From Newsgroup: sci.math
On 08/09/2026 08:40 PM, Thomas Heger wrote:
Am Sonntag000009, 09.08.2026 um 19:55 schrieb Ross Finlayson:
On 08/09/2026 07:18 AM, Beau Holmogorov wrote:
Thomas Heger wrote:
this stupid half german doesnt even know what a magnitude, direction >>>>> and a vector is, nor what a rate of change in that direction is. How >>>>> would you know all that without a reference, idiot
Actually I'm against vectors and wanted to replace them with
quaternions.
my man, quaternions without references, just another thing you dont
undrestand. Draw a quaternion on a piece of paper, tell me what you got. >>> They also are crashing for tan and cot, making them useless in physics
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"Triality is quadratic", is what Lounesto used to say,
and it's about that usual accounts of the "complex" or
"hypercomplex" analysis are two different things, though
one imagines that the imaginary terms are not alike,
that quaternions ijk are not the same as complex i,
then that as modeling "rotations and reflections" that
the usual use of the quaternion is more "Cartanian"
about reflections and rotations, than "deMoivre-Euler-Gaussian",
the usual accont of complex analysis, which is just a branch,
and makes for distinctness results, not uniqueness results.
I was actually playing around with the Dinkin diagram D_4, which is representing 'triality', and tried to connect that with physics.
look at:
https://en.wikipedia.org/wiki/Triality
and this file
file:///C:/Users/admin/Downloads/3FlavoursNeutrinoOscillationHiggsVEV.pdf
(and at my 'book':
https://docs.google.com/presentation/d/1Ur3_giuk2l439fxUa8QHX4wTDxBEaM6lOlgVUa0cFU4/edit?usp=sharing
)
Th
The other day I was watching a math stream and the fellow
was working out deriving Euler's identity. Anyway at
some point he got to working on establishing the "existence
and uniqueness of inverse multiplication" or division, in
complex numbers. So, then working through his example,
as he put it, "messy elimination" of that the existence
is easy to figure out yet the uniqueness of quotients
is an _axiom_ since they are left-complex and right-complex
quotients, to begin, the usual idea that complex division
is unique is as closed-minded as that there are no square
roots of negative numbers, nor even negative numbers,
nor even numbers.
So, Euler & Gauss is not the only game in town,
and Cartan has his own sorts of rules.
Usual accounts of the "almost" and "very" are
as much "not-quite" as "close-enough". Then,
people with their Hilbert problems and Millenium
problems and "solve the Riemann problem" make
for that maybe they want to "un-solve" it first.
Hm. There are lots of diagrams, about both closures, and openings.
For example, the usual complex diagram attains to a closure of
a sort, "complex analyticity", which is that it attains to
"real analyticity", about integers and counting and geometry and measure.
So, "zero" comes along as a "singularity", or opening.
Then here there's an "identity dimension" idea, which
is a diagrammatic way to basically take the right half of
the usual plane coordinate diagram, x >= 0 or x > 0,
and via transformation of coordinates, split the first
quadrant into octants by the identity line, then,
any function f(x) = x, or y = x, where x and y are
interchangeable, like y = 1/x, has that the interchanged
functions, are symmetric about the identity line.
Then, instead of making it more closed, the diagrammatic
setting, it makes it more open, since now the identity line
or identity dimension is a singularity like zero, and
then for "integral analysis" instead of "differential analysis".
Much like zero is a singularity, or about whether zero is
having that 1/zero is undefined, then for many sorts usual
and fundamental integral equations, like the linear fractional equation, Clairaut's equation, and d'Alembert's equation,
the "envelope" of these integral equations, which are like
singular boundaries in differential equations, is this
identity-line, that diagrammed in this identity-dimension-diagram,
have ways to transform the diagram, by transforming the dimensions,
instead of transforming the coordinates.
Then, since complex numbers have at least two definitions of
division, and all the positive numbers are in Quadrant I
in this diagram, then Quadrants II and IV have room for
complex-left and complex-right, or a complex-complex diagram.
They don't already have one that I've heard of, though
some accounts of the "semi-infinite" or "half-plane"
like Wigner or Witten, have their own kinds of developments
that can be written in a similar kind of way.
Anyways, mathematics sort of has "roots of zero" before "roots of
unity", then the identity dimension opens the singularity,
x = y = z = ... is a singularity.
(A spiral space-filling curve is a singularity, ....)
Singularities in a singularity theory are branches in a multiplicity
theory. Saying that complex numbers have unique quotients is like
saying real numbers have no imaginary components.
So, there are lots of diagrammatic settings, and various accounts
of transforms, then overlaying the diagrams and considering the
surfaces the same, Euler & Gauss is not the only game in town.
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