The Curated Internet

  Here’s a fun little experiment that anyone can conduct at home with a chocolate bar and a microwave oven. Remove turntable from microwave (the plate that rotates) Take a chocolate bar on a plate and place it inside microwave. Heat for for 30-60 seconds on high. You will notice that the chocolate would have melted in some regions and not in others (see image above) . But don’t worry this is supposed to happen                                                  Source   A microwave works by setting up a standing wave inside it. The size of the oven is chosen so that the peaks and troughs of the reflected microwaves line up with the incoming waves and form a “standing wave”. The above is a 1D analog of a standing wave, but a 2D standing wave looks like so: And there are nodes and anti-notes in three dimensions throughout the entire oven. At...

  This leads to Quantization of θ  because there are only specific values that become possible for  θ  when we impose that after n rotations it has to return back to its same starting point. ** And the following are those values: The problem of finding the values of θ for given values of n is more generally known as the N roots of unity.   We will leave it as an exercise, but the following animation plots all the possible the values of θ for integer values of n=2,3,4,.. from the above equation that we found:                             Source And those are the roots of the equation z^n  = z i.e if you start at these points on the unit circle and make n rotations you will get back to the same point that you started with. Have a good one! ** A more physical way to think about this is matching boundary conditions. ...

Below are some of the images from the observations made at the Lick Observatory using the Nickel Telescope during October 2018. (1) - M1 (Crab Nebula) (2) - M15 (3) - M16 (Eagle Nebula- Pillars of Creation) (4)&(5) - M17  (Omega Nebula) (6) - M42 (Orion Nebula) (7) - NGC6563 (Cat’s Eye Nebula) (8) - NGC7662 (Snowball Nebula)        

    Waaay back in 1054, a star in the constellation Tauras exploded in a dramatic fashion. The remnant of that massive explosion (supernova) is what we now call the Crab Nebula. But as a result of the explosion, the nebula is still expanding at a rate of over 1,000 kilometers per second. The above gif comparing the nebula in 1950 vs what it was in 1990 is one of the most epic ways to illustrate that fact.  

It can be casual to forget the magnificence of our planet and get lost in our tight-knit everyday lives. In the advent of a lunar eclipse (January 31, 2018 )  it is worth knowing that when it comes to eclipses, Earth holds a pristine status in our solar system.  To understand why, we need to shift our perspective a little bit and ask -”How would it be like if you were on Io (one of the moons of Jupiter) ?”                                            Image source : Deviant art The most startling thing about this experience would be that the Jupiter would appear 36 times larger than the full moon (from earth).  That’s HUGE!   Also since the moons of Jupiter lies in the same plane, you would be witnessing an eclipse every 42 hours …                   Moons - Io, Ganymede, and Callisto in solar eclipse In ...

  When one searches for Fourier series animations online, these amazing gifs are what they stumble upon. They are absolutely remarkable to look at. But what are the circles actually doing here?   Vector Addition Your objective is to represent a square wave by combining many sine waves. As you know, the trajectory traced by a particle moving along a circle is a sinusoid:   This kind of looks like a square wave but we can do better by adding another harmonic   We note that the position of the particle in the two harmonics can be represented as a vector that constantly changes with time like so:   And being vector quantities, instead of representing them separately, we can add them by the rules of vector addition and represent them a single entity i.e:                                                     ...

  The legend goes something like this: Gauss’s teacher wanted to occupy his students by making them add large sets of numbers and told everyone in class to find the sum of 1+2+3+ …. + 100. And Gauss, who was a young child (age ~ 10) quickly found the sum by just pairing up numbers:   Using this ingenious method used by Gauss allows us to write a generic formula for the sum of first n positive integers as follows: