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Reversing Conway's game of life is famously hard to compute. However, it's possible to approximate the inverse using gradient descent if we formulate GoL as a continuous computation.

Here's how it works. We represent the board as a grid of continuous values in [0,1]. To compute the next step, we first take a convolution with a 3x3 kernel equivalent to the summation of the neighboring living cells. Then we map the result into [0, 1] range using a continuous function corresponding to the alive rule. In this article it uses a narrow Gaussian centered at n=3. Then we can compute the gradient descent to figure out an approximation of inverse step.

Rotors are a rotation representation generalized from Quaternions and Complex numbers that works in any numbers of dimensions.

The article starts by pointing out that rotation is defined as a scalar property on a plane. The canonical representation of a plane is a bivector representing the outer product of two vectors. The bivector is in principle a fundamental concept just like the vector. Similar to how a vector represents a point with a scalar length, a bivector represents a plane with a scalar area (signed). The area of a bivector a∧b is |a||b|sin(angle). The reason we also see the term |a||b|sin(angle) from cross product is because cross product actually gives rise to a bivector instead a vector! Historically we've been confusing bivectors with vectors because they have the same representation in 3D.

The inner and outer product completes a geometric product of two vectors: ab=a⋅b+a∧b. Reflection R(a,v) is defined elegantly using geometric product: R(a,v)=-ava. Then it's noted that two reflections is equivalent to a rotation by twice the angle between a and b: R(a,R(b,v))=ba v ab. The "ab" here is known as a rotor. Applying a rotor on both sides of a vector rotates this vector in the plane a∧b by twice the angle between a and b. Quaternion is just a representation of Rotors in 3d: i:=y∧z, j:=z∧x, k:=k∧y. The scalar part (w in w+xi+yj+zk) corresponds to the inner product.

Keywords: bivector, geometric product, rotor, quaternion, rotation

An interactive introduction to four-color problem and zero-knowledge proof.

This article talks about PGA (projective geometric algebra), an alternative approach to do geometric transformation that is not using matrix.

A function graph plotter to explore complex functions by drawing on one complex plane and see the output on the other.

If you have two source of measurements, how can you combine them to get a more accurate measurement?

In short, you take the weighted average with the weight=1/variance.

Please also read https://news.ycombinator.com/item?id=36971975 for the missing (but important) bits.

A comprehensive note on functional analysis.

Nice visualization of dynamic math. Attractors, vector fields, fractal, flocking, and more.

An article about a fundamental philosophical question: why is there anything, instead of nothing?

Following deep down the rabbit hole, we can see that nothingness is actually not that trivial. Something naturally arise easily from nothingness. Logic, math, conciousness all have their root from nothing and yet are no nothing. A quite inspiring read.

Oversimplified:

- The period of x mod N, x2 mod N, x3 mod N, x4 mod N, … sequence can reveal the factors of the product of two prime numbers.
- The period is very difficult to find for large numbers.
- Quantum Fourier Transform enables the extraction the period in a fast way. (the only part that quantum mechanics involved and I still can't understand how)

A short summary on the method:

- construct an infinite sequence of numbers [f(0), f'(0), f''(0), ...] to represent a function f (note the the boundary conditions are specified directly)
- the sequence can constructd by self-reference (e.g. y = y'), compsition (+, -, negate, *, diff), analytic function composition (g f = g fa :> g' f * f', for log, exp, trig)
- By Taylor expansion, f(a+x) = sum(f[k] * x^k / k!), this allows us to evaluate f at any x.

Ask questions while you read mathematics grants deeper understanding. There is no point in aiming for speed. You must learn to read to learn how to write articles about mathematics. The article also includes a detailed example on writing about birthday paradox.

Another enticing list to dig into!

Amazing typesetting and visuals. How nice would it be if I have access e-books like this when I learn linear algebra.

Animated guide on tying various kinds of knots.

Set a function, draw on the z plane and watch the mapping out on the w plane.

<blockquote>starting point and ending point are</blockquote>