Squares and Triangles

November 1, 2018 Leave a comment

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:

 

puzzle.gif

 

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

January 9, 2018 Leave a comment

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

August 6, 2017 Leave a comment

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

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Fun Animation

July 10, 2017 Leave a comment

This is an adaptation of code from:here.

me

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Stretching and Folding

May 2, 2017 Leave a comment


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
msehenon

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

msehenon

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Harmony on a Donut

April 29, 2017 Leave a comment

“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.

msesynchr.gif

Visualizing as motion on a circle:

 
msekuramoto

Knots on a Donut (Doughnut)

April 28, 2017 Leave a comment

Professor Strogatz lecture on quasiperiodicity motivates this post. Just using the simple system:

 \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

msetrefolimsequasiperiod

 

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