[Fis] Preon and Helon Models

[Louis Kauffman]

I think you are referring to structure of multiplication of complex numbers. 
Any complex number 
z = R exp(i Theta) 
(R a real number non-negative) and multiplying two complex numbers corresponds to 
adding the angles and multiplying the radii.

Note that we have Euler’s formula

                                            exp(i Theta) = cos(Theta) + i sin(Theta).

So exp(i Theta)exp(i Phi) = [cos(Theta) + i sin(Theta)][cos(Phi) + i sin(Phi)]

                                          = [cos(Theta)cos(Phi) - sin(Theta)sin(Phi)] +i[sin(Theta)cos(Phi) + cos(Theta)(sin(Phi)]
                                               
                                           = cos(Theta + Phi) + i sin(Theta + Phi).

The fact that Lambek can write down a set of vectors in 4-space whose sums correspond to the particle interactions is not trivial.

The quaternions encode rotations of vectors in 3 and 4 space. There may be a rotational interpretation or extension of Lambek that uses these 
quaternion properties. I think Lambek expected something of this sort, but did not find it. 
Otherwise, he would have written about it. 
One can look for it.


> On Jan 30, 2025, at 11:49 AM, Walter Johnston <williampatonmalcolm at gmail.com> wrote:
> 
> Dear Luke,
> That sounds like a BRILLIANT observation to perhaps point to a vehicle for progress.
> Bless 
> 
> On Thu, 30 Jan 2025, 17:27 Luke Kenneth Casson Leighton <lkcl at lkcl.net <mailto:lkcl at lkcl.net> wrote:
> 
> 
> On Sunday, January 26, 2025, Luke Kenneth Casson Leighton <lkcl at lkcl.net <mailto:lkcl at lkcl.net>> wrote:
> 
> 
> On Sunday, January 26, 2025, Louis Kauffman <loukau at gmail.com <mailto:loukau at gmail.com>> wrote:
> Dear Folks,
> 
> that said: as you discovered, Lou: Braiding itself I believe has merit in that once placed into a loop (as Sundance does - creating in his theory a Mobius strip) it can represent both phase and position of contributing components, and as you progress down the strip, the interaction between components is clearly and naturaly shown in 2D (on paper) which makes understanding easy.
> 
> hilarious.
> 
> I just did some investigation into addition of unit vectors. it turns out that if the vectors being added are equal length, their addition is *directly equivalent* to the addition of their angles relative to a given common fixed axis [trigonometric quadrant shenanigans notwithstanding]
> 
> it's going to be something to do with trigonometric identities
> 
> Sine	sin(θ±α)=sinθcosα±cosθsinα
> Cosine	cos(θ±α)=cosθcosα∓sinθsin α
> 
> cos2θ+sin2θ=1 
> +sin2θ=1 
> and pythogorean identity sin2(a) + cos2(a)=1 and x^2 + y^2 = 1
> 
> combining when angles are restricted to clock positions, to give a near-embarrasing direct equivalence between unit vector addition and unit *angle* addition.
> 
> this mathematic elegance is responsible for dividing the Helon and Preon Model Community from the Standard Model Community for over 50 years!
> 
> reason: where Preons thought they were summing and subtracting particles of 1/3 charge, actually by a massive coincidence they were summing and subtracting *angles*!
> 
> aka *phases*!
> 
> face-palm moment :)
> 
> l.
> 
> p.s. this is why Lambek's Model looks like it does Vector arithmetic...
> 
> 
> 
> -- 
> ---
> geometry: without it life is pointless
> the fibonacci series: easy as 1 1 2 3
> 

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