Symmetry of the cube

Discussion in 'Physics & Math' started by arfa brane, Jan 22, 2019.

1. someguy1Registered Senior Member

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727
It just occurred to me what was the meaning of your challenge to "find" the powerset. I think you are making the point (no pun intended) that if we have six points that form a set (as the nodes of a graph do, per the Wiki article on graph theory), you are saying that without more information, I could not find the powerset.

You are actually correct on this subtle philosophical point. (Not in the case of a finite set, but in the general case). The thing is, I don't have to find them. They exist by virtue of the powerset axiom.

https://en.wikipedia.org/wiki/Axiom_of_power_set

That is an axiom of set theory that says that if you have a set, its powerset exists.

But note that is is a very powerful axiom that posits the existence of things we could never find: namely, ALL the subsets of a given set. For example take the set $\mathbb N = \{0, 1, 2, 3, \dots\}$, the familiar set of natural numbers.

By the powerset axiom, the set $\mathscr P(\mathbb N)$ exists. Its elements are ALL the subsets of the natural numbers.

We know from the time of Cantor that the powerset of the natural numbers is uncountable. What does this mean for our ability to "find" each of the subsets? Well, if we define "finding" a subset as writing down its elements, or writing down a formula or procedure or computer program for generating all its elements, then most subsets of the natural numbers are not findable! That's because there are only countably many Turing machines, computer programs, or finite-length descriptions of sets. So all but countably many subsets of the natural numbers are unfindable in this sense. Such subsets are officially called noncomputable.

But we know they exist. Why? Because the axiom of powersets says so. And this is a point of departure for mathematicians who believe in constructive math. They would say that ONLY those sets exist whose elements can be cranked out by a Turing machine. So there's a lot of serious philosophy lurking here.

https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics)

Perhaps this is the subtle issue that you were getting at when you asked me to find the powerset of a six element set. For finite sets that's no problem. But in the general case, it's impossible. Nevertheless, the powerset exists. Where does it exist? In Platonic heaven with all the other mathematical objects.

Last edited: Jan 27, 2019

3. someguy1Registered Senior Member

Messages:
727
Yes, but what does that have to do with anything we are talking about? That's not an argument, it's not a conversation. It's another change of subject.

5. arfa branecall me arfValued Senior Member

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7,692
I'll try to go over what, at least what I think, I've been talking about, whatever it is that you think you've been talking about.

7. someguy1Registered Senior Member

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727
If you keep it simple and stick to one little subtopic at a time, progress might be made. You did convince me about the triangulated sphere. I don't doubt you have interesting things in mind.

But since graph theorists are using tensor products in a different way than algebraists do (as far as I can tell) I no longer have confidence in that calculation of the iterated tensor products. It's correct algebraically but I don't know how graph theorists regard it.

8. arfa branecall me arfValued Senior Member

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7,692
No, the subject initially was the Riemann sphere, the projective plane, Platonic tilings, reflection groups, and a lot of stuff about a graph that I have.

In fact you've been changing the subject, but I'm not that concerned. What does concern me is your apparently poor grasp of graph theory, and graphs are pretty basic things, albeit there's a lot of theory.
You "correct" me repeatedly about the fact that all the groups I'm talking about, the group product I've notated with perfectly reasonable notation as <2,3>, is actually $Z_6$. Therefore you seem to also be saying the graph can't exist, there is no copy of this <2,3> group product, because it's bloody well is actually $Z_6$. That's what I have a graph of . . .?

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10. arfa branecall me arfValued Senior Member

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7,692
The bloody train is bloody late
You bloody wait, you bloody wait
You're bloody lost and bloody found
Stuck in fucking Chickentown

The bloody pies are bloody old
The bloody chips are bloody cold
The bloody beer is bloody flat
The bloody flats have bloody rats
The bloody clocks are bloody wrong
The bloody days are bloody long
It bloody gets you bloody down
Evidently Chickentown

11. someguy1Registered Senior Member

Messages:
727
That's the first thing you've said that makes a lick of sense to me. I had to look it up and it's a very cool song.

Last edited: Jan 27, 2019
12. arfa branecall me arfValued Senior Member

Messages:
7,692
Yes, it's really what you could also dub a topological invariant--the powerset is there without needing to characterise it.

13. arfa branecall me arfValued Senior Member

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7,692
Someguy1 has obviously been studiously correcting me about the sameness (isomorphisms) of certain groups.

I've been trying, just as studiously to correct him about his grasp of drawing graphs that then can be given a group structure.
Maybe someone will submit a paper proving that a graph of six decorated points on the same line is isomorphic to a graph of two lines with three decorated points on each line.

If someguy1 manages it, graph theorists around the world might well marvel at it, after they all stop laughing.

And that particular subject, a proof that two graphs are the same graph because each can be given the same group structure, is the train we're waiting for. It isn't coming though.

The bloody train is bloody late
You bloody wait, you bloody wait . . .

14. arfa branecall me arfValued Senior Member

Messages:
7,692
Here for example, an objection to pretty much everything I've posted up to the time this is posted:

Someguy1 seems to be trying to "correct" me about graph theory, which says, yeah sure, there is only one group of order 6. Big deal, there are many different (non-isomorphic) graphs that embed that group. How many? As many as you need.
So it seems someguy is projecting his own confusion, misunderstanding of graphs, and confidence with his own knowledge of groups, onto me.

Me!? What the hell did I say?

I would seriously like to see him tell a graph theorist they're using the wrong notation.

15. TheFroggerBannedValued Senior Member

Messages:
2,175
You are BOTH wrong.

16. arfa branecall me arfValued Senior Member

Messages:
7,692
Cars collide, colours clash
Disaster movie stuff
For a man with a Fu-Manchu moustache
Revenge is not enough

There's a dead canary on a swivel seat
There's a rainbow in the road
Meanwhile on Beasley Street
Silence is the code

. . .

17. arfa branecall me arfValued Senior Member

Messages:
7,692
Oh well, here's the rest of Beasley Street: take it away Jonno. (sorry about the relentlessly dark tone, the cynicism, etc, but I didn't write it)

Far from crazy pavements,
The taste of silver spoons
A clinical arrangement,
On a dirty afternoon
Where the faecal germs of Mr Freud
Are rendered obsolete
The legal term is "null and void", in the case of
Beasley Street

In the cheap seats where murder breeds,
Somebody is out of breath
Sleep is a luxury they don't need,
A sneak preview of death
Spend a year in a couple of hours,
On the edge of
Beasley Street

Where the action isn't,
That's where it is
Vacancies exist
In an X-certificate exercise
Ex-servicemen excrete
Keith Joseph smiles,
And a baby dies in a box on
Beasley Street

From the boarding-houses and the bedsits,
Full of accidents and fleas
Somebody gets it,
Where missing persons freeze
You can't see their feet
A riff joint shuts, opens up,
Right down on
Beasley Street

Cars collide, colours clash,
Disaster-movie stuff
For a man with a Fu Manchu moustache,
Revenge is not enough
There's a dead canary on a swivel seat,
There's a rainbow in the road
Meanwhile on Beasley Street,
Silence is the code

Hot beneath the collar,
An inspector calls
Where the perishing stink of squalor
Impregnates the walls
The rats have all got rickets,
They spit through broken teeth
The name of the game is not cricket, caught out on
Beasley Street

The hipster and his hired hat
Drive a borrowed car
Yellow socks and a pink cravat,
Nothing la-dee-dah
OAP, mother-to-be,
Watch the three-piece suite
When shit-stoppered drains
And crocodile skis, are seen on
Beasley Street

The kingdom of the blind,
A one-eyed man is king
Beauty problems are redefined,
The doorbells do not ring
A lightbulb bursts like a blister,
The only form of heat
Here a fellow sells his sister down the river on
Beasley Street

The boys are on the wagon,
The girls are on the shelf
Their common problem is
That they're not someone else
The dirt blows out,
The dust blows in,
You can't keep it neat
It's a fully furnished dustbin,
16 Beasley Street

Vince the ageing savage
Betrays no kind of life
But the smell of yesterday's cabbage,
And the ghost of last year's wife
Through a constant haze
Of deodorant sprays,
He says retreat
Alsatians dog the dirty days, down the middle of
Beasley Street

People turn to poison
Quick as lager turns to piss
Sweethearts are physically sick,
Every time they kiss
Each day repeats
On easy, cheesy, greasy, queasy,
Beastly Beasley Street

Look around for laughs
If I could have just one wish,
I would be a photograph
Of a permanent Monday morning,
Get lost or fall asleep
Where the yellow cats are yawning
Around the back of
Beasley Street

Last edited: Jan 27, 2019
18. TheFroggerBannedValued Senior Member

Messages:
2,175
Never heard of it!

I've heard of beagle-street.

19. arfa branecall me arfValued Senior Member

Messages:
7,692
Where have you been? I first heard this in the 70's at high school, we had an English teacher who was into poetry, lyrics, etc. You know, James K Baxter, Leonard Cohen, Bob Dylan.

Would it surprise you to hear I've never heard of The Frogger?

20. TheFroggerBannedValued Senior Member

Messages:
2,175
Who's that? You've never heard of him?? Are you telling lies again??? Who are you talking about???

Why is it such a great offence to you that I've never heard of something that you HAVE heard of??

I imagine there are things I have heard of that you haven't!!!

Last edited: Jan 27, 2019
21. arfa branecall me arfValued Senior Member

Messages:
7,692
I'm trying to assume here, that you aren't being serious?
Otherwise why imply I'm offended? Offended by what? By life?

22. arfa branecall me arfValued Senior Member

Messages:
7,692
You can all chill folks, I've decided to voluntarily admit myself to a math rehab clinic.

(Just kidding). I apologise in advance for the fact I have two degrees, one in IT, and for knowing a bit more than the average Joe Lunchbox about the problem I'm trying to solve. I could try with a little help, but I want it from my friends.

So I'm probably going to just bang on like I usually do, if you get lost because I seem to be inventing a new language, that's not my problem.
I may respond to one or two objections, unless I see how pointless and off-topic they are.

So, let's talk about graphs. Forget about groups, forget about fields, forget about algebra. A graph is a topological space, so there is no geometry (there is maybe, a pre-geometry). There is no notion of distance or direction, but there is "separation", and there is "adjacency".

Adjacency isn't a big problem (to define): any two points connected by an edge are adjacent. The fact that an edge "does this" to two points means the edge is the separation. There is still no notion of a length for edges, and no direction defined.
It therefore does not qualify as a metric space, but that does not mean it can't be given a metric; if it can, the metric does not have to correspond to a Euclidean distance.

Moreover, graphs can be added to graphs, and there are several ways to define a product of graphs (you might call one of them a tensor product, but then find out there's a clash of notation).

Similarly, there are various products of vectors and matrices. The inner product, cross product, outer product, and yeah, tensor product.

K?

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2,175