Applet
======

.. raw:: html

   <link rel="stylesheet" href="_static/ca_applet.css">
   <div id="ca-app"></div>
   <script src="_static/ca_applet.js"></script>

The library, running live in your browser. The update rules are the same
ones pyCA applies in Python — the lookup-table step of :class:`pyCA.ECA`,
the flipped outputs of :class:`pyCA.NoisyECA`, the broken clock of
:class:`pyCA.AsyncECA`, the heat-bath flips of :class:`pyCA.ICA`, and the
outer-totalistic B/S counts of :class:`pyCA.CA2D` — ported line for line
from the modules they document. Nothing is precomputed: the spacetime
diagram scrolling past is the history of an automaton being run as you
watch.

The applet is yours to take apart. It ships with the library as
`docs/_static/ca_applet.js
<https://github.com/EternalTime/CellularAutomata/blob/main/docs/_static/ca_applet.js>`_:
plain JavaScript with no build step and no dependencies, of which the first
half is the library itself — the rules and the measures, function for
function — and the rest is scenery.

The square panel plots the normalized Lempel--Ziv
complexity\ :footcite:`lempel1976` of
:func:`pyCA.measures.lz_complexity` against
:func:`pyCA.measures.entropy_rate` with :math:`k = 2`, computed on the
current state at every step, on equal axes. Both axes
estimate the entropy rate — :math:`h_2` sees only pair correlations,
:math:`C_{LZ}` regularity at every length — so the diagonal is the
reference, and the plot locates a rule on Wolfram's spectrum at a glance.
A frozen rule pins both measures to zero. A chaotic rule like 30 rides
the top right corner — each new cell is a fresh coin flip even knowing
its neighbor. A structured rule is the interesting case: rule 110 sags
below the diagonal, the gap being exactly the structure longer than two
cells that only the parse can see — the particles and the periodic
background they travel through.

The stochastic variants put a dial on that distinction. Feed rule 110 a
little noise and watch :math:`h_2` climb toward the coin-flip ceiling as
the correlations are eaten; slow the asynchronous clock and watch structure
survive surprisingly far. The Ising automaton stages the competition
directly — at low temperature the heat bath *orders* the lattice and both
meters fall; at high temperature it randomizes, and the rule fights to keep
:math:`h_2` down. For the two-dimensional family, type neighbor counts into
the B and S boxes — Life is B3/S23, and the meters read its thin ash of
still lifes and gliders as low entropy over a mostly empty lattice.

References
^^^^^^^^^^

.. footbibliography::