the importance of topology

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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

 

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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?

 

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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.