diffraction colors

peacock https://www.atoptics.co.uk/fz777.htm butterfly https://www.atoptics.co.uk/fz445.htm https://www.researchgate.net/blog/post/how-nature-uses-physics-to-create-blue http://www.yalescientific.org/2013/05/qa-what-causes-iridescence/

Vortex formation

http://hebergement.u-psud.fr/joovon/teaching/old/2012-2013/M2Nanomagn_Lecture6.pdf Vortex formation and dynamics of defects in active nematic shells https://iopscience.iop.org/article/10.1088/1367-2630/aa89aa The helicity and vorticity of liquid-crystal flows https://royalsocietypublishing.org/doi/full/10.1098/rspa.2010.0309 Static and dynamic behavior of the vortex– electrohydrodynamic instability in freely suspended layers of nematic liquid crystals https://aip.scitation.org/doi/pdf/10.1063/1.445600?class=pdf circular flow triggered by laser https://pubs.rsc.org/en/content/articlelanding/2016/sm/c5sm02098k#!divAbstract

用mendeley做書目引用

Mendeley搭配WORD寫論文 http://tul.blog.ntu.edu.tw/archives/5658 用Mendeley加入Word的書目，如何更換引用格式？ http://tul.blog.ntu.edu.tw/archives/5666 用Mendeley寫論文的最後一步：移除變數 http://tul.blog.ntu.edu.tw/archives/5776

機器人夢遊症

https://www.youtube.com/watch?v=VW7__lZsrZc

Fresnel Diffraction

http://www.arne-lueker.de/Objects/archives/MathcadMatlab/Fresneldiffration/Fresnel.html

有趣光學現象

https://www.itp.uni-hannover.de/fileadmin/arbeitsgruppen/zawischa/static_html/strange2.html

https://www.itp.uni-hannover.de/fileadmin/arbeitsgruppen/zawischa/static_html/strange2.html

Feynman Diagram

https://www.quantamagazine.org/how-feynman-diagrams-revolutionized-physics-20190514/

https://www.quantamagazine.org/why-feynman-diagrams-are-so-important-20160705/

Good article in physical meaning of topologically invariant

Dynamics of the physical systems are governed by differential equations. As such, usually it means that evolution is a continuous and unique function of initial or boundary parameters, and time. Trajectories do not cross, different solutions do not coincide etc. Topology is a part of mathematics which deals with continuous changes of objects. For example you may have a piece of paper and fold origami. As long as you do not use scissors and don't tear off paper, you are doing continuous transformations. Maybe even better you should think about rubber origami stretched and compressed, but not glued or teared. In such systems, if you start with some peculiar topological configuration, for example a ring ( think: closed, periodic trajectory) it may change very much, becoming even complicated and chaotic, but it is still a ring, folded in very complicated way. In fact one of the simplest topological conserved number is winding number counting how many times ring is winding itself ( think about the system where in the centre of the rings we are talking about is something which cannot be passed, like a point of zero energy in the system which cannot relax its energy to environment for example. Remember: we are talking about phase space, not configuration space! ). In such systems, winding number, let's say equal to 1, means trajectory revolves one time around such centre in phase space. Winding number equal to 2, revolves 2 times etc. It is possible, that different winding numbers means different energy! For example you can think about a steel belt. Winding means it form kind of helix. If winded enough time it may have some energy. Definitely different winding numbers will have different amounts of mechanical energy stored.

But as you agree that you have to wind it particular number of times ( probably because you have to tie its ends in some particular way, preventing unwinding) this system will have discrete energy spectrum! And energy bands separated one from another.

So topological invariants can lead to separation of trajectories in phase space, in such way that some additional order is created. It even is called topological phase transition, because if dynamics is changing, this order may break giving drastic change of system behaviour.

The band structure you are mentioned is one of the signals of such additional order. It may show as energy, momentum, space etc ordering/ bands.

Very similar phenomena can occur in crystal networks, semiconductors, quantum systems, nonlinear dynamics, polymer dynamics, low temperature phases where energy per particle is so small and systems are isolated so it cannot break quantum engagement etc

The most important thing is that topological conserved quantities usually are independent of various physical characteristics, depending usually only on very basic structure of the system: it's topological configuration. And of course, usually this depends on continuous relations defined in the system. If any of the parameters became non continuous for any reasons, topological conservation breaks.

https://physics.stackexchange.com/questions/373630/physical-meaning-of-topological-invariant/373684#373684

舉例寫法

https://www.hopenglish.com/how-to-use-for-example-such-as-and-like

Noether Theorem與守恆定律的根本原因

https://www.sciencenews.org/article/emmy-noether-theorem-legacy-physics-math

淺白數學講conservation law

https://www.hindawi.com/journals/amp/2016/9679460/

Heart-Shaped Faraday Motor

https://twitter.com/ilufie/status/1119623010497904640

short- and long-range order

short-range order: a few atoms (like unit cells)
long-range order: (like crystals) "lattice periodicity implied long range order"  知道一個就可以知道全部 ==> transnational symmetry

rotational symmetry???
Crystallographic restriction theorem
https://en.wikipedia.org/wiki/Crystallographic_restriction_theorem
旋轉對稱和平移對稱要相容
通常只會看到二重, 三重, 四重, 六重對稱

lattice structure ==> structure order
beside structure order
we have
charge ordering
spin ordering
magnetic ordering

細胞為什麼要排列成六角形或sine wave型

On growth and form



https://www.amazon.com/Growth-Form-DArcy-Wentworth-Thompson/dp/0521437768

https://blog.stephenwolfram.com/2017/10/are-all-fish-the-same-shape-if-you-stretch-them-the-victorian-tale-of-on-growth-and-form/

http://dev.biologists.org/content/144/23/4386

Crystalline Topological Phases

" The defect network picture works both for symmetry-protected topological (SPT) and symmetry-enriched topological (SET) phases, in systems of either bosons or fermions. "

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.99.115116

Classification of topological crystalline phases

https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.011040