[Fis] Preon and Helon Models

[Dr. Plamen L. Simeonov]

There is more of that beyond complex numbers, I think.
Charles Muses has found it. And Grothendieck too.
Hypernumbers.


On Thu, Jan 30, 2025 at 9:46 PM Louis Kauffman <loukau at gmail.com> wrote:

> 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
> wrote:
>
>>
>>
>> On Sunday, January 26, 2025, Luke Kenneth Casson Leighton <lkcl at lkcl.net>
>> wrote:
>>
>>>
>>>
>>> On Sunday, January 26, 2025, Louis Kauffman <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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