# Blog

A blog of Python-related topics and code.

## The Dottie number

The Dottie number is the (real) root of the equation $\cos x = x$. As the only fixed point of the cosine function, it is the number that is converged to by the iterated function sequence $$x, \cos(x), \cos(\cos(x)), \cos(\cos(\cos(x))), \ldots,$$ i.e. it is the number returned if you mash the COS button on your calculator enough (apparently the French professor after whom it is named did just this). Its value is about 0.7390851 and it is transcendental.

## Direct linear least squares fitting of an ellipse

Fitting a set of data points in the $xy$ plane to an ellipse is a suprisingly common problem in image recognition and analysis. In principle, the problem is one that is open to a linear least squares solution, since the general equation of any conic section can be written $$F(x, y) = ax^2 + bxy + cy^2 + dx + ey + f = 0,$$ which is linear in its parameters $a$, $b$, $c$, $d$, $e$ and $f$. The polynomial $F(x,y)$ is called the algebraic distance of any point $(x, y)$ from the conic (and is zero if $(x, y)$ happens to lie on the conic).

## Chaotic Balls

Inspired by this recent Numberphile video, here is a demonstration of chaos in a simple dynamical system: two balls, with near-identical starting conditions, bounce around elastically off a circular wall. After a short time, the balls' trajectories diverge completely.

## Isolines on a nuclide chart

As a short addendum to this blog post, the code below plots a chart of the nuclides showing four types of isoline:

## Nuclear fusion cross sections and reactivities

A quick update on this blog article with some extra cross sections and an additional plot of Maxwell-averaged reactivities. The processes covered are: