<div dir="ltr"><div dir="ltr"><div><div style="font-family:arial,sans-serif;color:rgb(7,55,99)" class="gmail_default">There is more of that beyond complex numbers, I think. </div><div style="font-family:arial,sans-serif;color:rgb(7,55,99)" class="gmail_default">Charles Muses has found it. And Grothendieck too.</div><div style="font-family:arial,sans-serif;color:rgb(7,55,99)" class="gmail_default">Hypernumbers. </div></div><br></div><br><div class="gmail_quote gmail_quote_container"><div dir="ltr" class="gmail_attr">On Thu, Jan 30, 2025 at 9:46 PM Louis Kauffman <<a href="mailto:loukau@gmail.com">loukau@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><div style="line-break:after-white-space">I think you are referring to structure of multiplication of complex numbers. <div>Any complex number </div><div>z = R exp(i Theta) </div><div>(R a real number non-negative) and multiplying two complex numbers corresponds to </div><div>adding the angles and multiplying the radii.</div><div><br></div><div>Note that we have Euler’s formula</div><div><br></div><div>                                            exp(i Theta) = cos(Theta) + i sin(Theta).</div><div><br></div><div>So exp(i Theta)exp(i Phi) = [cos(Theta) + i sin(Theta)][cos(Phi) + i sin(Phi)]</div><div><br></div><div>                                          = [cos(Theta)cos(Phi) - sin(Theta)sin(Phi)] +i[sin(Theta)cos(Phi) + cos(Theta)(sin(Phi)]</div><div>                                               </div><div>                                           = cos(Theta + Phi) + i sin(Theta + Phi).</div><div><div><br></div><div><div>The fact that Lambek can write down a set of vectors in 4-space whose sums correspond to the particle interactions is not trivial.</div><div><br></div><div>The quaternions encode rotations of vectors in 3 and 4 space. There may be a rotational interpretation or extension of Lambek that uses these </div><div>quaternion properties. I think Lambek expected something of this sort, but did not find it. </div><div>Otherwise, he would have written about it. </div><div>One can look for it.</div><div><br></div><div><br></div><blockquote type="cite"><div>On Jan 30, 2025, at 11:49 AM, Walter Johnston <<a href="mailto:williampatonmalcolm@gmail.com" target="_blank">williampatonmalcolm@gmail.com</a>> wrote:</div><br><div><div dir="auto">Dear Luke,<div dir="auto">That sounds like a BRILLIANT observation to perhaps point to a vehicle for progress.</div><div dir="auto">Bless </div></div><br><div class="gmail_quote"><div dir="ltr">On Thu, 30 Jan 2025, 17:27 Luke Kenneth Casson Leighton <<a href="mailto:lkcl@lkcl.net" target="_blank">lkcl@lkcl.net</a> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><br><br>On Sunday, January 26, 2025, Luke Kenneth Casson Leighton <<a href="mailto:lkcl@lkcl.net" rel="noreferrer" target="_blank">lkcl@lkcl.net</a>> wrote:<br><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><br><br>On Sunday, January 26, 2025, Louis Kauffman <<a href="mailto:loukau@gmail.com" rel="noreferrer" target="_blank">loukau@gmail.com</a>> wrote:<br><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><div style="line-break:after-white-space">Dear Folks,</div></blockquote><div><br></div><div>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.</div></blockquote><div><br></div><div>hilarious.</div><div><br></div><div>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]</div><div><br></div><div>it's going to be something to do with trigonometric identities</div><div><br></div><div><table style="color:rgb(68,68,68);font-family:sans-serif;font-size:16px"><tbody><tr><td align="right" style="padding-right:1em">Sine</td><td><span style="color:rgb(136,136,136)"></span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Element-43-Frame" role="textbox" aria-readonly="true" style="display:inline;line-height:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;border:0px;padding:0px;margin:0px"><spa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x;padding:0px 0px 0px 0.167em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Math;font-style:italic">α</span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;width:0px;height:2.766em"></span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0em;border-top-style:initial;border-right-style:initial;border-bottom-style:initial;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.368em;line-height:normal;overflow:hidden;width:0px;height:1.345em"></span></span></span></td></tr><tr><td align="right" style="padding-right:1em">Cosine</td><td><span style="color:rgb(136,136,136)"></span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Element-44-Frame" role="textbox" aria-readonly="true" style="display:inline;line-height:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;border:0px;padding:0px;margin:0px"><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-890" style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;width:18.766em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;width:15.369em;height:0px;font-size:19.52px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-891" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-895" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">cos</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-896" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-897" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">(</span></span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-898" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Math;font-style:italic">θ</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-899" style="display:inline;border:0px;padding:0px 0px 0px 0.222em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">±</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-900" style="display:inline;border:0px;padding:0px 0px 0px 0.222em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Math;font-style:italic">α</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-901" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">)</span></span></span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-902" style="display:inline;border:0px;padding:0px 0px 0px 0.278em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">=</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-903" style="display:inline;border:0px;padding:0px 0px 0px 0.278em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">cos</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-904" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"></span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-905" style="display:inline;border:0px;padding:0px 0px 0px 0.167em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Math;font-style:italic">θ</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-906" style="display:inline;border:0px;padding:0px 0px 0px 0.167em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">cos</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-907" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"></span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-908" style="display:inline;border:0px;padding:0px 0px 0px 0.167em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Math;font-style:italic">α</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-909" style="display:inline;border:0px;padding:0px 0px 0px 0.222em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">∓</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-910" style="display:inline;border:0px;padding:0px 0px 0px 0.222em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">sin</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-911" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"></span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-912" style="display:inline;border:0px;padding:0px 0px 0px 0.167em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Math;font-style:italic">θ</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-913" style="display:inline;border:0px;padding:0px 0px 0px 0.167em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">sin </span></span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;width:0px;height:2.766em"></span><span style="font-size:16px">α</span></span></span><span style="display:inline-block;border-width:0px 0px 0px 0em;border-top-style:initial;border-right-style:initial;border-bottom-style:initial;border-left-style:solid;border-color:initial;padding:0px;margin:0px;vertical-align:-0.368em;line-height:normal;overflow:hidden;width:0px;height:1.345em"></span></span></span></td></tr></tbody></table><table style="color:rgb(68,68,68);font-family:sans-serif;font-size:16px"><tbody><tr><td align="right" style="padding-right:1em"></td><td><span id="m_-7090359468671974232m_5920863208546019020MathJax-Element-16-Frame" role="textbox" aria-readonly="true" style="display:inline;line-height:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;border:0px;padding:0px;margin:0px"><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-223" style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;width:9.703em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;width:7.941em;height:0px;font-size:19.52px"><table style="font-size:16px;white-space:normal"><tbody><tr><td align="right" style="padding-right:1em"><br></td><td><span style="color:rgb(136,136,136)"></span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Element-16-Frame" role="textbox" aria-readonly="true" style="display:inline;line-height:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;border:0px;padding:0px;margin:0px"><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-223" style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;width:9.703em"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;width:7.941em;height:0px;font-size:19.52px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-224" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-227" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;width:1.766em;height:0px"><span style="border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-228" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">cos</span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;width:0px;height:2.766em"></span></span><span style="border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-229" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;font-size:13.8006px;font-family:MathJax_Main">2</span><span style="display:inline-block;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal;width:0px;height:2.254em"></span></span></span></span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-230" style="display:inline;border:0px;padding:0px;margin:0px;vertical-align:0px;line-height:normal"></span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-231" style="display:inline;border:0px;padding:0px 0px 0px 0.167em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Math;font-style:italic">θ</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-232" style="display:inline;border:0px;padding:0px 0px 0px 0.222em;margin:0px;vertical-align:0px;line-height:normal;font-family:MathJax_Main">+</span><span id="m_-7090359468671974232m_5920863208546019020MathJax-Span-233" style="display:inline;border:0px;padding:0px 0px 0px 0.222em;margin:0px;vertical-align:0px;line-height:normal"><span style="displa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pan></span></span></span></span></td></tr></tbody></table></div><div>and pythogorean identity sin2(a) + cos2(a)=1 and x^2 + y^2 = 1</div><div><br></div><div>combining when angles are restricted to clock positions, to give a near-embarrasing direct equivalence between unit vector addition and unit *angle* addition.</div><div><br></div><div>this mathematic elegance is responsible for dividing the Helon and Preon Model Community from the Standard Model Community for over 50 years!</div><div><br></div><div>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*!</div><div><br></div><div>aka *phases*!</div><div><br></div><div>face-palm moment :)</div><div><br></div><div>l.</div><div><br></div><div>p.s. this is why Lambek's Model looks like it does Vector arithmetic...</div><div><br></div><br><br>-- <br><div dir="ltr">---<br>geometry: without it life is pointless<div>the fibonacci series: easy as 1 1 2 3</div></div><br>
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