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

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