tetrahedra of the primal 5G. The 20G thus created
is shown in Figure 5. The 20G has icosahedral symmetry and is the fundamental unit of the aperiodic
quasicrystal described in the next section. The vertices of the 20G lie on the Dirichlet coordinate frame
and are hosted by two orthogonal D3 lattices, as described in a later section. The interested reader
is referred to the article by Fang et al. [1] for further discussions about the 20G.
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.
2.3. Emergence of the Dirichlet Quantized Quasicrystal
The 20G does not have aperiodic order and is therefore not a quasicrystal. Aperiodicity is
included by spacing the plane sets (classes) of the 20G prescribed by a family of Fibonacci chains made
of binary units 1 and φ as described in [1] (Fang et al. refer to this Fibonacci spaced quasi-lattice as the
Quasicrystalline Spin Network (QSN) with the goal of encoding the quantum geometry of space with a
suitable language. One of such attempts towards this goal of prescribing a suitable language in terms
of Clifford’s geometric algebra in Dirichlet space is an underlying objective of this current manuscript).
The resulting object is a quasicrystal (cf. Figure 6) and its vertices form a point set that also lives
in the Dirichlet coordinate frame (Since the space of Dirichlet integers is closed under addition and
multiplication, the spacing of tetrahedral vertices by 1 or φ in the appropriate direction, prescribed by
Dirichlet normalized shift vectors, map them to Dirichlet coordinate points). This can be illustrated
mathematically by recalling from Section 2.1.2 that the units of the Dirichlet quadratic ring are given
by ±φn and noting the following relation [18]:
φn = (−1)nFn−1 + φ(−1)n+1Fn,
(11)
where {Fn}n=0,1,2,... are the set of Fibonacci numbers. Equation (11) clearly shows that the Fibonacci
numbers are a special case of Dirichlet numbers, and hence, the space of vertex points generated by
the aforementioned Fibonacci spacing is embedded in the Dirichlet space. This Dirichlet quantized
quasi-lattice has a finite set of vertex types that are cataloged in detail in [1]. The detailed geometric
construction and properties of this emergent quasicrystalline structure are also described in [1].
The topology defined by the vertices of the 20G, along with additional properties such
as handedness, vertex types and the quasicrystalline diffraction pattern of the quasicrystal
(see Fang et al. [1]) invokes further understanding of the emergent geometry that is hosted by
Mathematics 2017, 5, 29
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the Dirichlet coordinate frame. We present a detailed exploration of the emergent Dirichlet geometry,
accompanied by a two-fold spawning of dimensions, in the following section.
Figure 6. The Dirichlet quantized quasicrystal referred to by Fang et al. [1] as the Quasicrystalline Spin
Network (QSN).
3. Emergence of the 6D Dirichlet Quantized Host, ΛD3
⊕
φΛD3
Recall that the vertices of the 20G (and the QSN) are Dirichlet coordinates, i.e., they are of
the form (x + φx′, y + φy′, z + φz′) where x, x′, y, y′, z, z′ ∈ Z. This is by construction true as described
in Sections 2.1.2 and 2.1.3 above and inherently linked to the icosahedral symmetry of the 20G.
Subsequently, the Dirichlet coordinates are written as ordered pairs of the form {(x, y, z), (x′, y′, z′)}
resulting in a collection of 6D coordinates (two sets of orthogonal 3D coordinates). The pairing
is comprised of a non-golden part and a golden part (the non-golden part of a + φb is a, and the
golden part is b), each embedded in one of two orthogonal 3D coordinate frames respectively. This is
illustrated in detail in Figures 7–9.
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 D3. 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 1φ 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.
The mapping between the 20G (vertices) that is embedded in the original lattice and
the inductively-generated component lattices embedded in the ΛD3
⊕
φΛD3 root system is bijective,
as the vertex pair (p, p′) uniquely associates with the corresponding vertex (p + φp′) of the 20G
in the original lattice. It is important to note that the resultant six-dimensional structure comprising
the golden and non-golden elements of the tetrahedral vertices is embedded in the ΛD3
⊕
φΛD3
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root system, but does not entirely fill the lattice space of ΛD3
⊕
φΛD3 . The embedding is simple
(the embedding is the marginal identity map p ? 1p∈D, where p ? 1 = p, p ? 0 = {} that maps all
points in the embedded space to itself; D is the embedded space with Dirichlet coordinates) and
is topologically invariant of ΛD3
⊕
φΛD3 . We refer to ΛD3
⊕
φΛD3 as the Dirichlet quantized host
(here, the word host is used to emphasize the important fact that the 6D ΛD3
⊕
φΛD3 hosts the 3D
quasicrystalline substrate as depicted pictorially in Figure 8).
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 ΛD3
⊕
φΛD3 .
Figure 9. Superposition of the golden and non-golden components of the Dirichlet coordinate frames
revealing the orthogonal relationship between them.
3.1. Vertices of 20G as Embeddings in ΛD3
⊕
φΛD3 Lattice
We present a brute force proof that the vertices of the 20G are embeddings in the ΛD3
⊕
φΛD3
lattice. The geometry of the D3 lattice is such that the sum of the coordinate points of a D3
lattice is even [19]. This requirement demands that the vertices of the 20G with coordinates
(x + φx′, y + φy′, z + φz′) must satisfy the following criteria:
(x + y + z) ≡ 2 (mod 0), and
(x′ + y′ + z′) ≡ 2 (mod 0),
(12)
where a ≡ b (mod n) is read as follows: a is congruent to b modulo n. In the Appendix, we provide
data of all of the coordinate points of the vertices of the 20G where it can be easily verified that the
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conditions listed in Equation (12) are satisfied. A more elegant proof will require a better understanding
of why the action of spinors on the tetrahedra results in new tetrahedra with vertices satisfying the
above conditions and will entail rigorous investigation using tools from the theory of algebraic rings
and combinatoric approaches. This is beyond the scope of the current manuscript.
In an identical manner, it can be verified that the coordinates of the QSN are embedded
in the ΛD3
⊕
φΛD3 lattice. It is also essential to emphasize that the vertices of the 20G and the QSN
are embeddings in the ΛD3
⊕
φΛD3 lattice. This means not all lattice points of ΛD3
⊕
φΛD3 are vertices
of the 20G and the QSN. Moreover, while the QSN is aperiodic in its point space, the golden and
the non-golden parts of its vertex coordinates are each embedded in a periodic D3 lattice.
3.2. Representation of ΛD3
⊕
φΛD3 and Its Root System
The Dirichlet quantized host, ΛD3
⊕
φΛD3 , has the following representation in Lie theory using
Dynkin graphs [20–22] as shown in Figure 10. The Dynkin diagram is clearly representative of the fact
that the 6D Dirichlet host lattice is comprised of two independent D3 component lattices.
Figure 10. Dynkin graph of ΛD3
⊕
φΛD3 .
Consequently, the Cartan matrix [20–22] can be deduced from the Dynkin graph shown
in Figure 10 as follows:
[
ΛD3
⊕
φΛD3
]
=

2 −1 −1
0
0
0
−1
2
0
0
0
0
−1
0
2
0
0
0
0
0
0
2 −1 −1
0
0
0 −1
2
0
0
0
0 −1
0
2

,
which can be easily recognized as a block diagonal form of two D3 Cartan matrices. The roots of
ΛD3
⊕
φΛD3 can be easily computed from the Cartan matrix [23] (Any set of roots is not unique.
The ones mentioned here are the ones that can be easily computed from the Cartan matrix). The simple
roots can be read off from the rows of the matrix
[
ΛD3
⊕
φΛD3
]
and are listed below:
α1 =

2
−1
−1
0
0
0

, α2 =

−1
2
0
0
0
0

, α3 =

−1
0
2
0
0
0

,
α4 =

0
0
0
2
−1
−1

, α5 =

0
0
0
−1
2
0

, α6 =

0
0
0
−1
0
2

.
Mathematics 2017, 5, 29
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The remaining roots are α1 + α2, α1 + α3, α1 + α2 + α3, α4 + α5, α4 + α6, α4 + α5 + α6 and
the negative copies of each of the roots listed above. In total, there are 24 roots of ΛD3
⊕
φΛD3 .
In subsequent sections, we present the action of a sequence of bounded linear operators
(transformations) on the Cartan matrix of ΛD3
⊕
φΛD3 and the concomitant changes in the topology
of the Dynkin graphs and the corresponding root system. We show that the response to this action
(sequel) spans the root system of higher dimensional lattice spaces, viz., SU(5), E6 and E8.
4. Emergence of SU(5), E6 and E8 from ΛD3
⊕
φΛD3
Several higher dimensional theories are of importance in theoretical physics. In this context,
we consider specifically SU(5), E(6) and E(8) [24–34]. To understand the emergent role of the aperiodic
Dirichlet quantized host, introduced in earlier sections, in connection with these higher dimensional
spaces, we restrict ourselves to the framework of Cartan sub-algebra [11].
We begin by presenting a flow chart (see Figure 11 below) depicting the emergence of a 6D
Dirichlet quantized host. The isolated green nodes associated with the Dynkin graphs for each
of the intermediary lattices, starting with that of the Dirichlet host and except for the graph of
E8, are hidden dimensions and can be regarded as the reflection of the internal Dirichlet space.
These hidden dimensions manifest as the physical Minkowski space that coincides with the connection
of the reflection space to the internal space, as shown schematically in Figure 11. The action of
the Dirichlet spinors and the transformation matrices Ti shown in Figure 11 is invariant in time.
Figure 11. A graphical representation of the emergence of lattices that constitute the unified world
space beginning with the Dirichlet quantized host lattice, ΛD3
⊕
φ3
6
6⊕
m=1
I Im,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
Ti,T̆2, i = {1, 2, 3} act on the corresponding Cartan matrices and regulate the emergence of higher
dimensional lattices.
Mathematics 2017, 5, 29
13 of 18
4.1. Transformations that Encode Emergence of Spacetime Fabric
The emergence of E8 is brought about by the action of a sequence of transformations on
the respective Cartan matrices as follows:
T̆2 ◦ T1 ◦
[
ΛD3
⊕
φΛD3
6⊕
m=1
I Im,0
]
= E4,
T2 ◦ T1 ◦
[
ΛD3
⊕
φΛD3
6⊕
m=1
I Im,0
]
= E6
6⊕
m=1
I Im,0,
T3 ◦
[
E6
6⊕
m=1
I Im,0
]
= E8
⊕
Π3,1,
(13)
where ◦ refers to a composition operation, and the Cartan matrices are given below. Furthermore, note
that A1 ≡ I I1,0 in terms of their Cartan representation. The associated Cartan matrix is given below:
[
ΛD3
⊕
φΛD3
6⊕
m=1
I Im,0
]
=

2 −1 −1
0
0
0
−1
2
0
0
0
0
−1
0
2
0
0
0
0
0
0
2 −1 0
0
0
0 −1
2
0
0
0
0
0
0
2

⊕

2 0 0 0 0 0
0 2 0 0 0 0
0 0 2 0 0 0
0 0 0 2 0 0
0 0 0 0 2 0
0 0 0 0 0 2

,
(14)
where:

a
b
0
0
c
d
0
0
0 0
p
q
0 0 m n
 ≡
(
a
b
c d
)⊕( p
q
m n
)
[
E4
]
=
[
A4
]
=

2 −1
0
0
−1
2 −1
0
0 −1
2 −1
0
0 −1
2
 ,
(15)
[
E6
6⊕
m=1
I Im,0
]
=

2 −1
0
0
0
0
−1
2 −1
0
0
0
0 −1
2 −1
0 −1
0
0 −1
2 −1
0
0
0
0 −1
2
0
0
0 −1
0
0
2

⊕

2 0 0 0 0 0
0 2 0 0 0 0
0 0 2 0 0 0
0 0 0 2 0 0
0 0 0 0 2 0
0 0 0 0 0 2

,
(16)
[
E8
⊕
Π3,1
]
=

2 −1
0
0
0
0
0
0
−1
2 −1
0
0
0
0
0
0 −1
2 −1
0
0
0
0
0
0 −1
2 −1
0
0
0
0
0
0 −1
2 −1
0 −1
0
0
0
0 −1
2 −1
0
0
0
0
0
0 −1
2
0
0
0
0
0 −1
0
0
2

⊕

2 0 0 0
0 2 0 0
0 0 2 0
0 0 0 2
 ,
(17)
Mathematics 2017, 5, 29
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The transformation matrices are listed as follows:
T1 =

1 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0
0 0 0
3
2
1
4
3
4
0 0 0 0
0 0 0 0 1 0 0 0 0 0
0 0 0 1
1
2
3
2
0 0 0 0
0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 1

⊕(1 0
0 1
)
,
(18)
T̆2 =

3
2
1
4
3
4
0
0
0 0 0
− 12
3
4 −
3
4
0
0
0 0 0
1
2 −
1
4
5
4 −
2
3 −
1
3
0 0 0
− 12 −
1
4 −
3
4
4
3
2
3
0 0 0
0
0
0
0
0
0 0 0
0
0
0
0
0
0 0 0
0
0
0
0
0
0 0 0
0
0
0
0
0
0 0 0

⊕

0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
 ,
(19)
T2 =

3
2
1
4
3
4
0
0
0
0 0
− 12
3
4 −
3
4
0
0
0
0 0
1
2 −
1
4
5
4 −
2
3 −
1
3 −
1
2
0 0
− 12 −
1
4 −
3
4
1
0
0
0 0
0
0
0
0
1
0
0 0
− 12 −
1
4 −
3
4
0
0
1
0 0
0
0
0
0
0
0
1 0
0
0
0
0
0
0
0 1

⊕

0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
 ,
(20)
and:
T3 =

1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
2
4
2
1
2
0
0
0
0
0
1
0
0
0
0
−1 −2 −3 −2
0 −2
0 − 12
4
3
8
3
4
7
3
2
3
3 − 12
0
−1 −2 −3 −2 −1 −2
1
0
− 23 −
4
3 −2 −
5
3 −
4
3 −1
0
1

⊕

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
 .
(21)
The transformations given by the notation Ti, i = 1, 2, 3 andT̆2 encode all of the information about
the evolution of the root-vectors and, hence, that of the emerging geometrical fabric of the spacetime
coordinates. In other words, they regulate the setting of the Dirichlet quantized host ΛD3
⊕
φΛD3 ,
thereby determining which geometry manifests itself.
4.2. Spectral Norm of Transformation Matrices
The spectral norm of an operator L is defined as:
||L||σ =
√
λmax(L∗L),
(22)
Mathematics 2017, 5, 29
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where L∗ is the conjugate transpose of L and λmax(L) denotes the maximum eigenvalue of L. The
spectral norms of the transformation matrices are given as follows:
||T1||σ = 1.6182,
||T̆2||σ = 2.6257,
||T2||σ = 2.7215,
||T3||σ = 11.1872.
(23)
The norm of T3 is an order of magnitude larger than the norms of all other transformation matrices.
4.2.1. Physical Interpretation of the Spectral Norms
Recall that the spectral norm of a transformation matrix reflects the largest singular value of
the matrix that entails the maximum extent of allowable deformation (stretching) of vectors in the
corresponding vector space [35]. Kleinert [36–39] used a linear perturbation of the coordinate frame by
a displacement field generating strain and consequently revealed that space-time with torsion and
curvature can be generated from a flat Euclidean space-time using singular coordinate transformations.
He argued in favor of comparing the above to a crystallographic medium filled with dislocations and
disclinations. Ruggiero and Tartaglia [40] showed that Kleinert’s singular coordinate transformations
are the spacetime equivalent of the plastic deformations, which lead to incompatible defect states
corresponding to the generation of mass, mass current and spin.
In our analysis presented above, T3 is a representation in the Cartan sub-algebra and coincides
with the bridging of the internal host lattice with that of the 4D Minkowski spacetime denoted by I I3,1.
The aforementioned spike in the spectral norm of T3 is a signature of the linear deformations (these
singular value deformations are a combination of rotation and stretching/scaling processes) in lattice
geometry proposed by Kleinert to account for gravity. The spike in the spectral norm coincides with
the link between the internal world and the Minkowski physical world denoted by the unimodular
lattice I I3,1.
5. Conclusions
In this paper, we have described a rigorous pathway for the emergence of the 12-dimensional
(12D) lattice from a 3D aperiodic substrate through an intermediary crystallographic host lattice of
the internal space in 6D. The mathematical framework is set in the language of Clifford’s geometric
algebra and representations from Lie theory. We have demonstrated that the spawning of the new
dimensions is inherently related to the symmetries of our base object, viz. the 20G that has icosahedral
symmetry. This has been possible by utilizing the concept of Dirichlet integers from the theory of
algebraic rings. Consequently, we have presented a sequence of transformations to illustrate the
emergence of spacetime geometry through the use of Dynkin graphs and Cartan algebra. The norms
of these transformation matrices have a special significance in relation to the underlying geometry
they induce.
Future work will include investigating the links between the quasicrystalline base and the higher
dimensional geometry from a dynamical point of view. This will enable the possibility of mapping our
understanding of dynamics in the measurable three dimensions with the dynamics encompassed by
the E8 Lie group. In a subsequent paper, we propose to perform detailed investigation of the Dirichlet
host from the perspective of the theory of algebraic rings. This will shed further light on the importance
of this intermediary host contributing toward a better understanding of the interconnections between
group symmetries of higher dimensions.
Acknowledgments: The first author would like to acknowledge several useful suggestions and discussions with
Carlos Castro Perelman.
Mathematics 2017, 5, 29
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Author Contributions: Amrik Sen formulated the research problem, designed the mathematical framework of the
scientific inquiry and wrote the paper, Raymond Aschheim performed computer calculations and generated most
of the figures, and Klee Irwin envisioned and initiated the research program, revised and edited the manuscript
and arranged all necessary financial and computational resources for the scientific work.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
Table A1. Vertex Coordinates of the 20G, (x + φx′, y + φy′, z + φz′).
x
y
z
x + y + z
x′
y′
z′
x′+y′+z′
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
Mathematics 2017, 5, 29
17 of 18
Table A1. Cont.
x
y
z
x + y + z
x′
y′
z′
x′+y′+z′
−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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