the importance of topology

[scruffy]

These are photonic orbitals.

They result from various energy levels.

These are "symmetries". They can be calculated algebraically, using group theory.

Group theory is the foundation of algebraic topology. You can use it to determine which shapes are "allowable".

Generally, any shape that's allowable under the symmetry group, will be observed.

These shapes also apply directly to neural networks. Memories have shapes. Shapes are recognized by the feature detectors in the cerebral cortex. Similar memories can be distinguished by their shapes.

New shapes of photons open doors to advanced optical technologies

Researchers from the University of Twente in the Netherlands have gained important insights into photons, the elementary particles that make up light. They 'behave' in an amazingly greater variety than electrons surrounding atoms, while also being much easier to control.

phys.org

 

[scruffy]

The one on the top left is especially blatant and interesting.

Notice how it's not "just" a ring. It has gaps. Why does it have gaps?

And more to the point, how do the gaps support the symmetry? Are the gaps "necessary"?

These are photons. Quantum theory tells us that the probability of finding the photon in the gap is near zero. What exactly are we looking at here? A probability "wave"?

If so, can we get 4:gaps instead of 8? How about 16? Or 64?

 

[scruffy]

It gets even weirder. By careful construction of waveguides, orbitals with non-integer dimensions can be achieved.

Non-integer means fractional (fractal) dimension.

For example, this one here has a dimension of 1.89.

It is a version of a famous fractal known as Sierpinski gasket.

Edge on, it looks like the gasket. But note the hole in the middle. It looks "different from" the rest of the pattern. It's not square and pointy, it's actually round

This the original gasket:

Here, the hole is large but it has the same shape as the others. Above, it doesn't.

Topological photonics in fractal lattices

Topological insulators are a new phase of matter unique for their insulating bulk and perfectly conducting edges. They have been at the forefront of condensed matter physics for the past decade, and more recently inspired the emergence of topological phases in many classical-wave systems, such...

phys.org

Some of you will appreciate the similarity to a Cantor dust. Spectral decomposition in a complex Hilbert space suggests a "pure point spectrum".

The mathematicians say;

If A=A⋆𝐴=𝐴⋆ is a densely-defined selfadjoint linear operator on a complex Hilbert space H𝐻, and if there is a complete orthonormal basis of H𝐻 consisting of eigenvectors of A𝐴, then it is true that the point spectrum σp(A)𝜎𝑝(𝐴) of A𝐴 is dense in σ(A)𝜎(𝐴). The converse is not true.

Density is crucial. The strange thing about the dust is, it has the same number of points as the original interval. It's as if removing 1/3 of the interval has no effect on the point count.

This counterintuitive behavior only occurs in fractals, It's because the dimension isn't an integer. It occurs because of the topology, each interval is homeomorphic to its parent.

So now, in a complex Hilbert space we do this same thing with functions instead of points. We remove 1)3 of the functions from an interval, and discover we still have the same number of functions as when we started.

Cantor proved this using number theory alone, without making reference to topology. In his world, these shapes are "projections" of non-integer dimensions into an integer (Euclidean) coordinate system . They "fool the eye", so to speak. But topologically, every time we remove an interval we're adding a hole, thus the gasket ends up with an infinite number of handles.

 

[Nobody911]

These are photonic orbitals.

They result from various energy levels.

View attachment 995328

These are "symmetries". They can be calculated algebraically, using group theory.

Group theory is the foundation of algebraic topology. You can use it to determine which shapes are "allowable".

Generally, any shape that's allowable under the symmetry group, will be observed.

These shapes also apply directly to neural networks. Memories have shapes. Shapes are recognized by the feature detectors in the cerebral cortex. Similar memories can be distinguished by their shapes.

New shapes of photons open doors to advanced optical technologies

Researchers from the University of Twente in the Netherlands have gained important insights into photons, the elementary particles that make up light. They 'behave' in an amazingly greater variety than electrons surrounding atoms, while also being much easier to control.

phys.org

Why topology matters? Yes, you can trick some scientists or mathematicians! lol.

 

[Sherlock Holmes]

These are photonic orbitals.

They result from various energy levels.

View attachment 995328

These are "symmetries". They can be calculated algebraically, using group theory.

Group theory is the foundation of algebraic topology. You can use it to determine which shapes are "allowable".

Generally, any shape that's allowable under the symmetry group, will be observed.

These shapes also apply directly to neural networks. Memories have shapes. Shapes are recognized by the feature detectors in the cerebral cortex. Similar memories can be distinguished by their shapes.

New shapes of photons open doors to advanced optical technologies

Researchers from the University of Twente in the Netherlands have gained important insights into photons, the elementary particles that make up light. They 'behave' in an amazingly greater variety than electrons surrounding atoms, while also being much easier to control.

phys.org

Why did you start this thread? what do you actually want to discuss?

 

[task0778]

Dang, I thought this thread was gonna be about boobs. With pics.

 

[scruffy]

Why did you start this thread? what do you actually want to discuss?

The gaps in the top left orbital.

 

[para bellum]

Chaos Theory meets the Standard Model for the first time. Cool.

 

[scruffy]

Chaos Theory meets the Standard Model for the first time. Cool.

Exactly.

That's my theory too.

The gaps in the orbitals (that is to say, their shapes and winding numbers) result from a pattern of attractors.

Chaos theory - Wikipedia

en.wikipedia.org

 

[para bellum]

Exactly.

That's my theory too.

The gaps in the orbitals (that is to say, their shapes and winding numbers) result from a pattern of attractors.

Chaos theory - Wikipedia

en.wikipedia.org

What got my attention was the fractional dimensions in your post #3. Something that the Standard Model does not even contemplate, but is a fundamental feature in Chaos Theory.

I see the Standard Model as sort of a dumbed-down universe- reduced to human comprehension levels. It is the boundary condition that Chaos Theory tells us is needed to make the universe predictable.

The Standard Model is half the picture, the other half is Chaos Theory. They were bound to collide eventually.

 

[Sherlock Holmes]

Exactly.

That's my theory too.

If only knew what chaos actually was, then you wouldn't make embarrassing errors like this.

 

[scruffy]

If only knew what chaos actually was, then you wouldn't make embarrassing errors like this.

I know what chaos is.

What it's NOT, is pop science.

Go look at the Lorenz attractor and tell me what you see.

Do you see a continuous curve, or do you see compact topologies?

There's all kinds of bullshit floating around on the internet, from people who don't know what they're looking at.

Be a careful observer, and tell me what you see.

 

[Sherlock Holmes]

I know what chaos is.

No you don't, you think it's random and it is not.

What it's NOT, is pop science.

Go look at the Lorenz attractor and tell me what you see.

Do you see a continuous curve, or do you see compact topologies?

There's all kinds of bullshit floating around on the internet, from people who don't know what they're looking at.

There's all kinds of bullshit getting posted on science debate forums too, look FFS, this is the APS, a site you've cited several times yourself.

The American Physical Society is a nonprofit membership organization working to advance physics by fostering a vibrant, inclusive, and global community dedicated to science and society. APS represents more than 50,000 members, including physicists in academia, national laboratories, and industry in the United States and around the world.

Here's what they say (and everyone else says) about the Lorentz attractor:

To the average layperson, the concept of chaos brings to mind images of complete randomness. Yet to scientists, it denotes stochastic behavior occurring in a deterministic system: namely, systems that are so sensitive to measurement that their output appears random, even though there is an underlying order. This seemingly paradoxical viewpoint was born when a mathematician turned meteorologist named Edward Lorenz made a serendipitous discovery that subsequently spawned the modern field of chaos theory and changed forever the way we look at nonlinear systems like the weather.

So everyone is wrong except you? seriously? I thought you once argued about consensus being important in science?

Be a careful observer, and tell me what you see.

I see determinism.

Can you show me a credible source that states that the Lorentz attractor is not deterministic? So far I can find nothing that agrees with you, so what am I to think? everyone except you is wrong?

 

[scruffy]

No you don't, you think it's random and it is not.

There's all kinds of bullshit getting posted on science debate forums too, look FFS, this is the APS, a site you've cited several times yourself.

Here's what they say (and everyone else says) about the Lorentz attractor:

So everyone is wrong except you? seriously? I thought you once argued about consensus being important in science?

I see determinism.

Can you show me a credible source that states that the Lorentz attractor is not deterministic? So far I can find nothing that agrees with you, so what am I to think? everyone except you is wrong?

You're being a leftard idiot.

Those morons would believe anything CNN tells them, even when they can see with their own eyes that bread costs 5 dollars a loaf.

 

[Sherlock Holmes]

You're being a leftard idiot.

Those morons would believe anything CNN tells them, even when they can see with their own eyes that bread costs 5 dollars a loaf.

So no source, nowhere on the worldwide web is there a source stating that chaotic systems are not deterministic, I wonder why that might be...

 

[scruffy]

So no source, nowhere on the worldwide web is there a source stating that chaotic systems are not deterministic, I wonder why that might be...

Because it's self evident.

No one thinks it requires any proof.

 

[Sherlock Holmes]

Because it's self evident.

That's what they said about Fermat's last theorem having no proof, as a mathematician you should never rely on self evident, but proof.

No one thinks it requires any proof.

That's what they said about Fermat's...

 

[scruffy]

That's what they said about Fermat's last theorem having no proof, as a mathematician you should never rely on self evident, but proof.

That's what they said about Fermat's...

We're not talking about proof, doofus.

We're talking about OUTCOMES.

Why can't you predict the outcome of a simple deterministic system?

Duh....

 

[Sherlock Holmes]

We're not talking about proof, doofus.

You asserted the following proposition: "There's nothing deterministic about chaos. That's why it's called chaos."

You said that in this post in the third sentence.

Is there any proof that you're correct? I certainly cannot find any websites on physics or mathematics that agree with what you say.

But I found lots of example of deterministic systems that are chaotic, lots of website that say so too.

What you meant to write was "There's nothing deterministic about randomness. That's why it's called random." if you did not mean that though, then you must explain what you understand the difference is between random and chaotic - can you do that even?

Please, no juvenile name calling, all that does is undermine your already questionable credibility.

Let's try this shall we, which of these assertions are true:

1. There's nothing deterministic about chaos. That's why it's called chaos.
2. There's nothing deterministic about randomness. That's why it's called random.

There are four possibilities here, both are true, none are true, 1. is true or 2. is true - which is it man, please make an effort to actually commit to an answer rather than accusing me of wrong doing for asking.

You should know better than to try to bamboozle Sherlock Holmes, I've dealt in many cases of charlatans in my career, I'm not easily fooled.

 

[scruffy]

You asserted the following proposition: "There's nothing deterministic about chaos. That's why it's called chaos."

You said that in this post in the third sentence.

Is there any proof that you're correct? I certainly cannot find any websites on physics or mathematics that agree with what you say.

But I found lots of example of deterministic systems that are chaotic, lots of website that say so too.

What you meant to write was "There's nothing deterministic about randomness. That's why it's called random." if you did not mean that though, then you must explain what you understand the difference is between random and chaotic - can you do that even?

Please, no juvenile name calling, all that does is undermine your already questionable credibility.

Let's try this shall we, which of these assertions are true:

1. There's nothing deterministic about chaos. That's why it's called chaos.
2. There's nothing deterministic about randomness. That's why it's called random.

There are four possibilities here, both are true, none are true, 1. is true or 2. is true - which is it man, please make an effort to actually commit to an answer rather than accusing me of wrong doing for asking.

You should know better than to try to bamboozle Sherlock Holmes, I've dealt in many cases of charlatans in my career, I'm not easily fooled.

I already gave you the formal mathematical treatment.

ANY system can be treated as a random system.

It is a DESCRIPTION, not a property.

This has been explained to you repeatedly, yet you still don't understand.

This is generally a problem with creationists.

In general.

This kind of behavior is rampant in the creationist community.

You want to argue with me about the "difference" between chaos and randomness. That proves you don't understand either term.

And I simply don't have the time to address your lack of understanding. Sorry.