In

there is a transform for (a,b)*(c,d) which is defined as (ac - bd, ad + bc).

For

I designed a transform based on the

transform in the following way:

(a,b,c) vx (d,e,f)

(ad - be - cf, ae + bd, af +cd, bf + ce) using

as a manipulation /construction space.

This i reduce to

(A,B,C,D) =(ad - be - cf, ae + bd, af +cd, bf + ce)

Using pairs from the construction bracket in the

transform i obtain 6 building blocks

1 (AA -BB, 2AB)

2 (AA -CC, 2AC)

3 (CC - BB, 2BC)

4(AA - DD, 2AD)

5 (BB - DD, 2BD)

6 (CC -DD, 2CD)

THE unary OPERATORS i and j are used to inform the manipulations so that i

^{2} = j

^{2} = -1 and (ij)

^{2} =+1.

Now my intention was to rotate the planes xy, xz, yz by this construction and i assumed that was what was happening until i rechecked the construction principles. The yz plane is not the same as the other two planes with the unary operators i and j operating on the axes. Under the

transform the yz plane is sheared to the xij plane whatever that is. It may be a vortex surface.

so the first constructed transform is mistaken in two counts. The manipulations were faulty and i will show the correct manipulations; but the design was mistaken as it was not tranforming to a map of geometrical space.

The expansions are as follows for the right handed form

AA = (ad)

^{2} + (be)

^{2} + (cf)

^{2} - 2abde - 2acdf + 2bcef

BB = (ae)

^{2} + 2abde + (bd)

^{2}CC = (af)

^{2} + 2acdf + (cd)

^{2}2AB = 2(a)

^{2}de + 2ab(d)

^{2} - 2ab(e)

^{2} - 2(b)

^{2}de - 2acef - 2bcdf

2AC = 2(a)

^{2}df + 2ac(d)

^{2} - 2ac(f)

^{2} - 2(c)

^{2}df - 2abef - 2bcde

2BC = 2(a)

^{2}ef + 2bc(d)

^{2} + 2abdf + 2acde

Now the first posted construction was based on combining blocks 1,2,3

supposedly giving

*{AA - BB + AA - CC, 2AB + BB - CC, 2AC + 2BC}

=> {AA - BB/2 - CC/2, AB + BB/2 - CC/2, AC + BC }.

[ in fact it should be {AA - BB/2 - CC/2, AB - BB/2 + CC/2, AC + BC } due to an error in the original formulation of block 3]

So clearly (when i expand it) my original manipulations were wrongly copied from page to page to screen.

But now i realise i have not combined like with like and so have to construct the following transform from blocks 1and 2 which i fear will be even less interesting than my mistaken one

{AA -BB/2 - CC/2. AB, AC}