At the end of the Universe there is Infinity. What shape does it have? No abstract mathematics is needed in order to answer this question. You need just Light. No algebraic topology, no Hopf algebras, no category theory. It is as simple as it can be - and yet imagining it can be challenging. And so here I am presenting you with this challenge. And I am accepting this challenge myself, since at present I do not have a clear answer.
So here it is.
We are in 4D, coordinates X,Y,Z,W. The infinity is given by the intersection of these two cylinders:
1) X2 + Y2 + Z2 = 1
2) Z2 + W2 = 1
We notice that necessarily X,Y,Z,W must be in the interval [0,1]. So our infinity is bounded, it is contained within the unit cube of the four-dimensional space.
Next thing we notice is that there are two equations for four variables. Therefore infinity is a two-dimensional surface (4-2=2). So it should be possible to visualize it in a 3D space in which we live. Of course there will be infinitely many ways of visualizing it, depending on our creativity. Some will be more appealing than others.
I will be working on this problem myself with my poor imagination. You can use yours.
The theory behind this challenging simple problem I will present in one of the next posts.
The fact is: This is the double cover of the conformal infinity of the Minkowski space as discussed by Roger Penrose - the mathematician who in 2020, was awarded one half of the Nobel Prize in Physics for the discovery that black hole formation is a robust prediction of the general theory of relativity. In the past I have already discussed it here. But today I doubt if my proposed solution is a good one. I think it may be either wrong or incomplete.
P.S.1. The challenge problem addresses a "toy infinity", good for (2+1)-dimensional spacetime, thus stripped of one space dimensions.
P.S.2. Is it a "twisted torus"? Whatever it means.....
P.S.3. Seeking for the solution
The Seekers - Walk With Me
P.S.4. We need to know ALL about this surface at infinity. It is infinitely important. Light can travel there in no time, and come back bringing us information. It is our past and our future. It is a homogeneous space for the Poincare group. Therefore a specie of elementary particles is living there. What are these particles? What do they do? What kind of a geometry this infinity carries? All these questions MUST be answered.
I have found a book "Surfaces in 4D space" by three authors Scott Carter, Seiichi Kamada and Masahico Saito. But it is too difficult for me. I do not know the techniques. And my surface is probably extremely simple: it has no knots etc. But I want to know all the topological invariants - if there are any non-trivial.
P.S.5. Laura is telling me that I should write a book about this Conformal Infinity. Perhaps. But first I need the results. A lot of them. And good quality results.
„We are in 4D, coordinates X,Y,Z,W. The infinity is given by the intersection of these two cylinders:
ReplyDelete1) X2 + Y2 + Z2 = 1
2) Z2 + W2 = 1”.
You ask about imagination... So look at something like this...
Take a cylinder S^1 x R. We may use the cylinder to construct a non-trivial vector bundle - Mobius Band. Let ~ be the equivalence relation (p, x) ~ (p + 2pi, -x) on S^1 x R.
The Mobius band M is defined as the quotient space:
M = (S^1 x R) / ~.
The fibre R is a vector space and the fibres are isomorphic to R.
You mention the torus. Let's make it:
Let there be S^1 x S^1. Now by introducing some equivalence relation we can analogously represent the Klein bottle as a quotient space. The Mobius band is an open subset of the Klein Bottle. It follows that the Klein bottle is not orientale (This property is important if we are looking for some deeper structure here).
Then, in a similar way, we can construct a Hopf fibration that allows us to view S^3 as a principal circle bundle over S^2.
Later, you look at how these structures twist, you watch their morphisms and each time you see quotient spaces appear. Are these quotient spaces a coincidence?
„You need just Light. No algebraic topology, no Hopf algebras, no category theory.”.
But what kind of infinity would that be? Some kind of flat one? And an infinity of what? Time, space, spacetime?
Well, we can assume that it is flat. Let it be. But then everything else must be twisted. Depends on how you look at it. A question of the reference system.
„You need just Light. No algebraic topology, no Hopf algebras, no category theory.”.
ReplyDeleteHowever, looking at it from another perspective this may indeed be the case....
Photo in the post looks similiar to the crop circle with double yin yangs: https://www.chinesefortunecalendar.com/YinYang/YY-CropCircle/YY-Farm5.JPG
ReplyDeleteIf I were to decipher this crop circle. I see it as a balance between the world of spirit and matter, divided into Densities. The circles immediately surrounding the yin yang symbols could mean 1 and 7 Density (at lower yin yang 1 Density and at upper yin yang 7 Density, for example).
The next circles (second in order from the yin yangs) connected to each other, they symbolize 4 Density - 4th Density naturally is itself a combination of matter and spirit, which is symbolized also by this connection between the circles.
Then the two outgoing ribbons would be for one yin yang the 2nd and 3rd Densities (material Densities), and for the other yin yang the outgoing ribbons would be the 5th and 6th Densities (spiritual Densities).
As for unity and infinity, the symbol of the "lying eight" (which is inside the photo, in this post) in a good way, I think, shows us that there is a balance between unity and infinity:
Infinity converges in Unity, and Unity dissolves into Infinity.
A single torus squeezed at two points?
ReplyDeleteLike torus. Symbol "lying eight" when turning around give a "donut" squeezed in central point. This is the best way to contemplate meaning of Oneness and Infinity that are complementing each other. I think.
ReplyDeletehttps://youtu.be/Wgo8ePtRLe0?t=193
ReplyDeleteThe Times They Are A-Changin' - but will it be conformal?
Nieskończoność (tę ograniczoną) widzę geometrycznie jako granicę podziału połówkowego dwóch linii przecinających się w punkcie. Jest to liczba powstająca rekurencyjnie w miarę zbliżania się do punktu przecięcia w którym osiąga maksimum. Poza punktem przecięcia jest już inny wymiar (liczby pozaskończone). :)
ReplyDelete"Poza punktem przecięcia jest już inny wymiar (liczby pozaskończone). :)".
DeleteOsobliwość?
OK. But we have a little problem here:Rdward Robak died in 2012
DeleteAch da się tak, ale jakie są definicje tych pojęć, które stosujesz?
DeleteNie wszystko co napisano:
Delete1) jest na temat
2) jest prawdą.
Czasem ludzie umierają na pozór.
Często definicje są redefiniowane i zapomina się o geometrii.
[AGEOMETRETOS MEDEIS EISITO]
Zdarza się, że ludzie nie wiedzą, bo nie chcą wiedzieć (świadomy wybór.)
Było miło.
Pani Matyldo. Zanim człowiek opracował matematyczne modele zera, to w powszechnym obiegu używano słów na wyrażenie NIC, dlatego proponuję najpierw opisać ten obiekt, który nazywa się powszechne słowem "nieskończoność", a gdy to już będzie uzgodnione, to można przypomnieć znane definicje. 🙂
Delete