Davidson Laboratories

Stevens Institute of Technology

711 Hudson Street

Hoboken NJ 07030

dchesley@stevens.edu

0: 1 2 3 0 9 14 20 23 27 28 08 0f 11 16 22 25 3b 3c 43 44 5a 5d 69 6e 70 77 1: 1 2 4 0 9 14 21 22 26 29 0b 0c 12 15 21 26 38 3f 40 47 59 5e 6a 6d 73 74 2: 1 2 5 0 10 13 19 24 27 28 03 04 1a 1d 29 2e 30 37 48 4f 51 56 62 65 7b 7c 3: 1 2 6 0 10 13 21 22 25 30 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d 4: 1 2 7 0 11 12 19 24 26 29 0a 0d 13 14 20 27 39 3e 41 46 58 5f 6b 6c 72 75 5: 1 3 4 0 11 12 20 23 25 30 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76 6: 1 3 5 1 6 14 17 23 27 31 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d 7: 1 3 6 1 6 14 18 22 26 32 01 06 18 1f 2b 2c 32 35 4a 4d 53 54 60 67 79 7e 8: 1 3 7 1 7 13 16 24 27 31 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76 9: 1 4 5 1 7 13 18 22 25 33 08 0f 11 16 22 25 3b 3c 43 44 5a 5d 69 6e 70 77 10: 1 4 6 1 8 12 16 24 26 32 00 07 19 1e 2a 2d 33 34 4b 4c 52 55 61 66 78 7f 11: 1 4 7 1 8 12 17 23 25 33 03 04 1a 1d 29 2e 30 37 48 4f 51 56 62 65 7b 7c 12: 1 5 6 2 5 14 17 21 29 31 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76 13: 1 5 7 2 5 14 18 20 28 32 0a 0d 13 14 20 27 39 3e 41 46 58 5f 6b 6c 72 75 14: 1 6 7 2 7 11 15 24 29 31 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d 15: 2 3 4 2 7 11 18 20 25 34 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d 16: 2 3 5 2 8 10 15 24 28 32 0b 0c 12 15 21 26 38 3f 40 47 59 5e 6a 6d 73 74 17: 2 3 6 2 8 10 17 21 25 34 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76 18: 2 3 7 3 5 13 16 21 30 31 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d 19: 2 4 5 3 5 13 18 19 28 33 00 07 19 1e 2a 2d 33 34 4b 4c 52 55 61 66 78 7f 20: 2 4 6 3 6 11 15 23 30 31 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76 21: 2 4 7 3 6 11 18 19 26 34 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76 22: 2 5 6 3 8 9 15 23 28 33 01 06 18 1f 2b 2c 32 35 4a 4d 53 54 60 67 79 7e 23: 2 5 7 3 8 9 16 21 26 34 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d 24: 2 6 7 4 5 12 16 20 30 32 08 0f 11 16 22 25 3b 3c 43 44 5a 5d 69 6e 70 77 25: 3 4 5 4 5 12 17 19 29 33 0b 0c 12 15 21 26 38 3f 40 47 59 5e 6a 6d 73 74 26: 3 4 6 4 6 10 15 22 30 32 03 04 1a 1d 29 2e 30 37 48 4f 51 56 62 65 7b 7c 27: 3 4 7 4 6 10 17 19 27 34 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d 28: 3 5 6 4 7 9 15 22 29 33 0a 0d 13 14 20 27 39 3e 41 46 58 5f 6b 6c 72 75 29: 3 5 7 4 7 9 16 20 27 34 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76 30: 3 6 7 31: 4 5 6 32: 4 5 7 33: 4 6 7 34: 5 6 7

Before illustrating the use of this table

Symbolically, let S be one of the sets of 16 special signmask values (occurring on, say, line N of the table above), and n any integer 0 ≤ n ≤ 127. Then:

a useful C program shows which triad (if any) is distinguished for a given choice of table row and hexadecimal signmask value.

For the quaternionic grouping of line N, using a signmask value of n = s XOR 2

As a first example of using the table, consider the top line, with index 0: the quaternionic grouping is described as 0 9 14 20 23 27 28,

Picking 6e as the signmask (again from line 0:) and relating its bits 1101110 to the triads gives the octonions with quaternionic subalgebras

The signmask value 3a, with bits 0111010, does

a representation of the

exemplifies the appearance of zero divisors, arising because the distinguished triad has the "wrong" handedness, so that the first e

e

(e

hence (using anticommutativity of any two distinct imaginary units)

Thus a pair of zero divisors can be constructed for each of the 96 exceptional vanishing associators; indeed, a second pair of zero divisors immediately follows from:

Note that each factor must contain 1 imaginary unit from the distinguished triad in order to produce the term with the "wrong" sign. Also note that a zero divisor multiplied by any real number remains a zero divisor. Geometrically, (some) zero divisors are thereby associated with lines through the origin. Another rearrangement shows

Hence a recipe for constructing zero divisors: pick any 2 imaginaries e

as desired. Note also that for e

- 35 quaternionic triads (hence signmask has 35 bits)
- 15 groupings of 7 triads (henceforth called
*heptads*), either octonions or twisted octonions (depending on signmask) - any 2 intersecting triads generate a heptad
- each imaginary unit is in 7 triads and 7 heptads
- each triad is in 3 heptads
- each triad intersects 18 others, is disjoint with 16 others
- each triad intersects any heptad in 1 or 3 basis elements

For the XOR-based multiplication tables, the observations itemized above can easily be derived from some further XOR properties. If c does not equal a XOR b, then a, b, c, a XOR b, a XOR c, b XOR c, a XOR b XOR c, and 0 are all distinct, and form a closed set under XOR. This generalizes to any set of N integers, no two of which XOR to give a third of the set - the set of all possible XOR combinations of 2 or more set elements (including 0 = a XOR a for any a in the set) is closed and of order 2

XOR closure maps to multiplicative closure in the representations considered here, simplifying the identification of triad membership in heptad subalgebras.

counts[0] = 4699455488 counts[1] = 9688596480 counts[2] = 10254827520 counts[3] = 6041190400 counts[4] = 2582200320 counts[5] = 817152000 counts[6] = 248299520 counts[7] = 25804800 counts[8] = 2211840 counts[9] = 0

Adding these up gives 2

An alternate, equivalent version is defined in Moreno [4]:

To see the equivalence of these two, recast in terms of adjoining a new imaginary j (corresponding to (0, 1) of the doubled algebra) to the existing lower-dimensional algebra, so that

Using ja = a*j, and (ab)* = b*a*, this becomes

(ac - db*) + (d*a + b*c*)j.

giving the form used by Moreno.

In the canonical mapping, for 1 ≤ i ≤ 2

Action on the octonions is more complicated to describe, due to the number of choices of octonions or twisted octonions with which to start, and the complexity of the resulting sedenions. In summary, a type of sedenion is obtained in which there are 7 twisted and 8 true octonion heptads. There are 7 different triads that are distinguished, one from each of the twisted heptads (note that in general it's possible for a single triad to be distinguished in more than one twisted heptad, but such is not the case here). These 7 together form one of the true octonionic heptads, and each of the other 7 true octonionic heptads contains exactly one of these distinguished triads. Surprisingly, this is true even if the Cayley-Dickson construction acts on twisted octonions.

It is assumed that the first crossterm must be just ac. Changing the sign of the bd term to + is well known to give split algebras with some "imaginary" units squaring to +1, and so won't be considered here. The 9 remaining binary choices then give 512 possible forms to consider. All but 32 (N.B. original paper said "16" here, ignoring terms like d*a +b*c*, in which b and d are both conjugated) are eliminated by not satisfying

(a, b)(c, d) = (ac - b*d, da* + bc) to the quaternions gives twisted octonions of the XOR representation with signmask 3a; comparing with the octonion representation table in section 1.3 shows that since 3a = 3b XOR 1, the distinguished triad is (e

- J. Baez
*The Octonions*, arXiv:math.RA/0105155 v4 23 Apr 2002 - S. Catto and D. Chesley
*Twisted Octonions and their Symmetry Groups*, Nuclear Physics B (Proc. Suppl.) 6 (1989) 428-432 - S. Catto
*Exceptional Projective Geometries and Internal Symmetries*, arXiv:hep-th/0302079 v1 (2003) - G. Moreno
*The Zero Divisors of the Cayley-Dickson Algebras over the Real Numbers*, arXiv:q-alg/9710013 v1 (1997)