SymPy Updated 2025-07-16
This is the dream cheating software every student should know about.
It also has serious applications obviously. www.sympy.org/scipy-2017-codegen-tutorial/ mentions code generation capabilities, which sounds super cool!
The code in this section was tested on sympy==1.8 and Python 3.9.5.
Let's start with some basics. fractions:
from sympy import *
sympify(2)/3 + sympify(1)/2
outputs:
7/6
Note that this is an exact value, it does not get converted to floating-point numbers where precision could be lost!
We can also do everything with symbols:
from sympy import *
x, y = symbols('x y')
expr = x/3 + y/2
print(expr)
outputs:
x/3 + y/2
We can now evaluate that expression object at any time:
expr.subs({x: 1, y: 2})
outputs:
4/3
How about a square root?
x = sqrt(2)
print(x)
outputs:
sqrt(2)
so we understand that the value was kept without simplification. And of course:
sqrt(2)**2
outputs 2. Also:
sqrt(-1)
outputs:
I
I is the imaginary unit. We can use that symbol directly as well, e.g.:
I*I
gives:
-1
Let's do some trigonometry:
cos(pi)
gives:
-1
and:
cos(pi/4)
gives:
sqrt(2)/2
The exponential also works:
exp(I*pi)
gives;
-1
Now for some calculus. To find the derivative of the natural logarithm:
from sympy import *
x = symbols('x')
print(diff(ln(x), x))
outputs:
1/x
Just read that. One over x. Beauty. And now for some integration:
print(integrate(1/x, x))
outputs:
log(x)
OK.
Let's do some more. Let's solve a simple differential equation:
y''(t) - 2y'(t) + y(t) = sin(t)
Doing:
from sympy import *
x = symbols('x')
f, g = symbols('f g', cls=Function)
diffeq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x)**4)
print(dsolve(diffeq, f(x)))
outputs:
Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)
which means:
To be fair though, it can't do anything crazy, it likely just goes over known patterns that it has solvers for, e.g. if we change it to:
diffeq = Eq(f(x).diff(x, x)**2 + f(x), 0)
it just blows up:
NotImplementedError: solve: Cannot solve f(x) + Derivative(f(x), (x, 2))**2
Sad.
Let's try some polynomial equations:
from sympy import *
x, a, b, c = symbols('x a b c d e f')
eq = Eq(a*x**2 + b*x + c, 0)
sol = solveset(eq, x)
print(sol)
which outputs:
FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a), -b/(2*a) + sqrt(-4*a*c + b**2)/(2*a))
which is a not amazingly nice version of the quadratic formula. Let's evaluate with some specific constants after the fact:
sol.subs({a: 1, b: 2, c: 3})
which outputs
FiniteSet(-1 + sqrt(2)*I, -1 - sqrt(2)*I)
Let's see if it handles the quartic equation:
x, a, b, c, d, e, f = symbols('x a b c d e f')
eq = Eq(e*x**4 + d*x**3 + c*x**2 + b*x + a, 0)
solveset(eq, x)
Something comes out. It takes up the entire terminal. Naughty. And now let's try to mess with it:
x, a, b, c, d, e, f = symbols('x a b c d e f')
eq = Eq(f*x**5 + e*x**4 + d*x**3 + c*x**2 + b*x + a, 0)
solveset(eq, x)
and this time it spits out something more magic:
ConditionSet(x, Eq(a + b*x + c*x**2 + d*x**3 + e*x**4 + f*x**5, 0), Complexes)
Oh well.
Let's try some linear algebra.
m = Matrix([[1, 2], [3, 4]])
Let's invert it:
m**-1
outputs:
Matrix([
[ -2,    1],
[3/2, -1/2]])
Uncomputable function Updated 2025-07-16
The prototypical example is the Busy beaver function, which is the easiest example to reach from the halting problem.
WormWideWeb Updated 2025-07-16
wormwideweb.org/
Browse freely moving whole-brain calcium imaging datasets
INNER JOIN Updated 2025-07-16
Microwave oven Updated 2025-07-16
Video 1.
How Microwaves Work by National MagLab (2017)
Source. A bit meh. Does not mention the word cavity magnetron!
Video 2. . Source. Cool demonstration of the standing waves in the cavity with cheese!
NADP+ Updated 2025-07-16
WebGL Updated 2025-07-16
HTML snippet:
new class extends OurbigbookCanvasDemo {
  init() {
    super.init('webgl', {context_type: 'webgl'});
    this.ctx.viewport(0, 0, this.ctx.drawingBufferWidth, this.ctx.drawingBufferHeight);
    this.ctx.clearColor(0.0, 0.0, 0.0, 1.0);
    this.vertexShaderSource = `
#version 100
precision highp float;
attribute float position;
void main() {
  gl_Position = vec4(position, 0.0, 0.0, 1.0);
  gl_PointSize = 64.0;
}
`;

    this.fragmentShaderSource = `
#version 100
precision mediump float;
void main() {
  gl_FragColor = vec4(0.18, 0.0, 0.34, 1.0);
}
`;
    this.vertexShader = this.ctx.createShader(this.ctx.VERTEX_SHADER);
    this.ctx.shaderSource(this.vertexShader, this.vertexShaderSource);
    this.ctx.compileShader(this.vertexShader);
    this.fragmentShader = this.ctx.createShader(this.ctx.FRAGMENT_SHADER);
    this.ctx.shaderSource(this.fragmentShader, this.fragmentShaderSource);
    this.ctx.compileShader(this.fragmentShader);
    this.program = this.ctx.createProgram();
    this.ctx.attachShader(this.program, this.vertexShader);
    this.ctx.attachShader(this.program, this.fragmentShader);
    this.ctx.linkProgram(this.program);
    this.ctx.detachShader(this.program, this.vertexShader);
    this.ctx.detachShader(this.program, this.fragmentShader);
    this.ctx.deleteShader(this.vertexShader);
    this.ctx.deleteShader(this.fragmentShader);
    if (!this.ctx.getProgramParameter(this.program, this.ctx.LINK_STATUS)) {
      console.log('error ' + this.ctx.getProgramInfoLog(this.program));
      return;
    }
    this.ctx.enableVertexAttribArray(0);
    var buffer = this.ctx.createBuffer();
    this.ctx.bindBuffer(this.ctx.ARRAY_BUFFER, buffer);
    this.ctx.vertexAttribPointer(0, 1, this.ctx.FLOAT, false, 0, 0);
    this.ctx.useProgram(this.program);
  }
  draw() {
    this.ctx.clear(this.ctx.COLOR_BUFFER_BIT);
    this.ctx.bufferData(this.ctx.ARRAY_BUFFER, new Float32Array([Math.sin(this.time / 60.0)]), this.ctx.STATIC_DRAW);
    this.ctx.drawArrays(this.ctx.POINTS, 0, 1);
  }
}
Web of Stories Updated 2025-07-16
Full channel title: "Web of Stories - Life Stories of Remarkable People".
1-2 to hour long interviews, the number of Nobel Prize winners is off-the-charts. The videos have transcripts on the description!
TODO what is their affiliation/who is behind it? There is nothing on the website.
webots.cloud Updated 2025-07-16
Not going to lie, this is some cool shit, robot simulation and 3D visualization in the browser.
Weekend Updated 2025-07-16
Days of the week where you don't do what you set out to do. And yet, it is in those days that you save your sanity, and possibly the world. Wait, this sounds exactly like a week day?
Figure 1.
Calvin and Hobbes "Oh No! I overslept! I gotta get up!" cartoon
. Source.
TODO find/create decent answer.
I think the best answer is something along:
A basic non-precise intuition is that a good model of reality is that electrons do not "interact with one another directly via the electromagnetic field".
A better model happens to be the quantum field theory view that the electromagnetic field interacts with the photon field but not directly with itself, and then the photon field interacts with parts of the electromagnetic field further away.
The more precise statement is that the photon field is a gauge field of the electromagnetic force under local U(1) symmetry, which is described by a Lie group. TODO understand.
This idea was first applied in general relativity, where Einstein understood that the "force of gravity" can be understood just in terms of symmetry and curvature of space. This was later applied o quantum electrodynamics and the entire Standard Model.
Bibliography:

There are unlisted articles, also show them or only show them.