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] .

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

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

$x,y,z\in L;x+y,y+z,x+z\in L\bar{s};x+y+z\in Ls$ (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.