How nano- and quantum technology, mathematics, and geometry are working together?

    How nano- and quantum technology, mathematics, and geometry are working together? 

Hofstadter's butterfly 

Researchers found Hoftadter's butterfly from the graphene. Hoftadter's butterfly is a butterfly-looking geometrical structure. Researchers can use that kind of structure to calculate the positions of the qubits. Or, sharper saying Hoftater's butterfly can be an effective tool for modeling the point where binary data transforms into the qubit. 

The area of Hofadter's butterfly can tell what is the right distance between the transmitter that transmits information into qubit. In that model, the qubit is multiple Hofstadter's butterflies that can transport information into the sensors. 

When energy hits to layer it can make Hofstadter's butterfly. The outside force can form that butterfly simultaneously if some force at corners pulls an energy field in that form where a circular energy field forms. That can used in a system that turns binary data into qubits. 

"Rendering of the butterfly by Hofstadter" Wikipedia/Hofstadter's butterfly

"Example of non-integer dimensions. The first four iterations of the Koch curve, where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses the scale factor (S = 3) and the number of self-similar objects (N = 4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26." (Wikipedia,Hausdorff dimension)

What would somebody do with the information about overlap points and lines? 

Or, What is the minimum mass of dust that can cover the entire paper? 

Do you know what is the Hausdorff's dimension? That commons the term dimension, which means Hausdorff's dimension can calculated and determine how much some group or pattern fills in dimensions. Hausdorff's dimension is the same thing, without depending on space or dimension 2 or 3D. 

"Imagine an endless piece of blank paper covered with a smattering of lines pointing every which way. A gust of wind comes and sprinkles dust on top of the paper — in effect covering the lines with points. Say a helpful mathematician tells you how much dust covers any one line. Based on that one piece of information, can you figure out how much dust is there in total?" (BigThink.com/Mathematicians Cross the Line to Get to the Point)

Another way to ask that thing is, what is the minimum number of sand bites that can cover the entire area? And what is the minimum number of lines that can connect them? 

What would somebody do about information about the distances of the lines and points? Or sharper what would somebody do about information about the minimum number of lines that are connecting a certain number of points that are randomly at level? 

And in that case, those points don't form stable geometrical structures. That information is one of the mathematical problems, and it is important when particles that form a system communicate with each other using coherent communication tools like lasers. This is one of the things that the modern technology turns interesting. 

When researchers create smaller and smaller quantum-scale structures they must have something that moves objects. The line can symbolize a laser- or other energy beam, and the point could be a particle that the system moves. 

When we think about the material and its smallest particles, we face the situation that every single particle is in its ball. The truth is that the quantum field around the particle is not the ball. It is a structure that form changes when electrons are changing their place around the atoms. 

That is the thing that makes it hard to make precise calculations about quantum gravity and extremely small-scale interactions. And those interactions are the most important things in quantum-scale technology. 

https://www.quantamagazine.org/a-mathematicians-guided-tour-through-high-dimensions-20210913/

https://www.quantamagazine.org/mathematicians-cross-the-line-to-get-to-the-point-20230925/

https://scitechdaily.com/ancient-graphite-reveals-a-quantum-surprise-scientists-discover-hofstadters-butterfly/?expand_article=1

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

https://en.wikipedia.org/wiki/Hofstadter%27s_butterfly

Are Ley-lines made accidentally, or are they made on purpose?

Image) The Malvern Hills in the United Kingdom, said by Alfred Watkins to have a ley line passing along their ridge (Wikipedia/Ley line)

Have you ever heard Ley lines? The fact is that many historical monuments are in straight lines. And the reason why those lines are made is a mystery. Are those lines made in purpose? Or is the location of those landmarks only an accident?

The Ley-lines are historic landmarks built into straight lines. Those landmarks are built at different times. And the question about those lines is are they built in purpose? Or are those locations in straight lines only a coincidence? The fact is that even if those historical structures are the built lines in purpose that thing might be made for some kind of positioning. 

We don't know, are Ley-lines made in purpose? 

1) If those Ley lines are made in purpose. The reason for that is that the old artificial landmark offers a good spot for surveyors. That point gives surveyors possible to put the buildings in the right place. 

2) If those lines are accidents. There is the possibility that the underground water flows are making the ground easier to dig in those lines. 

3) The Ley-lines might be shown route from somewhere to somewhere. They might be made landmarks for people who carried some documents or something like that. 

4) The most unbelievable explanation for the Ley-lines is that they are made for pointing the route of plasma balls. In that quite interesting idea, the plasma ball would make by using mirrors that heat the air. And then that plasma ball would start to travel in those lines. But as I wrote earlier that thing is theory. 

Like everything else that is written about this kind of thing. Nobody is sure why the Ley lines are made. Those lines can be a coincidence. Or they can be made for some unknown purpose. Maybe that purpose is to make the line that the traveler can follow. Or maybe those things are only landmarks for surveyors. Or maybe they are made above underground rivers. 

And that rising water keeps the air humid in those buildings. Or water ions that are mixed with some other ions are making the air easier to breathe inside those buildings. 

The old historic structures are offering a good landmark for surveyors. And that can give the natural explanation for those mystery lines. The fact is that the lines are true. But the thing that is a mystery is the reason why those sines are built. Are they built because the surveyors used some prehistoric monuments as a landmark? Or is there some other reason for that?

There is one non-proven claim about those lines.

Did somebody use them for creating electricity?

There is the possibility to make electricity by benefiting the magnetic field of Earth. The idea is that the long antenna can pull through the air. And the magnetic field of Earth inducts electricity into that antenna. In history, there was a radio receiver called "crystal radio". Those crystal radios were very popular before electricity turned more common. 

Crystal radios didn't need any other energy source than radio waves.  When electricity turned more common the amplifier radio ousted crystal radio. In amplifier radio. The sound was better than in crystal radios. And that made them more popular. But there is the possibility to make high-voltage electricity by using crystal radios. Those radios can connect to series or side-by-side. 

The crystal radios are turning topical again because of nanotechnology. The miniaturized crystal radios can use as a power source for nano-sized machines. And they can use to receive remote-control signals to those nanorobots. In nanotechnology, those systems can turn radio waves into electricity. And that electricity can use in nano-size engines and computers. 

Famous inventor Nikola Tesla introduced the idea that electricity can make by using Earth's magnetic field. In Tesla's idea, the only needed thing was the antenna that is long enough. So did those Ley-lines made to create electricity? The thing that is needed is long gold or iron wire. That inducts electricity from the magnetic field. 

One of the most interesting things in the claim is that the fire phenomenon is more common in the Ley lines than in other places. The theory claims that those lines are built for following the magnetic power lines of the Earth. That claiming might be just a coincidence. But in some theories, inside the stones of historic structures is iron crystals that can cause those fire phenomena. 

The idea is that. When the iron sticks are in a north-south position. The magnetic field of Earth will make them magnetic. And when the magnetic field is turning strong enough it starts to expand in that system. Sooner or later those magnetic crystals are starting to send the radio waves. If they cannot erupt their electric load to the ground. 

Those crystals can collect electricity from the magnetic field of Earth. Then they can send the radio waves through the air. If positive and negative radio waves are impacting each other. They are creating a stable electric channel. Then the magnetic field of Earth will pump energy to that radio bridge and sooner or later forms a plasma. The plasma ball travels in the magnetic power lines. The thing is that the ionized air between two towers is making perfect conditions for making the eruption channel to plasma ball. But can that thing explain Ley-lines? 

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

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

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

Image: https://en.wikipedia.org/wiki/Ley_line