Langtons ant was originaly created as a cellular automata by Christopher Langton.1 Recently in 2016 a new wave equation has been discovered. This 'One dimensional first order recurrence relation' when iterated over a closed surface, generates Langtons 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.

- The ant movement 'LEFT' or 'RIGHT' represents the positive or negative Electric displacement field of a charged particle. The Coulomb force.
- The ant 'UP' and 'DOWN' movement represents the inductive coupling between adjacent loops.
- The ant Rotation represents the Magnetic spin of the charged particle.
- The change in colour of the lattice square represent a Photon Event (see photo electric effect and radiation pressure ). The sum of these photon events on a given square represents the total Energy of the Photon when it is released as a quantised EM wave.

Copy and paste the following m-file into octave-online.net 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: gmail.com@graham.medland %;************************************************** E=256; A=0.5*(E+1); B=0.5*(E-1); K=pi/4; N=12288; t=1:1:65536; Psi_d(t)=1; k(t)=0; k(1)=32896; 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); Psi_d(k(t+1)+1)=real(exp(2*i*K.*(1+Psi_d(k(t+1)+1)))); endfor; imagesc(reshape(Psi_d,E,E)) %;************************************************* |

Both the octave script above and the maths below use the same variable names, and together they fully descibe 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.

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

Now we define some variables and constants.

Where K represents the orthogonal changes in direction (Electric / Magnetic field lines). | |

and | Are the complex conjugates of and which are the Langtons ant wave functions. |

E = n | (from Eulers identity above), 'E' represents the total number of charged particles in the system. |

This is the permitivity of free space. | |

This is the permeability of free space. | |

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

we define

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

Noting that,

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

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

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

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

Starting with a quadratic equation with and forming the eigen wave functions (roots of the quadratic) and and forming the quadratic term, we expand to get

Ths simplifies to

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

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

Langton, Chris G. (1986). "Studying artificial life with cellular automata".