Maths - Cayley Table - 16D

On this page we will generate various 16 dimensional Cayley tables using combinations of the following methods:

• From complex and double numbers combined using the (modified) Cayley-Dickson method.
• From Clifford algebras based on 4D vector algebra (each of which may square to +ve or -ve).
• From even subalgebras of 32D clifford algebras (again using various combinations of +ve and -ve squaring basis).

This continues the process started on this page and continued on this page, in the 4D case we saw that the Clifford and Cayley-Dickson were identical, when we doubled up to 8D we saw that the algebras started to diverge with the Clifford algebras being associative and the Cayley-Dickson having a conjugate (and therefore an inverse). On this page we double up again and the properties of the Cayley-Dickson algebras further degrade, no longer having an inverse.

The aim is to look for correspondences between the various types of algebras. I need to do more analysis to find the overlaps between these classes of algebras (which algebras are isomorphic).

As we have seen on this page the type of each entry will be common for all these methods so we only need to compare the sign. So to make the comparison clearer on the page we will only show the sign and colour code the entries so that the pattern will show:

16D Cayley-Dickson Algebras

DDDD	DDDC
+ + + + + + + + + + + + + + + + + + + + + + - - + + - - - - + + + - + - + + + + + + + + - - - - + - + - + + - - + + - - - - + + + - - - + - - - + + + + + + + + + - - - + - + + + + + + - - - - + + - + + - - - + - + - - + - + + + - + + - + + + - + - - + - + + - - - - - - - + - - - - - - - + - - - - - + + + - + + + + - - + + - + - - - - + - - - + + + + + + - + - - + + + - + + + + - - + + + + - + + + + - - - - - - - + + + + - + - - + - - - + + + + + - + - - + + + + + - + + - + - + - + - - + - - + + - + + - + -	+ + + + + + + + + + + + + + + + + - + - + - - + + - - + - + + - + - + - + + + + + + + + - - - - + + + + + - - + + - - + - + + - + - - - + - - - + + + + + + + + + + - + + + + - + - + - - + - + + + - + + - - - + - + - - + - + + - - - + + + - + + + + - - - - + - - - - - - - + - - - - - - - + + - + - + + - + + + - + - - + + + - + - - - - + - - - + + + + + - - - - + + - + + + - + - - + + + + + - + + + + - - - - - - - + - + - - - - + + + - + + - + - + - + - - + + + + + - + + - + - + + + + - - - + + - - - + + + +
DDCC	DCCC
+ + + + + + + + + + + + + + + + + - + - + - - + + - - + - + + - + - - + + + - - + + - - - - + + + + - - + - + - + - + - - + - + + - - - + - - - + + + + + + + + + + - + + + + - + - + - - + - + + + + - + - + + + - - + - + + - + - + + + + - + + + - - - - + + + - - - - - - - + - - - - - - - + + - + - + + - + + + - + - - + + + + - - - + + + - + + + + - - + - + + - + - + + + - + + - + - + + + + - + + + + - - - - - - - + - + - - - - + + + - + + - + - + - - + - + - - + + + - + - - + + + - - - - + - + - + + + + - -	+ + + + + + + + + + + + + + + + + - + - + - - + + - - + - + + - + - - + + + - - + + - - - - + + + + - - + - + - + - + - - + - + + - - - - + + + + + + + - - - - + + - + - - - + + - + - + - + - + + + - - + - - + - - + + - - + + - + + - - + - + + - - + + - - + - - - - - - - + - - - - - - - + + - + - + + - + + + - + - - + + + + - - - + + + - + + + + - - + - + + - + - + + + - + + - + - + + + + + - - - + - - - + + + + + - + - + + + - + + - + - + - + + - - + + - + + + + + - - + + - + + - - + + - + + - + + - - + +
CCCC = sedenion
+ + + + + + + + + + + + + + + + + - + - + - - + + - - + - + + - + - - + + + - - + + - - - - + + + + - - + - + - + - + - - + - + + - - - - + + + + + + + - - - - + + - + - - - + + - + - + - + - + + + - - + - - + - - + + - - + + - + + - - + - + + - - + + - - + - - - - - - - - + + + + + + + + + - + - + + - - - - + - + + - + + + - - - + + - + - - - - + + + - + + - + - + - - + - - + - + + + + + + - - - - + + + - - - - + - + - + + + - - - + - + - + - + - - + + - + + - - - + + - - + + + - - + + - + - + - - + + - -

how these results were generated.

As the above link explains, the table was generated by a computer program from the (modified) Caley-Dickson doubling process.

16D Clifford Algebras

That is, Clifford algebras based on 4 vector dimensions, I have tried the combinations of these dimensions squaring to positive and negative.

G 4,0,0	G 3,1,0
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G 2,2,0	G 1,3,0
+ + + + + + + + + + + + + + + + + - + - + - + - + - + - + - + - + - - + + - - + + - - + + - - + + + - - + + - - + + - - + + - - + - - + + - - + + - - + + - - + + + - - + + - - + + - - + + - - + + + + + + + + + + + + + + + + + - + - + - + - + - + - + - + - + - - + - + + - + - - + - + + - + + - - - - + + + + - - - - + + + + + + - - - - + + + + - - - - + - + - - + - + + - + - - + - + + + + + - - - - + + + + - - - - + - + - - + - + + - + - - + - + + - - + - + + - + - - + - + + - + + - - - - + + + + - - - - + +	+ + + + + + + + + + + + + + + + + - + - + - + - + - + - + - + - + - - + + - - + + - - + + - - + + + - - + + - - + + - - + + - - + - - + - + + - + - - + - + + - + + - - - - + + + + - - - - + + + + + + - - - - + + + + - - - - + - + - - + - + + - + - - + - + + - - + - + + - + - - + - + + - + + - - - - + + + + - - - - + + + + + + - - - - + + + + - - - - + - + - - + - + + - + - - + - + + + + + + + + + + + + + + + + + + - + - + - + - + - + - + - + - + - - + + - - + + - - + + - - + + + - - + + - - + + - - + + - -
G 0,4,0
+ + + + + + + + + + + + + + + + + - + - + - + - + - + - + - + - + - - + + - - + + - - + + - - + + + - - + + - - + + - - + + - - + - - + - + + - + - - + - + + - + + - - - - + + + + - - - - + + + + + + - - - - + + + + - - - - + - + - - + - + + - + - - + - + + - - + - + + - - + + - + - - + + + - - - - + + - - + + + + - - + + + + - - - - - - - - + + + + + - + - - + - + - + - + + - + - + + + + + + + + - - - - - - - - + - + - + - + - - + - + - + - + + - - + + - - + - + + - - + + - + + - - + + - - - - + + - - + +

how these results were generated.

As the above link explains, the table was generated by a computer program from the ordering of bases.

Even Subalgebras of 32D Clifford Algebras

That is, An even subalgebra of Clifford algebras based on 5 vector dimensions.

G+ 5,0,0	G+ 4,1,0
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G+ 3,2,0	G+ 2,3,0
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G+ 1,4,0	G+ 0,5,0
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EuclideanSpace

home