Lorentz covariance by Ciro Santilli 40 Updated 2025-07-16
Same motivation as Galilean invariance, but relativistic version of that: we want the laws of physics to have the same form on all inertial frames, so we really want to write them in a way that is Lorentz covariant.
This is just the relativistic version of that which takes the Lorentz transformation into account instead of just the old Galilean transformation.
Poisson's equation by Ciro Santilli 40 Updated 2025-07-16
Generalization of Laplace's equation where the value is not necessarily 0.
Light watch transverse to direction of motion. This case is interesting because it separates length contraction from time dilation completely.
Of course, as usual in special relativity, calling something "time dilation" leads us to mind boggling ideas of "symmetry breaking": if both frames have a light watch, how can both possibly observe the other to be time dilated?
And the answer to this, is the usual: in special relativity time and space are interwoven in a fucked up way, everything is just a spacetime event.
In this case, there are three spacetime events of interest: both clocks start at same position, your beam hits up at x=0, moving frame hits up at x>0.
Those two mentioned events are spacelike-separated events, and therefore even though they seem simultaneous to you, they are not going to be simultaneous to the moving observer!
If little clock one meter away from you tells you that at the time of some event (your light beam hit up) the moving light watch was only 50% up, this is just a number given by your one meter away watch!
Cyborg beetle by Ciro Santilli 40 Updated 2025-07-16
Video 1.
Singapore's Remote-Controlled Cyborg Insects by Vice Media (2018)
Source.
By Dr. Hirotaka Sato from Nanyang Technological University Singapore.
It is a bit rough, but kind of works. Flight control is only left or right though, they don't stimulate the wings directly.
Cycle graph (algebra) by Ciro Santilli 40 Updated 2025-07-16
How to build it: math.stackexchange.com/questions/3137319/how-in-general-does-one-construct-a-cycle-graph-for-a-group/3162746#3162746 good answer with ASCII art. You basically just pick each element, and repeatedly apply it, and remove any path that has a longer version.
Immediately gives the generating set of a group by looking at elements adjacent to the origin, and more generally the order of each element.
TODO uniqueness: can two different groups have the same cycle graph? It does not seem to tell us how every element interact with every other element, only with itself. This is in contrast with the Cayley graph, which more accurately describes group structure (but does not give the order of elements as directly), so feels like it won't be unique.
String polarization by Ciro Santilli 40 Updated 2025-07-16
This is about the polarization of a string in 3D space. That is the first concept of polarization you must have in mind!
Given the function :
the operator can be written in Planck units as:
often written without function arguments as:
Note how this looks just like the Laplacian in Einstein notation, since the d'Alembert operator is just a generalization of the laplace operator to Minkowski space.

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
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