LAWE Langtons ant wave equation

The Langtons ant wave equation.

Langtons ant was originally created as a cellular automata by Christopher Langton.1 Recently in 2016 a new wave equation was created by Graham Medland 2. This 'One dimensional first order recurrence relation' when iterated over a closed surface, generates Langton's ant.
The rules of langtons ant can be used to model the behaviour of charged particles under the influence of an electric and magnetic field.

This describes the basic laws of particle thermodynamics, the Langtons ant wave equation bridges the gap between quantum mechanics and 'thermodynamics' , demonstrating that you can iterate a well behaved and defined SHM wave equation and produce a rich source of CHAOS, which when bounded on a closed surface, produces almost Gaussian distributed thermal noise, Watch a quick video of this wave function where you can see the emergent thermodynamic properties.

Langtons ant Octave script

Copy and paste the following m-file into This will produce Langtons ant, feel free to change the Lattice size or number of iterations. You may need to click on the extended timer as it will take about 10 seconds to complete the 12,000 iterations required to produce Langtons ant!

%;* Last edited date: 1st June 2016 Ver 1.0
%;* m-file for langtons ant (closed form)
%;* Written by: Graham Medland
%;* email:
for t=1:1:N;
Theta = trace(diag(Psi_d));
Psi_t = exp(i*K*Theta);
X=(Psi_t + B*Psi_d(k(t+1)+1))*(conj(Psi_t) + A*Psi_d(k(t+1)+1))-(A*B);
k(t+1)=mod(k(t)+round(real( X-1 ) + imag( X-1 )),E^2);


Both the octave script above and the maths below use the same variable names, and together they fully describe the 'Langtons Ant Wave Equation.
Once you are familiar with the mechanics of the wave function you can begin to explore the Langtons Ant wave function using the LCEE online engine, allowing 5.5 x 10^7 ( 55 million ) iterations per second, thats 82 seconds to compute 4,294,967,295 langtons ant steps.

Thermo-dynamic wave equation.

Mapping the ant world to a one dimensional flat torus, using Eulers identity in two line notation we get,

eq 1.0

eq 1.1

This can now be recast using partial fractions into the form.

eq 1.6a

Now we define some variables and constants.
Wave number Where K represents the orthogonal changes in direction (Electric / Magnetic field lines).
Wavey number and wavey Are the complex conjugates of Psi delta and Psi theta which are the Langtons ant wave functions.
E = n (from Eulers identity above), 'E' represents the total number of charged particles in the system.
Epsilon This is the permitivity of free space.
Mu This is the permeability of free space.
i=sqrt(-1) Essential for any self respecting wave equation.

Now We create the first wave equation, this is the complex oscillator, it represents a photon event on the lattice and generates rotatations, substituting photon event 'oscillator'

we define
eq 1.6a

We also create the second wave equation, this is the magnetic torque and is a sum over all paths integral (see Feynman chessboard)
eq 1.7

Noting that,

eq 1.6a

Putting it all together, we get the general solution to a torsion based wave equation.

eq 1.6a

Now we multiply this by the 'photon event' wave equation, giving.

eq 1.6a

After 4 iterations of Psi Left, the function naturally changes direction and has the following equivalent function,

eq 1.6a

And returns back after another 4 iterations, thus the Ensemble of Langtons ant over the entire surface is a superposition of both rotors, which forms the Langtons ant Spinor equation

eq 1.12a

Quadratic form

Starting with a quadratic equation with photon event for magnetic field and photon event for electric field forming the eigen wave functions (roots of the quadratic) and Photon event Oscillator and Photon event Oscillator forming the quadratic term, we expand to get
eq 1.

Ths simplifies to
eq 1.

Therefore the Langtons ant wave equation can be expressed as a quadratic equation.
eq 1.

As an SHM equation we can remove the eq 1.12a, then make the roots complex (unobservable) and now equate both sides to give the Langtons ant 'residual' wave equation.

eq 1.

In my paper on Langtons ant I show how we can derive the next equation, which is the Lorentz contraction or relativistic version. where X represents how much effect the photon produces as it interacts with empty spacetime, and psi delta is the angular velocity of the particles matter wave, which is directly proportional the particles velocity in meters per seconds.

eq 1.

Which is the relativistic equation, from this we can deduce there is a direct correlation between a particles velocity and the matter wave frequency.

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    1. Langton, Chris G. (1986). "Studying artificial life with cellular automata". Physica D: Nonlinear Phenomena 22 (1-3): 120–149. ref:10.1016/0167-2789(86)90237-X.

    2. G. Medland, "Langtons Ant Wave Equation" in: SCTPLS Newsletter, Vol. 26, No. 2 .
    Society for Chaos Theory in Psychology & Life Sciences February 2019