What’s the Magic Behind Graphene’s ‘Magic’ Angle?

That being said, I got halfway through this article and I’m still scratching my head as to what is meant by “twisting” here.

From the discussion of sheets lining up with one another, I assume they mean rotating one sheet relative to the other along a shared normal axis. But when I hear “twisting,“ I think of twisting these sheets like you would do to make a Mobius strip.

Neither the article nor the nice-looking illustration make this clear, so I’m off instead on a Google hunt so that I can conceptualize what’s being worked on here.

Edit: OK, this is pretty cool.

It’s always exciting when new theories are required to explain experimental results— doubly so here since this could yield insight into the physics of superconductivity, which we don’t understand all that well yet.

Moire pattern: https://d2r55xnwy6nx47.cloudfront.net/uploads/2019/04/Magic-...

and https://d2r55xnwy6nx47.cloudfront.net/uploads/2019/04/Graphe...

Stupid question: could one recursively stack and twist these layers and ultimately join the head with the tail while keeping the property of "each layer is twisted in relation to the previous"?

I mean twisting each layer with the magic angle, and adding on top of the previous, and so on, until the very first layer is a suitable successor of the topmost layer.

This kind of stack could be made into a magically twisted graphene donut.

Maybe the construction would be completely useless, but a research paper with a title containing "magically twisted graphene donut" is a win nevertheless.

- What temperature range does this happen at?

- Is the resistance measured edge-to-edge across the sandwich, or between the layers?

https://link.springer.com/article/10.1007/s00004-015-0285-1

Vanderroost, Mike - Morphic Angles (2016)

I wanna know this, among other reasons, because no other paper cites this one, nor does any other paper describe these angles, yet, when one plugs the closest IEEE single/double float numeric approximation of the absolute numeric value of these angles into Google Scholar (in quotes), one can find numerous, but not staggeringly numerous, occurrences of these in the mathematics, physics, & biology literature.

All of these seem to so far have gone unidentified as linked to the rather generalized constructions from the above paper, because the obviousness of how to derive it seems somewhat unidirectional. I'd explain what I mean by that more specifically, but I actually struggle rather hard with finding the right words for it, so, for now I can only suggest trying it out yourself and seeing what you find!

I'll give you the IEEE float approximations I have at hand, which only cover the 2D case derived from the golden ratio, and not the 3D case derived from the plastic number (personally, I'd prefer if we started calling it the radiant ratio, but I don't think that'll catch on). Note that I don't have have single and double float values for them at hand right now, and that if one wanted to thoroughly check the literature for unnoticed occurrences of this, one'd have to factor in:

1. the two possible representations of each type of float. I forgot the term, but, the 'short' version typically returned by most but not all software, and the 'long' version actually stored, which /some/ software actually outputs. Both get accepted as the same value, of course)

2. each form of float rounding behavior

Anyway, here:

golden morphic angles (see the paper for why there exist two, tl;dr: intervals):

Circle, Ellipse:

degrees: 76.34541

radians: 1.3324789

turns: 0.2120706

Ellipse, Parabola, Hyperbola:

degrees: 126.869896

radians: 2.2142975

turns: 0.3524164

And just for completeness sake, the classic (not-generalizing-to-conic-sections) golden angle:

degrees: 137.50777

radians: 2.3999631

turns: 0.38196602