" tommy quaternion "

[tommy1729]

ok here is a 5 dim candidate

the coefficients are " magnitudes " ; nonnegative reals.


1 + a + b + c + d + e = 0.

so when " reduced " not all magnitudes can be nonzero.

mod ( 1 + a + b + c + d + e ) if you want.
( so the dimension reduces from 6 ( 5 letters and real ) to 5 )


a^2 = b^2  = ... = 1.

a b = c
a c = d
a d = e
a e = b

b c = e
b d = a
b e = d

c d = b
c e = a

d e = c

som and products are commutative.

som and product behave distributive.



regards

tommy1729

anyone considered it yet ?

***

a related 3d number is described here " triplex numbers " , and i comment a bit there ( under timmy1729 ) .

https://www.youtube.com/watch?v=dvI7dGXFgm8

maybe you like that.

***

another idea

Also notice that a + b X + c X^2 for a,b,c >= 0 where X = (-1)^(1/3) describes all complex numbers.

and uniquely so when you take mod 1 + X + X^2.

this is similar to the 5d case.


regards

tommy1729