## Squares and Triangles

This post is inspired by a puzzle I found on Twitter. I cannot find the source right now.  What is the area of the shaded triangle? The following gif provides the clue:

Note all basic elements are squares. The small upper squares have sides of unit length. Note the whatever the triangle the base as the  upper right unit square diagonal drawn diagonal always has a height of the drawn square with side 2.

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## Fun with Fractions

This post is inspired by tweet from @CutTheKnotMath and his post.

He posed this puzzle for the New Year:

Consider the list: $a_i, 1\le i \le n$

Consider the product: $\prod_{i=1}^n (a_i+1) -1$

For $n\ge 2$, replacing randomly chosen $a_i, a_j$ with $a_i+a_j+ a_i a_j$ in the expression does not change its value as $(a_i+1)(a_j+1)=(a_i+a_j+a_i a_j)+1$. So  repeating till there is one number allows you solve the problem, i.e. process ends a unique number. For the particular sequence this is particularly pleasing: $\prod ^n_{i=1}(1+1/i)-1=\prod^n_{i=1}\frac{i+1}{i} -1= n+1-1=n$.

Simulating a small example with Mathematica:

Categories: Mathematica, Mathematics

## A Bookmark

This post is just a bookmark for a “history of cardiac catheterization”.

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## Fun Animation

This is an adaptation of code from:here.

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## Stretching and Folding

This post explores the Henon map:
$x_{n+1}=y_n+1-a x_n^2\\\\ y_{n+1}=b x_n$

1000 iterations of a point

Exploring the “fate” of a disc after 10 iterations:

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## Harmony on a Donut

“Harmony” was my sloppy look at the Kuramoto model. Looking at 2 oscillators:
$\dot{\theta_1}=\omega_1+ k\sin (\theta_2-\theta_1)\\\\ \dot{\theta_2}=\omega_2+ k\sin (\theta_1-\theta_2)$

The following starts “knotted” 2:3 with $k=0$, i.e. no coupling and then increasing the coupling…another visualization of harmony.

Visualizing as motion on a circle:

## Knots on a Donut (Doughnut)

$\dot{\theta_1}=\omega_1\\\\ \dot{\theta_2}=\omega_2$
when $\omega_1/\omega_2$ is rational the trajectory is closed, when irrational quasiperiodic. In the following the first gif is $\omega_1=2,\omega_2=3$ and the second: $\omega_1=\sqrt{2},\omega_2=2$