Our daughter Amy suggested to me a book to read, "Eversion" by Alstair Reynolds
‘In mathematical framing, the definition is rather more rigorous,’ Dupin said. ‘But you have the crude sense of it. As a matter of fact, the problem is of interest to me. Consider the question of sphere eversion in a three-dimensional space—’ (...)
‘You can’t see it, but it’s possible. It’s a sphere, you see?’ He pinched the paper between two extremities, making it balloon out like a soggy attempt at a paper lantern. ‘It’s a sphere. I think it was always a sphere, and then something went wrong. That’s the topology.’
I looked to the land speeding by us on either side of Demeter. ‘Topography?’‘No. Topology. Surfaces and volumes. Transformations. Homotopic transformations. Can you see it, doctor?’"
We must regard the present state of the universe as the effect of its past and the cause of its future. Consider an intelligence which, at any instant, could have a knowledge of all forces controlling nature together with the momentary conditions of all the entities of which nature consists. If this intelligence were powerful enough to submit all this data to analysis it would be able to embrace in a single formula the movements of the largest bodies in the universe and those of the lightest atoms; for it, nothing would be uncertain; the future and the past would be equally present to its eyes. ~ Pierre Laplace
The Periodic Table of Elements. Pierre passed away on October 19, so the fact that I have received this beautiful gift a week after is an additional reason to ponder about the mysteries of love, densities and of time.
P.S.15. 08-11-23 8:29 Dreaming about the Ripples on the Dirac Sea
But I'm telling you: The infinite dense sea of
Bose-Einstein condensate of light
is even more intriguing!
- R.L. Forward "Negative matter propulsion".
- G.A. Landis "Comment on 'Negative matter propulsion'".
Negative mass the same as negative energy? I don't think so. What about "anti-photons?"
P.S.18 09-11-23 17:28 Changed notation. Replaced omega with E (for energy). Fixed Proposition 2.1. Added its Proof. Now it is time to verify and to find and correct possible errors.
P.S.22 20-11-23 11:54
P.S.23 21-11-23 13:13 Laura is telling me that it is not Elvis, who died in 1977, singing this song which was written by Leonard Cohen in the 80s. And no, it is not time travel either. It is Ron Jesse who is singing the song.
P.S.24 21-11-23 18:19 In stereographic projection coordinates the teleparallel affine connection and torsion come out beautifully! Somewhat different than those in the papers by Staruszkiewicz, but so much simpler, prettier and more elegant! Tomorrow will update the file. Haleluja!
"... Eva had come to the conclusion that there is something seriously amiss with her book.(...)Time, Eva now thought. It was time that contained the ambiguity. She had not come to terms with the concept of temporality in her book. She had not shown, had not even taken up the question, of how the systems of Plato and Spinoza, which presented reality as eternal and changeless, held by tenseless logic in the stasis of necessity, could accommodate even the notion of passing time. How is it that our sense of the flow of time, which so dominates our experience of thi world, can arise? To say that it is ultimately an illusion does not solve the problem. How can there be room for even this deception, in such systems as these?"
P.S.26 23-11-23 19:23 Added Sec 2.3.1. Stereographic projection
and 2.3.2 The connection coefficients. Added references to Staruszkiewicz's two papers. Updated the file.
P.S.27 25-11-23 17:15 Taking break for studying recent papers on the same or related subjects. Learning and thinking. Thinking and learning. Meditating. Brainstorming.
P.S.28 26-11-23 11:11 Laura's find on X:
Unable (temporarily) to answer another important question: "What is time?", I consider related question: What is (timeless) photon? And how it relates to the leptonic magnetic monopole of Georges Lochak? Clifford algebras are there.
P.S.28 26-11-23 11:36 Human consciousness is a local debugging unit with the ability to read the wave. It has also very limited wave writing capability. Most of these units are faulty. But that is how Nature works - creating abundances of everything (except for magnetic monopoles which seem to be very rare a specie).
P.S.29 26-11-23 12:47 Coincidence? After writing the previous P.S. I went home to get a lamp (I need more photons!) and found Laura in the laundry room reading this:
She did not read the previous P.S. But she must have read my mind Or I read her mind. Or simply colinearity - reading the same wave at the approximately the same time.
P.S.30 26-11-23 18:14 Added three new highly relevant references to the paper in progress. In the next couple of days will add new stuff with my comments about these references.
P.S.31 27-11-23. From "Single Photon Manipulation". IntechOpen 2020, p. 15
"During the last decades, an impressive technological development has been achieved permitting the manipulation of single photons with a high degree of statistical accuracy. However, despite the significant experimental advances, we still do not have a clear physical picture of a single photon state universally accepted by the scientific community, especially involved in quantum electrodynamics. "
P.S.32 27-11-23 15:08 And so, from today on, I become a member of the Editorial Board of the MDPI journal "Mathematics", section Mathematical Physics. Perhaps because I feel as being more a mathematician than a physicist... I respect and adore physics, but my true love is (applied) mathematics.
P.S.33 01-12-23 10:30
While reading Daniel Kastler, "Introduction à l'électrodynamique quantique", Dunod (Orléans, Impr. nouvelle), 1960, In "Avant propos":
"...Indépendamment de son utilite, l'unité des sciences physico-mathématiques nous semble être buts de l'humanisme scientifique."
P.S.34 02-12-23 12:07
P.S.35 02-12-23 12:27
P.S.36 02-12-23 18:26 Reading Christine Li, "5-minute self-discipline Exercises", p. 137
"List five things that you are thankful for:"
P.S.37 03-12-23 10:06 Reading Philip Goff, "Why? The Purpose of the Universe"
"(...) Philosophy for me isn’t just an abstract exercise but something I live out. During my ethical subjectivist teenage years, I kissed my best friend John’s girlfriend, telling him to his face that it was okay because ‘good’ and ‘bad’ depend on how you feel and it didn’t feel bad. For his part, John decided on a creative response. With the help of a few friends, he kidnapped me, tied me up, and bleached my hair bright white. Even in the absence of moral objectivity, there are ways of stopping people doing shitty things. (...)"
Ibid:
"My favourite interpretation of quantum mechanics is the most boring and unsexy of all of them: the pilot wave interpretation."
I criticized it in my Quantum Fractals (perhaps partly because I was jealous?) . But now, when I read that it is "boring and unsexy" - perhaps will change my views, reconsider. It happens once in a while that what we at some point of time dislike and even hate, later on becomes our favorite!
11:37: I still think that it can be happily merged with EEQT. It will then become more sexy and less boring (discrete events!), but without losing its attractive features. Or, better, add the pilot-wave feature to EEQT! Should be doable! In EEQT between jumps the classical system follows the classical mechanics. But it may as well be governed by the pilot-wave guided evolution. Except that it will be now dissipative.
P.S.38 05-11-23 19:39 While reading "Non-Commutative Geometry and Measurements of Polarized Two Photon Coincidence Counts" by 7 authors (two of them I have met in the past)
There are anti-photons!
Bjab should be happy!
P.S.39 06-12-23 18:36 Philip Goff is is overusing the Ockham razor principle. As Laura has noticed, when we were discussing the subject,: if Nature would use the Ockham razor, we would not be here discussing the question "Why?".
P.S.40 10-12-23 Failure! Defeat! I am depressed. Tried to read the paper
by Bertram Kostant, Nolan Wallach
After five or six pages the mathematics there became completely unknown to me. And I know that it is a very important paper - necessary for understanding consciousness. But in the paper we find category theory, exact sequences, algebraic topology, harmonic analysis - and lot of other things that I have rough idea about, but not working knowledge. Disaster. It will take me a month or more to learn, to fill out the holes in my mathematical education. They say:
Perhaps. Should I feel inspired by my failure? I should and I am.
P.S.40 15-12-23 19:42 Finally finished preparing my complete list of publications with links to all published papers. Here it is. It has a title line in Russian, because it was prepared for Math-Net.Ru.
And maybe add a pinch of Free Will. Piece of cake ;)
ReplyDeleteJust a pinch. I am for!
DeleteCycles within cycles within cycles.... Stopinders short and long. And perhaps one in every trillion of stopinders a real ooportunity arises to exercise our Free Will. But before we realize it the moment is usually already gone.
DeleteDon't forget the p-adic numbers, they have free will too! Some are more 'power'-ful than others!
DeleteOne can have long lasting regret over long gone moments even if doing what you regret not doing could easily have changed more important future moments like finding this blog (technically it was first the Quantum Future site, then the Cass site, then a Polish blog, then an English one, back to Polish then to this English one I think).
Delete@Natus Videre
DeleteOK. Suppose we are friends with p-adic numbers. But still question remains: what is time? How many dimensions time has? Why future and past seem to have unequal status?
@John G
DeleteThe past is for learning on mistakes, the present is for shaping the future. Forget regret!
@Arkadiusz Jadczyk
DeleteMaybe time can be interpreted as fractals of space. Our entire reality is replicated on each iteration (which would be uncountably infinite). Each 'snapshot' of space and all its infinite sub-components have an informational 'address' along with an electro-magnetic signature. This means the universe 'knows' and keeps a record of all its activities (which we perceive as time). Constantly affected by consciousness and free will, these 'snapshots' of space can attract and repel each other, leading to 'fuzzy,' ever-changing outcome probabilities. Gravity acts as a 'paradox resolver,' handling contradictory timelines and striving for balance.
Depending on our state of awareness, it is possible to 'jump' from one fractal continuation to another (i.e. when learning life lessons). Some sets of 'snapshots' are only accessible from a higher awareness level. Past, present and future can coexist at the same 'time.' When you look at a tree or an animal, you are looking at the past. Your point of view is the present, 'your' present. When you interact with a being that has accumulated more knowledge and wisdom, you are seeing the future.
Still the question remains: "what is space?"
DeleteHere are three definitions of space that I can think of right now:
Delete(1) That which, on some level, is materially indivisible.
(2) The minimal material building block(s) of the universe.
(3) A unit or a set of units which can be assembled to create more complex material structures.
For example, let's pretend that the minimal units are electrons, protons, and neutrons. There would be no such thing as a 'half-electron', 'quarter-proton,' or 'split-neutrons.'
The elements, atoms, molecules would be geometric combinations of whole electrons, protons, and neutrons. This is what I mean by 'indivisible' units.
So space has a limited selection of components (electron, proton or neutron, in this example), but could have infinitely many combinations of these components.
Now, just for fun, let's suppose electrons, protons, and neutrons can be split up infinitely many times, so the minimal set would be infinite. Then, you could take just one electron and cover all the universe with its infinite 'dust.' The universe would then be a massive grid where a very small number such as 10^(-10000000000000) would have a material meaning, as it would be mapped to a tiny part of an electron. Would this make any sense? Maybe, maybe not.
If such infinite splitting exists, then I think it must have an immaterial origin (anti-space, pure energy), because castles are not made up of dust! I can easily imagine a castle with a height of 10^(-10000000000000) meters, but materializing it may not be possible.
After all, maybe balance cannot be achieved without less restrictive, immaterial units 'wrapping' material units.
I there aew separated units, say electrons and protons, what is it that separates them?
DeleteI could certainly see spacetime in the basis vector sense as being the building blocks for everything. XY could be a rotation graviton and XT a boost graviton for example. For translations and conformal structures, you would need at least a 6-dim spacetime and up to 8 can give you other force particles. Using an odd number of building blocks could give you fermions.
DeleteYou can also certainly make an infinite grid with spacetime. Things like protons and 2nd/3rd generation particles might actually be 2 or 3 things built with spacetime basis vectors. The Dirac sea fermion cloud might be a really large number of built things surrounding a single grid location maybe in a complex spacetime structure.
Fermions both with their spacetime components and the structure of the 8 1st generation particles is spacetime-like so having fermions as a spacetime basis vector building block stand-in isn't impossible to think with. It's kind of a time vs space structure that separates electrons and protons (their 3 quarks).
For two entities to be distinct, they must be independent (not expressible in terms of each other), and stable (not degradable).
DeleteLet's consider prime numbers. Say the electron equals 2, and the proton is set to 3. No matter how many times you multiply or divide 2 with itself, you will never get 3.
But 2, 3, and the rest of the primes which also represent different independent 'tracks,' are all divisible by 1. So the electron and the proton can originate from or be made of the same 'substance' even though they are 'independent.' It's like an XY plane with an origin that is everywhere. The intersection of the axes is the binding point, and the intersection can be anywhere on the grid depending on the point of view. There is a strong link between the origin and all possible points.
Separation springs from an origin, a core 'substance.'
Stability is a property acquired by movement, more specifically angular momentum. If a soccer ball is given a great spin when it is hit, then the wind has negligible effects on its trajectory—the rotating ball creates its own areas of low and high pressure. Electrons and protons have different rotational dynamics which makes them 'stick' to each other. It is the birth of 'polarity.' Like pieces of a puzzle, they have well defined interfaces that can 'attach' to their complements. Thus, their arrangements must be geometric—there is no randomness here. Disturbances provoke changes in their geometric arrangements which always aim to preserve balance.
Separation causes space. In other words, space arises from the attractive/repulsive tension between fundamental building blocks originating from a common source.
@Natus Videre
DeleteBut we have a serious problem. There are different deparations. There are small separations and large separations. What is it that make some separations small and some large? Are there atoms of separations? What are they? They are certainly not those protons and electrons that are separated by separations....
@John G.
DeleteThe problem here that I see is that while there are REAL things like tables and chairs (it hurts when you kick them), it is not that simple with "space". Is space something real? Or is it just an "organizing principle"? If so, to which category it belongs? Let's forget time for a while (perhaps time is just an imaginary space, whatever it means.
Space (with time) as basis vectors would kind of just be math; it's not even attempting to be anything yet. Bosons and fermions aka the things not in your EEQT center algebra are certainly the things of hurt when you kick them. That center algebra aka differential geometry with metric, g-structures, differential forms, etc. is maybe kind of all space and time-ish, the metric being the most 4-dim space and time-ish as we perceive it.
DeleteOne vertex of a spacetime grid could be Cl(8) and Cl(8)xCl(8)xCl(8)... could be all vertices of your current state grid. So yeah the center algebra would somehow be responsible for organizing that tensor product into a 4-dim grid and maybe it was 8-dim before a symmetry break and maybe at the Planck scale with no symmetry breaking, it's just the 1-dim tensor product basically the natural numbers.
A conformal group g-structure in the center algebra could make spacetime seem 4-complex dimensional. Course there's also conformal group bosons which could kind of steal reality from those tables and chairs. Is your 4-dim anti-spacetime taking the place of Kaluza-Klein extra dimensions or does it come from conformal infinity without needing extra dimensions?
@Arkadiusz Jadczyk
DeleteIndeed, that's a serious problem. Space is not only made of "that which is," but also made of "that which is not [seen]." It reminds me of fractals. Sometimes they produce a pattern, sometimes they produce an anti-pattern which is just as mysterious as the 'visible' pattern. Some areas of space are entirely 'geometrically' avoided as if there was an organizing principle.
When scientists try to create a 'vacuum' by sucking out all the air, they don't 'disappear in a vortex, their lab doesn't collapse, the spatial structure of the experiment remains intact. We reach a point where we cannot further explain reality without considering the antithesis of matter. So electrons and protons may hold an internal balance when they are present, but when they are not there, something else fills space in their absence.
But is there really a vacuum? What holds space in place if there is 'empty' space?
The material world is limited by its smallest building blocks. However, the non-material world is not limited—it can make fractal divisions ad infinitum that can be much smaller than the smallest material unit, because uncountable infinity can address all points in space. We then have a sea of matter, and an even bigger sea of states (frequencies). Thus, an energetic grid permeates space. The finer matter becomes, the closer it is to the energetic boundary, and the stronger its connection is to the non-material world and to the information field.
We are mostly made of water. Water has a triangular/tetrahedral shape. It is a polar molecule, it allows an energetic accumulation to occur. We are not as balanced as we think we are. What is the other 'half' or the dual of a tetrahedron? Another tetrahedron! "A: Double tetrahedron in 3D is hexagon in 4D." (Session 4 July 2020). Hexagon tilling is very space efficient. We have also seen that right triangles can create squares/cubes. 4D space is a 'squared' space; in other words, balanced. So, in our 3D world, matter is paired with its complement (dual) which is not visible to us, because the pairing is completed in a higher-dimensional space.
"Coordinates (p1, p2, p3, p0) = (p, p0), p2 = p2 − p02."
ReplyDeletePrzeciwnie:
p2 = p02 − p2
i dalej nie:
p2 + m2 = 0
a:
p2 = m2
You were clear enough.
DeleteIs it a matter of choice?
50/50?
Have you flipped a coin?
I think our not anti material world needs signature (- - - +)
In P.S.7. I added a (outdated) table of signs conventions. About 30 percent of books on gravitation uses +++- signature. The rest is using +---. In quantum field theory the great majority is using +---. The table lists Weinberg's book on quantum field theory. First he used +++-, but later on changed to +---.
DeletePersonally I prefer to pay attention to only one sign rather than to three (and these signs flip when we convert between vectors and co-vectors. This personal preference is based on experience. Space is space and is Euclidean. Time came later as an extra dimension of a different nature than space. It is still a mystery.
Nie rozumiem tej tabelki (i chyba nie chcę jej rozumieć).
DeleteJedno co z niej wynika to to, że powstał duży bałagan.
"I prefer to pay attention to only one sign rather than to three"
DeleteThis is an argument for comfort (laziness), but does it go hand in hand with understanding?
Your choice forces the equation (you wrote):
Deletep² + m² = 0,
something must be then urojone.
In this table table we are concerned only with the first column, "g-sign". "+ " means signature +++- (or -+++) , "-" means signature (---+) (or +---),,where my "or" means that "time" is the zero-th coordinate.
DeleteIndeed it was all much better at the very beginning of the special relativity theory, when there were only plusses and "time"-coordinate was imaginary.
Czy ja też mogę uzyskać w komentarzu bold?
DeleteYes, you write, for instance < b> p < /b>
Deleteand you get bold p.
Sorry, my fault. E > m and that is OK.
DeleteBack to the signature conventions. If we use the +--- convention, when I see the ∇ symbol I have to check whether the derivatives there are with respect to contravariant or covariant coordinate indices. And when I see a.b, I have to remember what the bold symbols mean and what their scalar product means.
DeleteWith -+++ convention I only have to remember that the energy is p^0 and NOT p_0.
So why did they converted themselves to +--- ?
DeleteIt is like why 90% of scientific publications are nowadays in English language? Probably the bankers so decided. For reasons known only to them.
DeleteAdded Proposition 2.1 to the photon's file. Will add the proof and examplea next.
ReplyDelete"Negative mass the same as negative energy? I don't think so. What about "anti-photons?""
ReplyDeleteThis is a very interesting issue. I am at the stage of thinking that in some formulas mass is an algebraic quantity (+ or -) and in some other formulas there must be an absolute value of mass.
Why do bottle caps have the same handedness?
ReplyDeleteWhy do DNA strands have the same handedness?
Why does the vector product have the same handedness?
Does matter have the same handedness?
And photons?
Why? Why? Why?
DeleteAnd why is Bjab asking such questions?
Because Bjab is observing and thinking. Bjab has curiosity.
Why some people are curious about reality and some are not?
Questions lead to even more questions.
There must be a second world, invverted (or everted) with repsect to the one we happen to be in.
Finally finished Sec. 2.2.
ReplyDelete
ReplyDeleteis determined p. ->
is determined by p.
(20) mianownik.
Hilbert of sections ->
Hilbert space of sections
Thank you. Will fix it in a new version coming soon. Will add a section on stereographic projection. Working on it right now.
DeleteWhat about (20)? The denominator is E_m^2, which is p^2+m^2. I do not see a problem here ....
DeleteI can see only m^2 there (not omega^2_m).
DeleteOnly now I realized that it is showing the old version of Nov. 8. Will fix it later today. Thank you.
DeleteBut (20) in this version is correct. If you calculate the inverse matrix of (19) you will find that there comes, as the result, indeed m^2 in the denominator!
It shows now the most recent version. Previous (20) is now (21).
DeleteYour EEQT center algebra and all the classical things you've been doing for a while do scream pilot wave to me. You do need that discrete very local classical transaction to know the far away in spacetime in order to be a pilot wave but I think that comes along for free with differential geometry but I'm not sure exactly how to think of it.
ReplyDeleteIt's one little degree of freedom. For Bradonjic, it's an affine one-form that's a generalized proper time. For Tony, it's a Feynman propagator phase. Elsewhere it seems related to a solder form/tautological one-form that is binding velocity to momentum.
Your quantum jump could then prune every classical branch made since the previous jump except one with a timing probably very GRW-like. Free will pruning could be interesting to think about as some advanced Zeno/anti-Zeno effect of sorts.
Still questions remain: what is Randomness? And Who is playing the dice?
DeleteEverything that is is dynamic deformations of the aether.
ReplyDeletePerhaps. But how that would help with time travel, free will, and telepathic communication?
DeleteThere is no time travel, free will is illusory, some telepathy is telecommunication.
DeleteNo time travel? Then what about UAP?
Deletehttps://www.youtube.com/watch?v=lcrCMLVk614
Are we going to ignore the data, because they do not fit our pet theory? Telepathy may well act backward in time. How aether dynamics is going to deal with that?
Moreover, ether dynamics without aether dynamics is just a buzz word. It like when you are getting a book with a nice title and covr image, but nothing but empty pages for you to fill inside!
@Bjab
DeleteGravito-magneto-hydro-dynamics - that would sound even better!
Is it us collecting the data or is there a man/equipment in the middle?
Deletehttps://www.livescience.com/18603-error-faster-light-neutrinos.html
"Telepathy may well act backward in time."
DeleteTja.
My alarm clock will ring at seven tomorrow. - My pocket telephaty.
"Moreover, ether dynamics without aether dynamics is just a buzz word."
DeleteSome aether dynamics is well known - for example: Maxwell equations .
@Bjab
Delete"Is it us collecting the data or is there a man/equipment in the middle"
There is always something in the middle. For instance Google, or ISP. Or just air, or thoughts that come from we do not know where from. Nevertheless AFP need to be taken into account quite seriously. Rare events, but extremely significant.
"There is always something in the middle."
DeleteSo we should check everything in different ways.
" thoughts that come from we do not know where from."
I know - from the brain.
AFP - what do you mean?
@Bjab
Delete"Some aether dynamics is well known - for example: Maxwell equations "
That is a good beginning. But which Maxwell equations. The devil is in the details.
@Bjab
Delete"I know - from the brain."
Again: good beginning. But how do they form in the brain? The devil is in the details.
@Ark:
Delete"But which Maxwell equations."
The ones:
https://en.wikipedia.org/wiki/Maxwell%27s_equations
(useful in the range where the aether behaves linearly)
@Bjab
DeleteWhat about this sentence in the linked Wikipedia article?
"Some observed electromagnetic phenomena are incompatible with Maxwell's equations."
@Ark: "What about this sentence ..."
DeleteSo, as I wrote, these strange phenomena are cases when the aether becomes non-linear (in 3 D).
@Bjab
DeleteAnd what to do then? Wikipedia suggests using quantum electrodynamics in such cases. Is there some other alternative?
Or should we follow the advice of Wikipedia?
For instnce Wikipedia also writes:
Deletehttps://en.wikipedia.org/wiki/Born%E2%80%93Infeld_model
"Born–Infeld electrodynamics displays good physical properties concerning wave propagation, such as the absence of shock waves and birefringence. A field theory showing this property is usually called completely exceptional, and Born–Infeld theory is the only [5] completely exceptional regular nonlinear electrodynamics."
@Ark:
ReplyDelete"Bjab should be happy!"
Happy? With what?
With negative energy? - Oh, no.
Moving backward in time? - Ridiculous.
My guess is that the momentum is opposite to the antiphoton velocity.
@Bjab
Delete"Moving backward in time? - Ridiculous."
"People used to think Copernicus was a raving lunatic for saying the earth revolved around the sun. What modern day "ridiculous" theory might end up being common knowledge in a few hundred years?"
https://www.quora.com/People-used-to-think-Copernicus-was-a-raving-lunatic-for-saying-the-earth-revolved-around-the-sun-What-modern-day-ridiculous-theory-might-end-up-being-common-knowledge-in-a-few-hundred-years
However, after Copernicus and until now, the Sun has not stopped rotating around the Earth. Everyone can easily experience this every day.
DeleteWhen I read about geometry, I find that people are mostly interested in the properties of shapes (angles, perimeters, areas, volumes, etc) and "clever" equations relating these properties to themselves. But I have the impression that a "unified" language or approach is missing. For example, "adding" a triangle to a pentagon would yield a new equation representing the algebraic "mixture" of these two polygons.
ReplyDeleteThe motivation behind building such a "framework" is that it would allow us to reveal the properties of "that which passes through" the shape. Instead of only analyzing the properties of a geometric object, we would look at the properties of the energetic/magnetic distortion generated by the object. Maybe we would find that a triangle has a strong energetic accumulation (vortices?) at its corners due to the intersection of the lines? A simple experiment could be to magnetize a regular polygon and observe the change in the magnetic field lines when another polygon is added to the "mixture."
Have any of you tried to find algebraic equations of triangles, pentagons, or hexagons such that these shapes can be "assembled" in a general, common framework, like polynomials? Can all shapes be defined in terms of rotations (spinors?), just like three circles can make an equilateral triangle?
Very interesting point. I an not aware of a general theory of shapes, probably there are papers and monographs on this subject. But closely related is the theory of knots, where topology plays important role.
Deletehttps://www.youtube.com/watch?v=kxaWqKM5JyQ
Quantum jumps:
ReplyDeletehttps://twitter.com/Rainmaker1973/status/1711815437988565440
Quantum jumps:
ReplyDeletehttps://www.quantamagazine.org/quantum-leaps-long-assumed-to-be-instantaneous-take-time-20190605/