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Constructing numbers in quantum gravity: infinions
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IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012008
IOP Publishing
doi:10.1088/1742-6596/1194/1/012008
1
Constructing numbers in quantum gravity: infinions
Raymond Aschheim, Klee Irwin
Quantum Gravity Research, Los Angeles, California, USA
E-mail: raymond@quantumgravityresearch.org
Abstract. Based on the Cayley-Dickson process, a sequence of multidimensional structured
natural numbers (infinions) creates a path from quantum information to quantum gravity.
Octonionic structure, exceptional Jordan algebra, and e8 Lie algebra are encoded on a graph with
E9 connectivity, decorated by integral matrices. With the magic star, a toy model for a quantum
gravity is presented with its naturally emergent quasicrystalline projective compactification.
1. Introduction
This paper is dedicated to the two centuries of the eight squares theorem, published in
latin as "Adumbratio demonstrationis theorematis arithmici maximale universalis" by Carl
Ferdinand Degen, (October 7th, 1818)[1] and the one century of the Cayley-Dickson process,
"On quaternions and their generalization and the history of the eight square theorem" published
by Leonard Eugene Dickson in March, 1919 [2]. Inspired by the Nag Hammadi codex VI (from
3rd century AD) "The eighth reveals the ninth", we take the ninth-dimensional vision of the
lattice of octonion integrals, the E9 affine Lie algebra, and get a new quasicrystalline view on
quantum gravity.
1.1. Two hundred year old sedenion multiplication table
Figure 1. First sedenion
multiplication table, named
"16 serierum", published in
1818 by C. F. Degen. This
table omits the K4 coeffi-
cients, "factoribus facile sup-
plendis (et etiam tamdiu sig-
nis) omissis" the sign fac-
tor
is omitted, because
it
is easy to compute.
He
named p, pI , ...pXV the units
e0, e1, ...e15 and implemented
the same xor table with 1 =
e0, a = e1, b = e2, c = e4, d =
e8.
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The eight squares theorem, upon which octonions were discovered 28 years later, claimed a
possible generalization to any power of two, and gave the table of sedenions (see Figure 1), but
omitted the computation of the sign coefficients. There is no possible combination of coefficients
K4 realizing an 16 squares theorem, therefore the claim is wrong, but the intuition for the
Infinions is there, several decades before Hamilton, Graves, Cayley and Dickson. In the next
section we present the generalization to an infinite dimension, that we name Infinion, or simply
I, of the "16 sererium" from Degen, with a similar product rule eiej = ki j ei xor j and we give
an algorithm to compute ki j .
1.2. 9D coordinates
The roots of the affine Lie algebra E9 constitute the even unimodular lattice of E8, which is
also the weight lattice of the E8 Lie algebra. It is natural to represent the coordinates of the E9
roots in 9 dimensions. We give them explicitly hereafter, and show how this is also convenient
to represent the E6 sublattice in the same basis. Section 3 will describe a natural action on the
E8 algebra obtained from the Tits magic square, using Jordan algebra and the 9D coordinates.
Section 4 will show a quasicrystalline compactification of E9 to a 4-dimensional quasicrystal,
with an isomorphism between the 9D and the 4D coordinates.
An Simplex lattices are naturally expressed in n + 1 dimension coordinates satisfying
n+1
k=1 xk = 0,
E8 lattice is the superposition of three A8 lattices: E8 =
2
i=0A
i
8,
72 of its roots are permutations of {31,31, 07}, = 0[3], 3A08,
84 of its roots are P(23, 16), = 1[3], 3A18,
84 of its roots are P(23,16), = 2[3], 3A28,
E6 lattice is the superposition of three A8 lattices satisfying
3
k=1 xk =
6
k=4 xk =
9
k=7 xk = 0,
18 of its roots are P(31,31, 07), = 0[3], 3A08,
27 of its roots are P(23, 16), = 1[3], 3A18,
27 of its roots are P(23,16), = 2[3], 3A28,
E8 lattice is the superposition of two D8 lattices: E8 =
1
j=0D
j
8,
112 of its roots are P(22, 06), = 0[2], 2D08,
128 of its roots are EP(18), = 1[2], 2D18,
8
k=1 xk
= 0[4].
1.3. Group Theoretic View
E8 = SU(9) + 84 + 84.
(1)
The relationship between the E8 lattice and the Simplex lattice, E8 = 3A8, is illustrated
and will be extended to exceptional periodicity algebras [3, 4],
exceptionally 84 = 3C9 3-form and 84 = 6C9 6-form in SU(9) [5],
or generally 84 = 28 + 56 = 2C8

3C8 2-form and 3-form, and 84 = 56 + 28 =
6C8

5C8 6-form and 5-form in Cl(8).
2. Infinion
Infinion1 is the infinite limit of the algebra of dimension 2n built by the Cayley-Dickson process.
1 Some results presented in this section were first presented in the talk "Non associative quantum gravity" at
the conference on non-associative algebra in Coimbra in 2011. No proceedings, but the book of abstracts were
available [6].
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The Cayley-Dickson process is based on the two recursion rules,
(a, b) = (a,b),
(a, b)(c, d) = (ac db, bc+ da).
(2)
Binary xor operation arises from the iteration of equation 2. The organization of numbers on
a cubic dice illustrates the binary xor operation, where any two opposite faces sum to seven, but
also xor to seven: up xor bottom = 7 = 1 xor 6 = 2 xor 5 = 3 xor 4 = 001 xor 110 = 010 xor 101
= 011 xor 100 =111; two similar bits annihilate, two different bits create: 4 xor 7 = 100 xor 111
= 011 = 3, 4 xor 5 = 1... a xor b xor a = b, a xor a = 0, a xor 0 = a, xor is commutative and
associative. The unique canonical basis is ordering the units such that the index of a product is
the xor of the indices,
eiej = si jei xor j , si j S0 = {1, 1},
e0 = 1, s0 j = si 0 = 1, si i = 1 + 20i ,
si j = sj i(1 2(ji +
0
i +
j
0) + 4(
0
i
j
0).
(3)
Theorem 1. Giving Kn = (ki j = 2si j1)|0 i < 2n & 0 j < 2n, and, similarly Cn = (ci j),
Dn = (di j), Rn = (ri j), the Cayley-Dickson process (equation 2) is equivalent to the recursion
relation Kn+1 =
(
Kn Rn
Cn Dn
)
& K0 = 1, where
ci j =
j
0 (1
j
0)ki j ,
di j = ji + (1
j
i )(
0
i (1 0i )ki j),
ri j =
j
0 (1
j
0)di j .
(4)
Proof. Formula 5 shows how to build KN+1 from KN with the help of three binary matrices of
dimension 2N , the row i j =
0
i , the column i j =
j
0, the diagonal matrix i j =
j
i .
The following recursion relation, also illustrated graphically in Figure 2,
Kn+1 =
(
Kn Rn
Cn Dn
)
=
(
Kn
(Kn + n).n + n
Kn + n
(Kn + n).n
)
,
(5)
is equivalent to the relations defining ci j , di j , ri j in (4), themselves implying realization of the
recursive equation (2) while in the canonical basis (3).
Implementation code and examples are given in the Appendix.
Figure 2.
Left: K2 , K3 and K4,
White= 1 (+), black= 0 (-),
Right: K3 is built from K2,
top: K2,K2, 2, C2,
middle:K2, 2, (K2 + 2),2, D2,
bottom: D2, 2, R2.
Applying our theorem, we can compute the multiplication table of Infinion, restricted to any
power of two. We represent eiej = si jei xor j by a 4 by 4 bitmap, where the central 2 by 2
square is either white if si j is positive, or black otherwise. The 12 bits around encodes in binary
the value i xor j. This is clearly visible on the sedenion table given in figure 3. Figures 4 and 5
show self-similarity and a fractal behavior structured from the xor product.
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Figure 3. Sedenion table.
Figure 4. 256-nion table.1
1 It was selected at the art-science
event T.O.E [7]
Figure 5. Kilonion table.1
1 It was exhibited as a poster in
Coimbra[6]
3. Magic star
The Gosset polytope of the E8 roots is projected to 13 vertices drawing an hexagram. This
Magic star [8] in figure 6 (projected using [9]) is slightly rotated to show the central E6 and the
three Jordan pairs around.
3.1. Jordan algebra
Figure 6. Magic Star in e6 Coxeter plane at the left, rotated to g2 Coxeter plane on the right.
Projective forms H and V are given in table 1, columns e6 and g2 , such that x = H R and
y = V R for each root R.
Jordan Matrix :
Each E8 vertex holds an exceptional Jordan [10, 11] matrix J M38,
10D Minkowski Spacetime with a transversal octonion o as
J2 =
(
t x8
o = x0e0
7
k=1 x
kek
o = x0e0 +
7
k=1 x
kek
t+ x8
)
M28 = SL2(O),
(6)
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Table 1. H and V forms.
e6
g2
e6
h4
H
V
H
V
H
V
H
V
0.00
0.00
-0.06
-0.25 0.00
0.00
-0.30 0.05
0.00
0.00
0.11
0.07
0.00
0.00
-0.11 0.19
0.00
0.00
-0.06 0.12
0.00
0.00
0.08
0.05
0.00
0.00
0.24
-0.07 0.00
0.00
0.11
0.29
0.00
0.00
0.00
0.00
0.12
0.00
-0.46 0.80
-0.70
-0.41
-0.68
-0.39
-0.70
-0.47 0.58
0.33
0.70
-0.41 0.68
-0.39 0.70
-0.47 0.46
0.13
0.00
0.82
0.00
0.78
0.00
0.74
0.34
0.33
Central cross encoding scalar and Spin(9, 1) spinor =
(
+

)
,
J =
t x8 + = 7k=0 k+ek
o
+
2t
=
7
k=0
k
ek
o

t+ x8
M38 = SL3(O),
(7)
Jordan product: J1 J2 = 12(J1J2 + J2J1) [10],
Freudenthal product: J1J2 = 12(2J1 J2Tr(J1)J2Tr(J2)J1+I(Tr(J1)Tr(J2)Tr(J1
J2))) [12],
Associator: [J1, J2, J3] = (J1 J2) J3 J1 (J2 J3) [13],
Left quasi multiplication: Lx : Lx(y) = x y,
Quadratic map: Ux = 2L2x Lx2 [14],
Linearized map: Vx,y : Vx,y(z) = (Ux+z Ux Uz)(y) [15],
Trilinear map: {x, y, z} = Vx,y(z) = 2(Lx.y + [Lx, Ly])(z),
Axioms: A1 :UxVy,x = Vx,yUx,A2 :UUxy = UxUyUx,
Jordan pair: x, y|A1 & A2 & VUxy,y = Vx,Uyx.
Discrete Jordan Matrix:
Each octonion o in J (see figure 7), induced by lattice coordinates, can be restricted to integer
[16] and can be encoded by its 9D coordinates in a 3 3 matrix, by applying the rotation (11).
Figure 7. An A2 plane cuts several shells of the
E8 lattice, each shell being associated to a color
for the vertex. The Jordan matrix can become
integral, and represented as a 9 9 matrix in Z
J =
t x8
+
o
+
2t

o

t+ x8
.
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3.2. F4 action, E6 action
F4 action: F4 action is a derivation [17] on M
3
8:
An element of 52D algebra F4 is represented by two traceless H+ and H,
Its action [18] on J=H + is
F4(H+, H)(J) = J = [H+, J,H],
(8)
Invariants are I1 = Tr(J), I2 = Tr(J2), I3 = Det(J) = 13Tr(J J J).
E6 action: E6 action is a derivation [17] on M
3
8 C:
An element of 78D algebra E6 is represented by H1, H+ H Tr0(M38),
Its action [18] on J is
E6(H1, H+, H)(J) = J = [H+, J,H] + e1H1 J,
(9)
Invariants are I2 = Tr(J2), I3+I 3 = 3Det(J) = Tr(J (JJ)), I4 = Tr((JJ)(JJ)).
E6(26) action: An action on the reduced structure group is proposed in [19],
J = + + ,
S = 18Tr

dd(
+
+
).
3.3. E7 action, E8 action
E7 action: An action of E7 by a Freudenthal triple system on E8 was proposed in [20] :
56D representation of E7 as M227=(M38).
E8 action: E8 proposed action is a derivation on M
3
8 O:
The action is extrapolated from Tits-Rosenfeld-Freudenthal magic square [21] expressed by
Vinberg [22] as:
L(A, J3(B)) = Der(A)

Im(A)

Tr0(J
3(B))

Der(J3(B)).
(10)
4. Quantum gravity
Our model is based on the E8 Lattice decorated with a Jordan matrix at each vertex. We
compute the 9D coordinates of each vertex by the following rotation [23]:
R = 1
6

5 1 1 1 1 1 1 1 2
1 5 1 1 1 1 1 1 2
1 1 5 1 1 1 1 1 2
1 1 1 5 1 1 1 1 2
1 1 1 1 5 1 1 1 2
1 1 1 1 1 5 1 1 2
1 1 1 1 1 1 5 1 2
1 1 1 1 1 1 1 5 2
2 2 2 2 2 2 2 2 2
.
(11)
4.1. Induced Fano plane
As illustrated by figure 8, each vertex, like the vertex {1, 1, 1, 1, 1,2,2, 1,2} of the blue A8,
3A18 (having 8D coordinates {0, 0, 0, 0, 0,2,2, 0} in 2D08), is the center of a magic star. It
intertwines the down-pointing red triangle of 3A28 and the up-pointing green triangle of 3A
0
8.
An enclosed green hexagon gives the 6 g2 elements, and the green and red tips of the regular
tetrahedrons completing the triangles give the two other matrices to complete the set of 9 Jordan
matrices (the central one being traceless) and 6 scalars needed to define an element of the e8
algebra.
The correspondence between the Fano plane shown in figure 9, and the seven vertices in the
magic star of figure 8 is obvious, and gives the key to the use of Im(A) in equation (10).
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Figure 8. Three A8 lattices.
Figure 9. Induced Fano plane.
4.2. E8 Quasi-lattice compactification
A golden selective projection operates the E8 to H4 folding. The rings in figure 10 will project
to 4D in the Elser-Sloane quasicrystal[24] as the green points represented in figure 11. Around
the central 600-cell, there are 120 other 600-cells, whose centers are the vertices of a larger
600-cell, thanks to the self-similarity of the quasicrystal. In figure 11 we represent each of the
121 600-cells by only one 30-ring, made of 30 tetrahedrons (while they are made of 20 similar
rings), to keep our diagram readable.
Figure 10. Rotation of E8 from the E6 Coxeter plane close to the magic star, to the H4 Coxeter
plane where 4 of the 8 rings of 30 roots will match a 600-cell. Projective forms H and V are
given in the table 1, columns e6 and h4 , such that x = H R and y = V R for each root R.
4.3. Quasi-lattice action
A, the observer, chooses a tetrahedron, selects a vertex in it, selects an operation:
Operation F4 involves two E8 vertices and updates one E8 vertex,
Operation E6 involves three vertices and updates two,
Operation E7 needs choosing a top in the tetrahedron, involves seven vertices,
Operation E8 involves the full magic star.
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Figure 11. Elser-Sloane quasicrystal triacontagonally projected.
B, the observed Jordan matrices affected to lattice vertices are initially blank.
P the observation, is operated as follows:
Once the observer and its operation are chosen, selected vertices, if blank, are initialized,
The operation is performed and vertices are updated.
Figure 12 shows one 30-ring from a perspective where the facing tetrahedron, having big blue
spheres as vertices, belongs to only one A18 (the blue one), while its opposite belongs to the green
A08. The magic star can now be identified in the quasicrystal in a similar way that it is in the
crystal in figure 8.
Figure 12.
30-ring:
a
ring of 30 of the 600 tetra-
hedrons from the 600-cell,
projected from 4D to 3D.
Colors indicate to which A8
lattice each vertex belongs
before projection from 8D
to 4D.
5. Concluding remarks
We have given the canonical explicit construction of the algebra we name infinion, properly gen-
eralizing the division algebras. The table used is based on xor and is different from the octonion
table often used in physics. Infinion opens the way to new Jordan matrices M32n to M
3
which
could be useful to describe exceptional periodicity [4].
We have proposed a lattice model based on the E8 lattice, where each vertex belongs to one
of three copies of the A8 simplex lattice. The explicit transformation is given, and also the
procedure to find a magic star attached to any vertex and build an exceptional action from the
geometric neighborhood where each vertex has a Jordan matrix. Projecting it to a quasi-lattice
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model on the Elser-Sloane quasicrystal, we have realized a new type of compactification from
an 8D integral point set to a dense 4D quasicrystal.
We expect to use these new mathematical tools to build a theory of Quantum Gravity sharing
some properties (E8, compactification) with String Theory and some (discreteness, topology)
with Loop Quantum Gravity.
Appendix
Listing 1 gives the code to compute signs for any n, as a bitmap. Listing 2 shows how to use
the bitmap to get the signs. Listing 3 implements multiplication of Infinions expressed as a list.
The algorithm uses the basic operators on a square bitmap:
ColorNegate: reverse each pixel between white and black
ImageMultiply: apply a logic and, or a binary multiplication (where white is 1, black is
0). The resulting pixel will be white only if it is white in both operated images.
ImageAdd: apply a logic or, or a binary addition (where white is 1, black is 0). The resulting
pixel will be black only if it is black in both operated images.
Listing 1. Recursive algorithm computing the matrix of signs Kn = (ki j) as an image.
ImDoub [ n ,
i
,
l
,
c
]
:= ImageAssemble [{{ a ,
ImageAdd [ ImageMultiply [ ImageAdd [ ColorNegate@a ,
l ] , ColorNegate@i ] , c ]} ,
{ImageAdd [ ColorNegate@a , c ] ,
ImageMultiply [ ImageAdd [ ColorNegate@a ,
l ] ,
ColorNegate@ i ] } } ]
ImDub [ a ]
:= ImDoub [ a ,
Image@IdentityMatrix [ First@ImageDimensions@a ] ,
Image@SparseArray [{{1 ,
i } > 1} , ImageDimensions@a ] ,
Image@SparseArray [{{ i
, 1} > 1} ,
ImageDimensions@a ] ]
KImage [ n ]
:= KImage [ n ] = I f [ n == 0 , Image [{{1}} ] ,
ImDub [ KImage [ n 1 ] ] ]
Listing 2. Code example computing the matrix K4 from KImage.
( k = Round/@(2 ImageData@KImage [4 ] 1)) //MatrixForm
Listing 3. Multiplication of two infinions represented as lists.
pad [ z ]
:= PadRight [ I f [ Length@z> 0 , z , {z } ] ,
2 Ceiling [(1050+Log [Max[ 1 , Length@z ] ] ) / Log [ 2 ] ] ]
Mul1 [ a , b ]
:= Sort@Flatten [ Table [{BitXor [ i 1, j 1] , a [ [ i ] ] b [ [ j ] ] k [ [ i , j ] ] } ,
{ i , 1 , Length@a } ,{ j , 1 , Length@b } ] ]
Mul2 [ a , b ]
:= ( # [ [ 2 ] ] &/@( Total/ @Part i t ion [ Mul1 [ a , b ] , Length@a ] ) )
Mul [ a , b ]
:= Mul2 [ PadRight [ pad@a , Max[ Length@pad@a , Length@pad@b ] ] ,
PadRight [ pad@b , Max[ Length@pad@a , Length@pad@b ] ]
Listing 4. Multiplication of two quaternions, two conjugate octonions and two sedenion zero-
dividors (e4 + e10)(e7 e9).
Mul [{0 , 1} ,{0 , 0 , 1} ]
Mul[{1 ,1 ,1 ,1 ,1 ,1 ,1 ,1} ,{1 ,1 ,1 ,1 ,1 ,1 ,1 ,1} ]
Mul [{0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 1} ,{0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 1} ]
{0 ,0 ,0 ,1}
{8 ,0 , 0 , 0 , 0 , 0 , 0 , 0}
{0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0}
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