## Harmony on a Donut

“Harmony” was my sloppy look at the Kuramoto model. Looking at 2 oscillators:

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

Visualizing as motion on a circle:

## Multiplication

This is my homage to Mathologer’s wonderful video. The following is multiplication (2x to 100 x) modulo 200 represented on circle:

## Hewing Prime Ladder from Zeta

## Trade-offs

The brachistochrone problem is classical example used to illustrate the use of calculus of variations. In some ways it illustrates the trade-off between shortest distance (straight line) and greatest speed for a time minimization problem. The solution curve is a cycloid. In addition to the time minimization, no matter where start on the curve you will reach the end point in the same time.

## Breaking Sticks

This post is based on a probability puzzle. What is the probability that breaking a straight stick at 2 random points will result in 3 segments that can form a triangle.

Consider the resulting segment lengths: , . Let the stick length =1.

Now,

To form a triangle

for all

These constraints reduce to:

You can visualize this as follows. As there are only 2 degrees of freedom, consider only space of . The space of allowable is the region of the unit square: . Within this space the values of that form a triangle with lengths is the region satisfying the constraint above.

The probability is then: (1/8)/(1/2)=1/4.

Just for fun:

The light blue triangle is the region of that can form triangle and the pink triangle encloses the allowable stick segments.

The triangles on the right have been rescaled to be circumscribed by the unit circle. The relative lengths pertain not the absolute lengths.

## Australian Mortality Data

The Human Mortality Database provides access to world mortality data. This post looks at life expectancy, hazard rates, survival data for Australia from 1921 to 2011 (from the HMD Australian data).

The following plot shows the time course of “approaching” the 2011 life expectancy curve by area between curves referenced to area between 2011 and 1921 curves. There seems to have been “accelerated” progress around 1970’s.