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The last episode of 3Blue1Brown's "lockdown math" series posed the problem: given two numbers, $x$ and $y$, chosen randomly from the uniform distribution between 0 and 1, what is the proportion that their ratio, when rounded down to the nearest integer, is even? That is, what is $P(\lfloor \frac{x}{y} \rfloor \mod 2 = 0)$?

A Babyonian spiral (OEIS A256111) is the figure formed by starting with a zero-vector at the origin and concatenating vectors such that each subsequent vector is the next one longer than the previous one that also lands on a position with integral Cartesian coordinates. That is, the $i$th vector has integer components $x_i$ and $y_i$ satisfying, $x_i^2 + y_i^2 = n_i^2 > n_{i-1}^2$. Each vector is chosen such that it minimizes the angular separation from the previous one.

Following on from this earlier post, here is a class, `TruchetHexes`

, which generates a pleasing weave-like pattern by tiling the following hexagon shapes in random orientations.

The following code implements a Quadtree in Python (see the previous blog post). There are three classes: `Point`

represents a point in two-dimensional space, with an optional "payload" (data structure associating the `Point`

with more information, for example the identity of an object). The `Rect`

class represents a rectangle in two-dimensional space through its centre, width and height. There are methods to determine if a given `Point`

object is inside the `Rect`

and to determine if the `Rect`

intersects another `Rect`

.

Following on from this post about the simple double pendulum, (two bobs connected by light, rigid rods), this post animates the double *compound* pendulum (also called a double complex or physical pendulum): two rods connected to each other, with their mass distributed along their length. The analysis on Wikipedia provides the dynamical equations for the case of equal-mass and equal-length rods. Here, the more general case of rods with lengths $l_1$ and $l_2$ and masses $m_1$ and $m_2$ is considered.