On Discrete Hopf Fibrations, Grand Unification Groups, the Barnes-Wall, Leech Lattices, and Quasicrystals

Carlos Castro Perelman

Bahamas Advanced Science Institute and Conferences, Long Island, Bahamas.

DOI: 10.4236/jhepgc.2024.104096                

Abstract

A discrete Hopf fibration of S¹⁵ over S⁸ with S⁷ (unit octonions) as fibers leads to a 16D Polytope P₁₆ with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Λ₁₆. It is argued (conjectured) how a subsequent 2-1 mapping (projection) of P₁₆ onto a 8D-hyperplane might furnish the 2160 vertices of the uniform 2₄₁ polytope in 8-dimensions, and such that one can capture the chain sequence of polytopes $2_{41},2_{31},2_{21},2_{11}$ in $D=8,7,6,5$ dimensions, leading, respectively, to the sequence of Coxeter groups $E_8,E_7,E_6,SO\left(10\right)$ which are putative GUT group candidates. An embedding of the $E_8\oplus E_8$ and $E_8\oplus E_8\oplus E_8$ lattice into the Barnes-Wall Λ₁₆ and Leech Λ₂₄ lattices, respectively, is explicitly shown. From the 16D lattice $E_8\oplus E_8$ one can generate two separate families of Elser-Sloane 4D quasicrystals (QC’s) with H₄ (icosahedral) symmetry via the “cut-and-project” method from 8D to 4D in each separate E₈ lattice. Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC’s $\mathcal{Q}\times \mathcal{Q}$ spanning an 8D space. Similarly, from the 24D lattice $E_8\oplus E_8\oplus E_8$ one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC’s) $\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}$ with H₄ symmetry and spanning a 12D space.

Keywords

Division Algebras, Hopf Fibrations, Barnes-Wall Lattice, Leech Lattice, Exceptional Lie Algebras, Grand Unification, Quasicrystals

1. Introduction

Three decades ago, Dixon proposed an Algebraic design of Particle Physics based on the Division Algebras: Octonions, Quaternions, Complex, Real Numbers [1]. Over a decade ago Baez and Huerta [2] showed how the Standard Model Group $SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$ coincides with the intersection of $SU\left(5\right)$ (Georgi-Glashow model [3]) and $SU\left(4\right)\times SU\left(2\right)\times SU\left(2\right)$ (Pati-Salam [4] model) inside Spin (10). A recent extensive discussion of an algebraic roadmap of Particle Theories, based on a sequence of reflections associated with Division algebras, starting with the Spin (10) model, and exploring the full set of six familiar particle Physics models (Georgi-Glashow; Pati-Salam, …), all the way to the Standard Model (post-Higgs) can be found in [5] [6].

The book by Sirag [7] showed how the ADE Coxeter graphs unify at least 20 different types of mathematical structures which are of great utility in grand unified field theories, string theory, catastrophe theory, gravitational instantons, knots, links, braids, twistors, conformal field theories, elliptic curves, the Monster group; qubits, black holes, the holographic principle, … Since the noncommutative and nonassociative algeba of the octonions was instrumental in the discovery of the exceptional groups $G_2,F_4,E_6,E_7,E_8$ , which are an integral part of the ADE Coxeter graphs, the aim of this work is to go beyond the algebraic design of Particle Physics, and search for a geometric framework underlying (compatible with) the latter algebraic design of Particle Physics.

We shall explore the discrete Hopf fibration of S¹⁵ over S⁸ with S⁷ (unit octonions) as fibers and which leads to a 16D Polytope P₁₆ with 4320 vertices obtained from the convex hull of the 16D Barnes-Wall lattice Λ₁₆. We then conjecture how a subsequent 2-1 mapping (projection) of P₁₆ onto a 8D-hyperplane might furnish the 2160 vertices of the uniform 2₄₁ polytope in 8-dimensions, and such that one can capture the whole chain sequence of polytopes $2_{41},2_{31},2_{21},2_{11}$ in $D=8,7,6,5$ dimensions, leading, respectively, to the sequence of Coxeter groups $E_8,E_7,E_6,SO\left(10\right)$ which are putative GUT (grand unified theory) group candidates. The double cover of SO(10) is Spin (10) which is the starting point of the algebraic road map of the aforementioned six Particle Physics theories [5] [6].

Consequently, the geometrical properties of the 16D Polytope P₁₆ encode a wealth of (discrete) symmetries that are very relevant to construct grand unified theories of Particle Physics. If this is feasible one would have found a nice geometric framework of grand unified model groups, polytopes and discrete Hopf fibrations of (hyper) spheres which are deeply connected to the existence of the four normed division algebras: real, complex, quaternion and octonions.

In the remaining of this work, we discuss lattices with relevant physical application. In particular, we display explicitly the embedding of the $E_8\oplus E_8$ lattice (essential in the construction of the Heterotic string) and the $E_8\oplus E_8\oplus E_8$ lattice into the Barnes-Wall Λ₁₆ and Leech Λ₂₄ lattices, respectively. From the 16D lattice $E_8\oplus E_8$ one can generate two separate families of Elser-Sloane 4D quasicrystals (QC’s) with H₄ (icosahedral) symmetry via the “cut-and-project” method from 8D to 4D in each separate E₈ lattice.

Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC’s $\mathcal{Q}\times \mathcal{Q}$ spanning an 8D space. Similarly, from the 24D lattice $E_8\oplus E_8\oplus E_8$ one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC’s) $\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}$ with H₄ symmetry and spanning a 12D. We finalize with some concluding remarks.

2. Discrete Hopf Fibrations of S¹⁵ Lead to the Polytopes Associated with E₈, E₇, E₆, SO(10)

Given the four Hopf fibrations

$S^1\to S^1,S^3\to S^2,S^7\to S^4,S^{15}\to S^8$ (1)

Dixon [8]-[11] discussed two specific Hopf lattice fibrations resulting from the discrete Hopf fibrations of S⁷ over S⁴, and S¹⁵ over S⁸ [8]-[11]. One of them is the Hopf lattice fibration of the E₈. lattice over the Z⁵ cross-polytope (with 2 × 5 = 10 vertices) where the fibers were provided by the 24 root vectors of the D4 lattice so that one generates the 10 × 24 = 240 roots of the E₈ lattice.

Related to the last of the four Hopf fibrations, Dixon also discussed the Hopf lattice fibration of the 16-dim Barnes-Wall lattice Λ₁₆ [12] over the cross-polytope (orthoplex) Z⁹ with the E₈ lattice as fibers. Given the 240 root vectors of the E₈ lattice for fibers, and the cross-polytope (orthoplex) Z⁹ as the base, with 2 × 9 = 18 vertices, leads to a total of 18 × 240 = 4320. lattice sites which matches the kissing number of the Λ₁₆ Barnes-Wall lattice. Namely, the centers of the 4320 spheres packing the 16D space at each lattice site correspond to the 4320 vertices associated with the 4320 minimal vectors of the Λ₁₆ lattice of norm 4.

It is well known (to the experts) that the 240 real roots of the E₈ Gossett 4₂₁ polytope in 8D can be projected to two Golden-ratio scaled copies of the 120 root H₄ 600-cell quaternion in 4D, see [13] and references therein. The 600-cell in 4D has 120 vertices that correspond to the 120 roots of H₄. This very specific projection from 8D to 4D is possible due to the fact that the 8 simple roots of E₈ can be geometrical “folded” into two Golden-ratio scaled copies of the 4 simple roots of the Coxeter non-crystallographic group H₄ in 4-dim [13] (240 = 2 × 120).

A convex polytope P₁₆ in 16D can be geometrically obtained by taking the convex hull of the 4320 vertices associated to the 4320 minimal vectors of the Λ₁₆ lattice. There is a uniform 8D polytope 2₄₁ [14] with E₈ for its Coxeter group and which has 2160 vertices and 17,520 = 240 + 17,280 7-faces. 240 of those 7-faces are comprised of uniform 2₃₁ polytopes with E₇ for their Coxeter group, and the other 17,280 7-faces are 7-simplices (higher dim version of the tetrahedron).

It is known that any finite simply-laced Coxeter-Dynkin diagram can be folded into $I_2\left(h\right)$ where h is the Coxeter number (height) which corresponds geometrically to the projection to the Coxeter plane. The number of roots is equal to the rank times the height. For example, in the case of E₈ one has 240 = 8 × 30, leading to 8 polygons with 30 vertices. Because none of the Coxeter groups in 16D, $A_{16},B_{16},C_{16},D_{16}$ , can be geometrically “folded” into E₈, it is very unlikely that one will be able to project the P₁₆ polytope to two Golden-ratio scaled copies of the uniform 2₄₁ polytope in 8D, and which would have been consistent with the 2160 + 2160 splitting of the 4320 vertices of the parent 16D polytope P₁₆.

However, it is still plausible that the P₁₆ polytope admits enough reflection symmetries such that one could find a judicious 8D-hyperplane through the centroid of P₁₆, with the right orientation, and perform a 2-1 map (projection) from 16D to 8D of all the 4320 vertices of P₁₆, and obtain the sought-after 2₄₁ polytope with its 2160 vertices for the 8D projection. In other words, does the P₁₆ polytope admit at least one 8D hyperplane for a “mirror” such that its 4320 vertices are symmetrically arranged into 2160 pairs with respect to this 8D “mirror”?

In a given coordinate system, the 2160 vertices of the 8D polytope 2₄₁ can be defined as follows [14]: there are 16 (2⁴) vertices obtained from permutations of

$\left(\pm 4,0^7\right)=\left(\pm 4,0,0,\cdots ,0\right)$ (2)

where 0⁷ denotes seven zero entries. There are 1120 ($16\times C_4^8=16\times 70$ ) vertices obtained from permutations of

$\left(\pm 2,\pm 2,\pm 2,\pm 2,0,0,0,0\right)$ (3)

and 1024 (2⁷ × 8) vertices of the form

$\left(\pm 3,\pm 1,\pm 1,\cdots ,\pm 1\right)$ (4)

where the 1’s must have an odd number of minus signs. The total number of vertices is 2160 and lie on a S⁷ hyper-sphere of radius 4. In section 2 we shall explicitly display the coordinates of the 4320 minimal vectors of the Barnes-Wall lattice Λ₁₆ of length-squared equal to 4 such that the tips of all the vectors (vertices) lie on a S¹⁵ hyper-sphere of radius 2. By joining the tips of all these vectors in S¹⁵ one constructs the convex polytope P₁₆. By a simple inspection, one finds that a rescaling of P₁₆, followed by an orthogonal projection to 8D will not generate the 2-1 map yielding the 2160 vertices of 2₄₁ displayed in Equations (2)-(4).

However, this goal might be attained, firstly, by performing a rescaling of the vertices V of P₁₆: $V\to V^{\prime }=\lambda V$ , with $\lambda >1$ , followed by a SO(16) rotation of these rescaled vertices , $V^{\prime }\to V^{″}$ , and a SO(8) rotation of the vertices W of 2₄₁: $W\to W^{\prime }$ , and finally, one projects onto an 8D hyperplane the rescaled and rotated vertices of P₁₆. This projection $\pi$ can can be realized in terms of a 8 × 16 rectangular matrix M that maps the 16 entries of $V^{″}$ into the 8 entries of $W^{\prime }\in {2^{\prime }}_{41}$ . By a prime in ${2^{\prime }}_{41}$ one means that the original polytope 2₄₁ with coordinates given by Equations (2)-(4) has been rotated. The SO(16) rotations can be implemented via the use of the 120 bivectors ${\Gamma }^{mn}$ of a Clifford algebra Cl(16) in 16D. While the SO(8) rotations can be implemented via the use of the 28 bivectors ${\gamma }^{ab}$ of a Clifford algebra Cl(8) in 8D. In doing so, one has

$\left(V^{″}\right)=\lambda \left({\text{e}}^{i{\theta }_{mn}{\text{Γ}}^{mn}}V{\text{e}}^{-i{\theta }_{mn}{\text{Γ}}^{mn}}\right),V\in P_{16},m,n=1,2,\cdots ,16;\lambda >1$ (5a)

where the Clifford vectors are $V\equiv X_m{\text{Γ}}^m$ , $V^{″}\equiv {X^{″}}_n{\text{Γ}}^n$ . From Equation (5a) one can obtain the transformation of the coordinates ${X^{″}}_n={X^{″}}_n\left(X_m\right)$ . Because the 120 bivector ${\Gamma }^{mn}$ generators do not commute (in general) one cannot factorize the exponential in Equation (5a) into a product of exponentials. The SO(8) rotations involving the vertices W of 2₄₁ are given by

$\left(W^{\prime }\right)=\left({\text{e}}^{i{\theta }_{ab}{\gamma }^{ab}}W{\text{e}}^{-i{\theta }_{ab}{\gamma }^{ab}}\right),W\in 2_{41},a,b=1,2,\cdots ,8$ (5b)

with $W\equiv X_a{\gamma }^a$ , $W^{\prime }\equiv {X^{\prime }}_b{\gamma }^{\sigma }$ . There are 28 bivector generators in 8D and from (5b) one obtains the transformation of the coordinates ${X^{\prime }}_b={X^{\prime }}_b\left(X_a\right)$ .

Consequently, the combined rescaling-rotation-projections leads to equations of the form

$\pi \left(V^{″}\right)=\lambda \pi \left({\text{e}}^{i{\theta }_{mn}{\text{Γ}}^{mn}}V{\text{e}}^{-i{\theta }_{mn}{\text{Γ}}^{mn}}\right)=\left({\text{e}}^{i{\theta }_{ab}{\gamma }^{ab}}W{\text{e}}^{-i{\theta }_{ab}{\gamma }^{ab}}\right)=W^{\prime }$ (6)

such that the end result is that pair of vertices $V_1,V_2\in P_{16}$ are mapped to a single vertex W of the 2₄₁ polytope. It is in this way how the 2-1 map from P₁₆ to the 2₄₁ polytope could be constructed, if possible. At first sight, as one scans through all the 4320,2160 vertices of P₁₆,2₄₁, respectively, one encounters an over-determined system of equations whose number is much larger compared to the $28+120+128+1=277$ parameters at our disposal. However one must not forget that not all of the equations are independent due to the very large number of symmetries.

There are 120 antisymmetric parameters ${\theta }^{mn}$ associated with the SO(16) rotations implemented by the 120 bivectors ${\Gamma }_{mn}$ of the Clifford algebra Cl(16) in 16D. There are 8 × 16 = 128 parameters associated with the 8 × 16 entries of the rectangular matrix M implementing the 16D → 8D projection. The total number is 120 + 128 = 248 which agrees with the dimension of the $e_{8\left(8\right)}$ algebra comprised of 128 non-compact $Y_{\alpha }$ (spinorial) generators and 120 compact $X_{\mu \nu }$ generators. A chiral spinor $S_{+}$ in 16D has 128 entries. The (anti) commutators are $\left[X_{\mu \nu },X_{\rho \sigma }\right]={\eta }_{\mu \sigma }{X}_{\nu \rho }\pm$ permutations. $\left[X_{\mu \nu },Y_{\alpha }\right]~{\Gamma }_{\mu \nu \alpha }^{\beta }{Y}_{\beta }$ , and $\left\{Y_{\alpha },Y_{\beta }\right\}~{\text{Γ}}_{\alpha \beta }^{\mu \nu }{X}_{\mu \nu }$ , with $\mu ,\nu =1,2,\cdots ,16$ , and $\alpha ,\beta =1,2,\cdots ,128$ .

The fact that 128 spinorial generators $Y_{\alpha }$ of the $e_{8\left(8\right)}$ algebra are linked to the above construction of the 2-1 map of P₁₆ to 2₄₁ might be related to the fact that the spin group is the double cover of the rotation group. This property of spinors was crucial in the construction of E₈ from a Clifford algebra in 3D by [15]. The H₃ Coxeter group in 3D admits a natural lift to H₄ in 4D, by simply adding one node in the Coxeter diagram, and in turn, the H₄ can be geometrically “unfolded” into E₈ via the reverse mechanism explained earlier: the 8 simple roots of E₈ can be geometrically folded into two Golden-ratio scaled copies of the H₄ roots.

One may ask, why focus our attention to the 2₄₁ polytope in 8D with 2160 vertices, half as many as the 4320 vertices of P₁₆? One of the reasons why the 2₄₁ polytope is important is because the centroids of 240 of its 7-faces (comprised of uniform 2₃₁ polytopes with E₇ for their Coxeter group) are precisely positioned at the 240 vertices of the Gosset 4₂₁ polytope in 8D. As its 240 vertices represent the root vectors of the simple Lie group E₈, this Gosset polytope is sometimes referred to as the E₈ root polytope. There are a total of $2^8-1=255$ uniform polytopes with E₈ symmetry in 8D¹.

Another very important and salient feature is that there is a chain-sequence of three polytopes $2_{41},2_{31},2_{21}$ in $D=8,7,6$ dimensions whose Coxeter groups are $E_8,E_7,E_6$ , respectively. In particular, the 7-dim facets of 2₄₁ contains 2₃₁ polytopes (and 7-simplices), and in turn, the 6-dim facets of 2₃₁ contains 2₂₁ polytopes (and 6-simplices). One can proceed further by noticing that the 6-dim 2₂₁ polytope has for 5-facets: 1) 27 2₁₁ polytopes (5-orthoplexes, cross polytopes) with D₅ as their Coxeter group, and 2) 72 5-simplices with A₅ for their Coxeter group. Therefore, one may descend still further along the chain of polytopes $\cdots 2_{21}\to 2_{11}$ leading to $E_6\to E_5=D_5=SO\left(10\right)$ .

Note that there is also the sequence of three polytopes $4_{21},3_{21},2_{21}$ in $D=8,7,6$ dimensions whose Coxeter groups are $E_8,E_7,E_6$ , respectively. And there is also the sequence of $1_{42},1_{32},1_{22}$ polytopes in $D=8,7,6$ dimensions whose Coxeter groups are $E_8,E_7,E_6$ , respectively. However, our focus in this work is the chain-sequence of four polytopes $2_{41},2_{31},2_{21},2_{11}$ in $D=8,7,6,5$ dimensions, respectively, stemming from the polytope 2₄₁ in 8D and resulting from the projection of the 16D Polytope P₁₆ down to 8D, and which was obtained from the discrete Hopf fibration of S¹⁵. The unit S¹⁵ is associated with the 15 imaginary units of the Sedenions which very recently have been used to construct a satisfactory model of three generation of fermions [16].

One can see that these chain-sequences of polytopes are very relevant in constructing extensions of the Standard Model of particle physics because the groups $E_8,E_7,E_6,SO\left(10\right)$ are among the many candidates to construct grand unified theories (GUT) [17]-[23] beyond those based on the groups $SU\left(5\right)$ and $SU\left(4\right)\times SU\left(2\right)\times SU\left(2\right)$ (Pati-Salam). From $SO\left(10\right)$ there are two natural bran-ching routes to the standard model group $SO\left(10\right)\to SU\left(5\right)\to SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$ , and $SO\left(10\right)\to SU\left(4\right)\times SU\left(2\right)\times SU\left(2\right)\to SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$ .

Another physical application is that there are polytopes whose number of vertices has a one-to-one correspondence with the number of fundamental particles associated to the GUT model one hopes to construct. For instance, Boya [24] found a natural correspondence among the vertices of the self-dual 24-cell (the octacube) in 4D and the particle content of the minimal supersymmetric standard model that requires 128 bosons and 128 fermions in two different sets, the ordinary particles and their supersymmetric partners.

To sum up: starting from the 16D Polytope P₁₆ with 4320 vertices (obtained from the convex hull of the Barnes-Wall lattice Λ₁₆), we conjecture that a 2-1 projection onto a judicious 8D-hyperplane could exist, implementing the adequate reflection symmetry, in order to furnish the 2160 vertices of the uniform 2₄₁ polytope in 8-dimensions, so that one can then capture the chain sequence of polytopes $2_{41},2_{31},2_{21},2_{11}$ in $D=8,7,6,5$ dimensions, leading, respectively, to the sequence of Coxeter groups $E_8,E_7,E_6,SO\left(10\right)$ , and which are putative GUT group candidates. All these findings resulted from the discrete Hopf fibration of S¹⁵ over S⁸ [8]-[11] with S⁷ (unit octonions) as fibers. And, in doing so, we hope to answer Dixon’s question of whether or not his construction of the Barnes-Wall lattice Λ₁₆ has any physical applications [8]-[11].

3. The Barnes-Wall, Leech Lattices and the Cartesian Products of Quasicrystals

3.1. The Barnes-Wall Lattice

The Barnes-Wall lattice Λ₁₆ is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vectors. It is the sublattice of the 24-dim Leech lattice fixed by a certain automorphism of order 2, and is analogous to the Coxeter-Todd lattice [12].

There are 480 vectors obtained from permutations of

$\frac{1}{\sqrt{2}}\left(\pm 2,\pm 2,0^{14}\right)$ (7)

where 0¹⁴ denotes 14 consecutive zero entries. And 3840 vectors obtained from permutations of

$\frac{1}{\sqrt{2}}\left(\pm 1,\pm 1,\pm 1,\cdots ,\pm 1,0^8\right)$ (8)

where 0⁸ denotes 8 consecutive zero entries. All the minimal vectors have norm 4 (these vectors are not roots) whereby norm one means the length squared of the vectors. It is worth pointing out an interesting numerical coincidence with these numbers of {480, 3840} vectors. There are 480 = 2 × 240 octonionic multiplication tables and 3840 = 16 × 240 split-octonionic multiplication tables [8]-[11]. Adding the numbers of vectors yields $2\times 240\times \left(1+8\right)=4320$ . We shall see below that in the case of the 24D Leech lattice one has $3\times 240\times \left(1+16+{16}^2\right)=196560$ minimal vectors of norm 4 (these vectors are not roots).

The E₈ lattice is constructed from 112 vectors ($\frac{2^2\times 8\times 7}{2}=112$ ) obtained from permutations of

$\left(\pm 1,\pm 1,0^6\right)$ (9)

after taking an arbitrary combination of signs and an arbitrary permutation of coordinates. And 128 vectors (2⁷ = 128) obtained from permutations of

$\frac{1}{2}\left(\pm 1,\pm 1,\pm 1,\cdots ,\pm 1\right)$ (10)

with the condition that one takes an even number of minus signs². All roots have norm 2. The E₈ lattice is related to 240 integral octonions [25].

The purpose now is to embed the rank-16 lattice $E_8\oplus E_8$ directly into a rescaling of Λ₁₆ and establish a one-to-one correspondence among the 480 = 240 + 240 roots of $E_8\oplus E_8$ with 480 of the rescaled 4320 minimal vectors of the Λ₁₆ lattice. The 16-dim lattice $E_8\oplus E_8$ was instrumental in the construction of the 10D Heterotic string (there is also the 16-dim lattice $\Lambda \left(D_{16}\right)$ corresponding to SO(32)). Firstly, one performs a rescaling of the vectors in Equations (7) and (8) by a factor of $\frac{1}{\sqrt{2}}$

$\frac{1}{\sqrt{2}}\left(\pm 2,\pm 2,0^{14}\right)\to \frac{1}{2}\left(\pm 2,\pm 2,0^{14}\right)=\left(\pm 1,\pm 1,0^{14}\right)$ (11)

$\frac{1}{\sqrt{2}}\left(\pm 1,\pm 1,\pm 1,\cdots ,\pm 1,0^8\right)\to \frac{1}{2}\left(\pm 1,\pm 1,\pm 1,\cdots ,\pm 1,0^8\right)$ (12)

And then one embeds the vectors in 8D into 16D by arranging the 8 entries of the 8D-vectors in the following two ways

$\left(\pm 1,\pm 1,0^6|0^8\right),\left(0^8|0^6,\pm 1,\pm 1\right)$ (13)

And

$\frac{1}{2}\left(\pm 1,\pm 1,\pm 1,\cdots ,\pm 1|0^8\right),\frac{1}{2}\left(0^8|\pm 1,\pm 1,\pm 1,\cdots ,\pm 1\right)$ (14)

where we indicate by $0^8$ an array of 8 extra zeros separated from the slot of the initial 8 entries in order to perform the embedding. In this way the entries in Equations (11) and (12) have the same structure as the entries in Equations (13) and (14), and by direct inspection one can see that the entries (after permutations in the appropriate slot) of Equation (13) describe 112 + 112 of the vectors of $E_8\oplus E_8$ , while the entries (with an even number of minus signs) of Equation (14) describe the other 128 + 128 vectors of $E_8\oplus E_8$ , and such that 240 vectors of one copy of E₈ are orthogonal to the 240 vectors of the second copy of E₈. Therefore, in this straightforward way one has embedded the rank-16 lattice $E_8\oplus E_8$ into a rescaling of the Λ₁₆ lattice. The E₈ lattice provides the maximal packing of spheres in 8D. The Leech lattice yields the maximal packing in 24D [26] [27]. For further details of the mathematics of E₈ see [28].

3.2. The Leech Lattice

The Leech lattice is an even unimodular lattice in 24-dimensional Euclidean space. The minimal vectors of the 24D Leech lattice Λ₂₄ [12] consists of: 1) 97,152 (2⁷ × 759) vectors obtained from permutations of

$\frac{1}{\sqrt{2}}\left(\pm 1^8,0^{16}\right)$ (15)

and an even number of minus signs. 2) 1104 (2 × 24 × 23) vectors obtained from permutations of

$\frac{1}{\sqrt{2}}\left(\pm 2^2,0^{22}\right)$ (16)

and 3) 98304 (2¹² × 24) vectors obtained from permutations of

$\frac{1}{2\sqrt{2}}\left(\mp 3,\pm 1^{23}\right)$ (17)

The total number of vectors is 196,560 which is the kissing number of the Leech lattice. The vectors have norm 4³.

Because the Λ₁₆ Barnes-Wall lattice is a sublattice of the 24-dim Leech lattice L₂₄, one can embed the rank-24 lattice $E_8\oplus E_8\oplus E_8$ into a rescaling of the Leech lattice by the same factor of $\frac{1}{\sqrt{2}}$ . One now embeds the vectors in 8D into 24D by arranging the 8 entries of the 8D-vectors in the following three ways (involving the cyclic permutations of slots)

$\left(\pm 1,\pm 1,0^6\left|0^8\right|0^8\right),\left(0^8\left|0^8\right|0^6,\pm 1,\pm 1\right),\left(0^8|0^6,\pm 1,\pm 1|0^8\right)$ (18)

and

$\begin{array}{l}\frac{1}{2}\left(\pm 1,\pm 1,\pm 1,\cdots ,\pm 1\left|0^8\right|0^8\right),\frac{1}{2}\left(0^8\left|0^8\right|\pm 1,\pm 1,\pm 1,\cdots ,\pm 1\right),\\ \frac{1}{2}\left(0^8|\pm 1,\pm 1,\pm 1,\cdots ,\pm 1|0^8\right)\end{array}$ (19)

A simple inspection of Equations (18) and (19) and Equations (15) and (16) shows that one has an embedding of the rank-24 lattice $E_8\oplus E_8\oplus E_8$ into a rescaled Leech lattice L₂₄ by a factor of $\frac{1}{\sqrt{2}}$ .

The Leech lattice was instrumental in the 24-dimensional orbifold compactification of the 26-dim bosonic string down to two dimensions. The automorphism group of the string twisted vertex operator algebra is the Monster group as shown by [29] [30], and whose order is close to 10⁵⁴.

The 120 elements of the group of icosians [12] are provided by 120 unit quaternions whose coefficients are comprised of elements of the form $a+b\tau$ belonging to the Golden field $Q\left[\tau \right]$ , with $a,b$ rationals and $\tau =\frac{1}{2}\left(1+\sqrt{5}\right)$ is the Gol-den ratio, and $\sigma =\frac{1}{2}\left(1-\sqrt{5}\right)=1-\tau =-\frac{1}{\tau }$ is its Galois conjugate. An example of an icosian is the following unit quaternion

$q=\frac{1}{2}\left(\tau e_1+\sigma e_2+e_3\right)\iff \frac{1}{2}\left(0,\tau ,\sigma ,1\right)=\frac{1}{2}\left(0,\tau ,1-\tau ,1\right)\Rightarrow q\overline{q}=1$ (20)

where the icosian $x=\alpha e_o+\beta e_1+\gamma e_2+\delta e_3$ is represented by $x=\left(\alpha ,\beta ,\gamma ,\delta \right)$ , and each entry belongs to $Q\left[\tau \right]$ .

There are two norms for such vectors [12]. The quaternionic norm $QN\left(x\right)=x\overline{x}$ which is a number of the form $u+v\sqrt{5}$ , with $u,v$ rational. And the Euclidean norm $EN\left(x\right)=u+v$ . With respect to the quaternionic norm the icosians belong to a four-dim space over the Golden field $Q\left[\tau \right]$ . But with respect to the Euclidean norm they lie in an eight-dim space. The latter Euclidean norm was instrumental in the Turyn-type construction for the Leech lattice based on the three-dim lattice over the icosians $\left(x,y,z\right)$ [12].

Instead of using icosians to construct the Leech lattice, one can use octonions instead. To our knowledge, the first one to use octonions in order to represent the Leech lattice over O³ was Dixon [8]-[11]. Wilson, later on [31] provided the following representation of the Leech lattice over O³: If L is the set of octonions with coordinates on the E₈ lattice, then the Leech lattice is the set of triplets $\left(x,y,z\right)$ such that

$x,y,z\in L;x+y,y+z,x+z\in L\overline{s};x+y+z\in Ls$ (21)

with

$s=\frac{1}{2}\left(-e_1+e_2+e_3+e_4+e_5+e_6+e_7\right)$ (22)

where $e_1,e_2,\cdots ,e_7$ are the seven imaginary octonionic units squaring to −1.

The Dixon and Wilson’s representations are actually equivalent as shown by [8]-[11] [32]. The end result is that inner shell of Λ₂₄ containing the minimal vectors is broken into three subsets with orders 3 × 240; 3 × 240 × 16; 3 × 240 × 16², respectively, the sum of all three orders being $3\times 240\times \left(1+16+{16}^2\right)=196560$ which is the kissing number of the Leech lattice. The first subset with 3 × 240 = 720 vectors has a one-to-one correspondence with the 720 roots of the $E_8\oplus E_8\oplus E_8$ lattice as shown above corresponding to the canonical embedding of $E_8\oplus E_8\oplus E_8$ into a rescaling of Λ₂₄ after a cyclic permutation of the entry slots as displayed by Equations (18) and (19).

An intuitive explanation of the above 16, 16² factors is the following. Since 24 = 8 + 16, there are many ways to perform the embedding of an 8D basis frame of vectors into 24D. The 240 roots of E₈ are given by integer linear combinations of the 8 simple roots ${\beta }_1,{\beta }_2,\cdots ,{\beta }_8$ which comprise the 8D basis frame of vectors. There is room to perform translations of this 8D basis frame of vectors along the 16 transverse dimensions (to the 8 dimensions) in 24-dimensions. And also one can perform $GL\left(16,Z\right)$ transformations of this basis frame in the extra 16-dimensions. This simplistically explains the origins of the 16, 16² factors in the above counting of minimal vectors. 16 for translations and 16 × 16 for $GL\left(16,Z\right)$ transformations. The 16 discrete translations and $GL\left(16,Z\right)$ transformations can be combined into $GA\left(16,Z\right)$ , the general affine group over the integers. There is still an extra factor of 3 (in 3 × 240) that escapes us but it might be related to the triality property of SO(8).

Octonions and icosians can also be used to construct regular and uniform polytopes. The 600-cell in 4D has 120 vertices and H₄ is the Coxeter group. The coordinates of the locations of those 120 vertices in 4D can be represented in terms of the entries of 120 icosians (unit quaternions). Given the one-to-one correspondence between a vertex V and an icosian $\iota$ , one can define the group composition $V_1\ast V_2$ of two vertices in terms of the quaternionic product of the two icosians as follows

$V_1\ast V_2=V_3\iff {\iota }_1{\iota }_2={\iota }_3\iff V_3$ (23)

The upshot of establishing this vertex-icosian correspondence is that one can generate the positions of all the 120 vertices of the 600-cell from the composition law described by Equation (23) simply by starting with the quaternionic product of two icosians and generating the rest by successive iterations. An excellent video of the construction of the 120 vertices of the 600-cell based on the product of icosians can be found in [33].

The E₈ lattice [28] is also closely related to the nonassociative algebra of real octonions O. It is possible to define the concept of an integral octonion analogous to that of an integral quaternion. The integral octonions naturally form a lattice inside O [8]-[11] [25]. This lattice is just a rescaled E₈ lattice. (The minimum norm in the integral octonion lattice is 1 rather than 2). Embedded in the octonions in this manner the E₈ lattice takes on the structure of a nonassociative ring [28].

A similar construction of the 120 vertices of the 600-cell in 4D works for the 240 vertices of the E₈ Gosset 8D-polytope based on the integral octonions of norm 1. Because the octonions are a noncommutative and nonassociative normed division algebra, these 240 vertices have a multiplication operation which is no longer a group but rather a loop, in fact a Moufang loop [34]. In other words, the subset of unit-norm integral octonions is a finite Moufang loop of order 240, and which has a one-to-one correspondence with the 240 vertices of the E₈ Gosset polytope.

The octonions are nonassociative but alternative. On the other hand, the sedenions are not associative nor alternative, and are not a normed division algebra because they have 84 zero divisors⁴. As a result the norm of a product of two sedenions is not equal to the product of their norms. And because of this fact, it would be difficult to generate the coordinates of the locations of the vertices of polytopes in 16D from the products of unit sedenions.

Nevertheless, we should not overlook the importance of sedenions. More recently, the authors [16], building on previous work were able to find an algebraic realization of three fermion generations within the complex Clifford algebra $Cl\left(8,C\right)$ by incorporating an unbroken $U{\left(1\right)}_{EM}$ gauge symmetry. The complex Clifford algebra $Cl\left(8,C\right)$ is the multiplication algebra of the complexification of the Cayley-Dickson algebra of sedenions S.

We finalize this work with some remarks about lattices and Quasicrystals. From the 16D lattice $E_8\oplus E_8$ one can generate two separate families of Elser-Sloane 4D quasicrystals (QC’s) with H₄ (icosahedral) symmetry via the “cut-and-project” method from 8D to 4D in each separate E₈ lattice [35]. Therefore, one obtains in this fashion the Cartesian product of two Elser-Sloane QC’s $\mathcal{Q}\times \mathcal{Q}$ spanning an 8D space. Because E₈ is a crystallographic group, and there are no non-crystallographic groups in $D>4$ , one cannot obtain an 8D QC via the “cut-and-project” method of the 16D Barnes-Wall Λ₁₆ lattice down to an 8D model set. Instead one obtains the Cartesian product $\mathcal{Q}\times \mathcal{Q}$ of two 4D QC’s with H₄ symmetry and spanning an 8D space. Similarly, from the 24D lattice $E_8\oplus E_8\oplus E_8$ one can generate the Cartesian product of three Elser-Sloane 4D quasicrystals (QC’s): $\mathcal{Q}\times \mathcal{Q}\times \mathcal{Q}$ with H₄ symmetry and spanning a 12D space.

A family of quasicrystals of dimensions 1,2,3,4 governed by the E₈ lattice was constructed by [36]. The icosian ring associated with the unit quaternions with coefficients in the Golden field $Q\left[\tau \right]$ , and the standard “cut-and-projection” method from R^2d to R^d was instrumental in the construction. Nested sequences of quasicrystals formed systems whose symmetries were all derivable from the arithmetic of the icosians. The use of Coxeter diagrams clarified the relationship of E₈ and quasicrystal symmetries and lead to the fundamental chain $A_1\times A_1\subset A_4\subset E_6\subset E_8$ that underlies five-fold symmetry in quasicrystals. The role of the non-crystallographic Coxeter groups $H_2\subset H_3\subset H_4$ in $D=2,3,4$ dimensions, respectively, was essential.

Quasicrystalline compactifications of string theory based on a class of asymmetric orbifolds were constructed long ago by [37]. New non-supersymmetric tachyon-free string theories using a quasicrystalline orbifold in 4D have recently been constructed by [38]. The set of points of a one-dimensional cut-and-project quasicrystal or model set, while not additive, was shown to be multiplicative for appropriate choices of acceptance windows. This permits the introduction of Lie algebras over such aperiodic point sets [39]-[41]. More recently, (nonassociative) Jordan Algebras over Icosahedral cut-and-project QC have been constructed by [42].

The most immediate project is to test the existence of a 2-1 map (projection) of P₁₆ (with 4320 vertices) into a judicious 8D hyperplane leading to the 2₄₁ polytope with 2160 vertices. If this is feasible one would have found a nice geometric framework of grand unified model groups, polytopes and discrete Hopf fibrations of (hyper) spheres which are deeply connected to the existence of the four normed division algebras: real, complex, quaternion and octonions [1]. Furthermore, it is worth exploring further the arguments of [7] related to how the ADE Coxeter graphs unify Mathematics and Physics.

Acknowledgements

The author is indebted to M. Bowers for assistance.

NOTES

¹One may notice that 255 is the number of generators of the Clifford Cl(8) algebra excluding the unit generator.

²The requirement of having an even number of minus signs reduces the number from 2⁸ to 2⁷.

³As a reminder, the norm of a vector is defined as the length squared.

⁴84 = 14 × 6, where 14 is the dimension of the g₂ algebra associated with G₂ which is the automorphism group of the octonions. And the factor of 6 = 3! corresponds to the order of the symmetric group S₃.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1]	Dixon, G.M. (1994) Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. Kluwer.
[2]	Baez, J. and Huerta, J. (2010) The Algebra of Grand Unified Theories. Bulletin of the American Mathematical Society, 47, 483-552. https://doi.org/10.1090/S0273-0979-10-01294-2
[3]	Geogi, H. and Glashow, S. (1974) Unity of All Elementary Particle Forces. Physical Review Letters, 32, 438-441. https://doi.org/10.1103/PhysRevLett.32.438
[4]	Pati, J. and Salam, A. (1974) Lepton Number as the Fourth “Color”. Physical Review D, 10, 275-289. https://doi.org/10.1103/PhysRevD.10.275
[5]	Furey, N. (2023) An Algebraic Roadmap of Particle Theories, Part I: General Construction.
[6]	Furey, N. (2023) An Algebraic Roadmap of Particle Theories, Part II: Theoretical Checkpoints.
[7]	Sirag, S.P. (2016) ADEX Theory: How the ADE Coxeter Graphs Unify Mathematics and Physics, Series in Knots and Everything. Vol. 57, World Scientific. https://doi.org/10.1142/9502
[8]	Dixon, G.M. (2014) Division Algebras, Lattices, Physics, Windmill Tilting.
[9]	Dixon, G.M. (1995) Octonions: Invariant Leech Lattice Exposed.
[10]	Dixon, G.M. (1995) Octonions: Invariant Representation of the Leech Lattice.
[11]	Dixon, G.M. (1995) Octonions: E8 Lattice to Λ16.
[12]	Conway, J.H. and Sloane, N.J. (1993) Sphere Packings, Lattices and Groups. A Series of Comprehensive Studies in Mathematics Vol. 290, Springer Verlag. https://doi.org/10.1007/978-1-4757-2249-9
[13]	Moxness, J.G. (2023) The Isomorphism of H4 and E8.
[14]	Polytopes. https://en.wikipedia.org/wiki/E8_polytope https://en.wikipedia.org/wiki/4_21_polytope https://en.wikipedia.org/wiki/2_41_polytope https://en.wikipedia.org/wiki/600-cell
[15]	Dechant, P.P. (2017) The E8 Geometry from a Clifford Perspective. Advances in Applied Clifford Algebras, 27, 397-421. https://doi.org/10.1007/s00006-016-0675-9
[16]	Gourlay, L. and Gresnigt, N. (2024) Algebraic Realization of Three Fermion Generations with S3 Family and Unbroken Gauge Symmetry from Cl(8, C).
[17]	Bars, I. and Gunaydin, M. (1980) Grand Unification with the Exceptional Group E8. Physical Review Letters, 45, 859-862. https://doi.org/10.1103/PhysRevLett.45.859
[18]	Gursey, F., Ramond, P. and Sikivie, P. (1976) A Universal Gauge Theory Based on E6. Physics Letters B, 60, 177-180. https://doi.org/10.1016/0370-2693(76)90417-2
[19]	Lisi, G. (2007) An Exceptional Simple Theory of Everything.
[20]	Dray, T., Manogue, C. and Wilson, R. (2022) Octions: An E8 Description of the Standard Model. Journal of Mathematical Physics, 63, Article ID: 081703. https://doi.org/10.1063/5.0095484
[21]	Dray, T., Manogue, C.A. and Wilson, R.A. (2024) A New Division Algebra Representation of E6. Journal of Mathematical Physics, 65, Article ID: 031702. https://doi.org/10.1063/5.0216648
[22]	Dray, T., Manogue, C.A. and Wilson, R.A. (2024) A New Division Algebra Representation of E7. Journal of Mathematical Physics, 65, Article ID: 031703. https://doi.org/10.1063/5.0199098
[23]	Chester, D., Marrani, A. and Rios, M. (2023) Beyond the Standard Model with Six-Dimensional Spacetime. Particles, 6, 144-172. https://doi.org/10.3390/particles6010008
[24]	Boya, L. (2008) Supersymmetry and Polytopes. Proceedings, Workshop on Geometry and Physics: Special Metrics and Supersymmetry, Bilbao, 29-31 May 2008, 28-34. https://doi.org/10.1063/1.3089204
[25]	Conway, J.H. and Smith, D. (2003) On Quaternions and Octonions. AK Peters Ltd. https://doi.org/10.1201/9781439864180
[26]	Viazovska, M. (2017) The Sphere Packing Problem in Dimension 8. Annals of Mathematics, 185, 991-1015. https://doi.org/10.4007/annals.2017.185.3.7
[27]	Viazovska, M. (2021) Almost Impossible E8 and Leech Lattices. European Mathematical Society Magazine, 121, 4-8. https://doi.org/10.4171/mag/47
[28]	E8 Lattice. https://en.wikipedia.org/wiki/E8_lattice
[29]	Frenkel, I., Lepowsky, J. and Meurman, A. (1989) Vertex Operator Algebras and the Monster. https://books.google.com
[30]	Lepowsky, J. (2007) Some Developments in Vertex Operator Algebra Theory, Old and New. https://doi.org/10.1090/conm/442/08536
[31]	Wilson, R.A. (2009) Octonions and the Leech Lattice. Journal of Algebra, 322, 2186-2190. https://doi.org/10.1016/j.jalgebra.2009.03.021
[32]	Leech Lattice. https://en.wikipedia.org/wiki/Leech_lattice
[33]	Quaternions to the 4D 600-Cell. https://www.youtube.com/watch?v=KnJRtxJ5taM
[34]	Moufang Loop. https://en.wikipedia.org/wiki/Moufang_loop
[35]	Elser, V. and Sloane, N.J.A. (1987) A Highly Symmetric 4D Quasi-Crystal. Journal of Physics A, 20, 6161-6168. https://doi.org/10.1088/0305-4470/20/18/016
[36]	Moody, R.V. and Patera, J. (1993) Quasicrystals and Icosians. Journal of Physics A: Mathematical and General, 26, 2829-2853. https://doi.org/10.1088/0305-4470/26/12/022
[37]	Harvey, J., Moore, G. and Vafa, C. (1988) Quasicrystalline Compactifications. Nuclear Physics B, 304, 269-290. https://doi.org/10.1016/0550-3213(88)90627-X
[38]	Baykara, Z.K., Tarazi, H.C. and Vafa, C. (2024) New Non-Supersymmetric Tachyon-Free Strings. https://doi.org/10.1007/JHEP04(2024)121
[39]	Fairlie, D., Twarock, R. and Zachos, C. (2006) Lie Algebras Associated with One-Dimensional Aperiodic Point Sets. Journal of Physics A: Mathematical and General, 39, 1367-1374. https://doi.org/10.1088/0305-4470/39/6/010
[40]	Twarock, R. (2000) Aperiodic Virasoro Algebra. Journal of Mathematical Physics, 41, 5088-5106. https://doi.org/10.1063/1.533393
[41]	Patera, J., Pelantova, E. and Twarock, T. (1998) Quasicrystal Lie Algebras. Physics Letters A, 246, 209-213. https://doi.org/10.1016/S0375-9601(98)00495-2
[42]	Corradetti, D., Chester, D., Aschheim, R. and Irwin, K. (2023) Jordan Algebras over Icosahedral Cut-and-Project QC.