The Mystery of Langton's Ant and the Cardioid

Directions:
• You can deform the grid by selecting the menu item "Deform Grid/Subdivide." Then by clicking on a quadrilateral with green edges, the program will subdivide the quadrilateral into three quadrilaterals preserving the "checkered" property of the grid. The program will similarly subdivide the quadrilateral attained by reflection across the y-axis.
• You can move around the grid by using the "Change view" menu.
• If you select translate, you can drag the plane around
• If you select zoom, you can click and drag to zoom in or away from the point you first click
• After you have deformed the grid to your liking, click "Ant/Start" to see the grid start to become colored by the Ant
• You can speed up and slow down the Ant using the "Speed" menu
• If you select one of the "Fast" Speeds the Ant will quickly leave your viewing angle. Select "View/Global" to see the Ant's drawing from a distance- each square will appear as a pixel and the green square represents the region you were alowed to deform (after some time, the outline should converge to a cardioid)
How the Ant moves:
• The "Ant" has two properties: a location and a direction, he starts on a quadrilateral just to the "east" of the y-axis facing "east"
• Think of the grid as being numbered by elements of the cyclic group Z4. To denote this numbering, we use the colors [0]=Black, [1]=Blue, [2]=White, [3]=Yellow.
• The Ant proceeds by repeating the following pattern:
• If the square he is standing on is numbered [0] or [1] he turns left and otherwise he turns right.
• He increments the numbering of the square by adding one modulo 4.
• He walks forward.
New Features:
• New Views:
• Immediate Change/Global Immediate Change: Shows the squares of type [1] and [3] in red (the ones that will change next time the ant passes them)
• Visits: Shows the number of times the Ant has visited a square (colored mod 500)
• Visits Under Square Root: Changes what looks like a cardioid into what looks like a circle
• New Deform Grid Options:
• Selecting two adjacent squares after selecting Cut and Reglue (I/II) removes the two squares and reglues them in two distinct ways. (I) is essentially the inverse operation to Subdivision. (II) adds handles to the grid.
• The first four grid coloring options allow you to paint the region you are able to edit in such a way as to not mess up the symmetry of the Ant. You should do any physical grid deformations first (or else symmetry might become messed up)
• The last two coloring options allow for experimentation, but using these will most likely ruin the symmetry.
• Random Jumps:
• If from the Deform Grid menu, you check Random Jumps, when he completes a full loop, he will restart at some other randomly choosen square on the axis of symmetry. (does not work well with deformations yet)
What is known:
• Under these circumstances, the ant returns to the square where he starts facing the same direction as he started infinitely often. At these points when he returns, the coloring of the grid exhibits reflective symmetry.
• The Ant's path is unbounded, thus the pattern cannot repeat.

(*The conjecture:
• Consider any tiling by squares of an oriented once punctured surface that admits reflective symmetry and a checkered pattern. If in a nieghborhood of the puncture, the tiling looks like the ususal tiling of plane by squares with some finite number removed, then the asymptotic limit of the set of all squares visited by the Ant in time t can be taken to be a cardioid. (Probably not rigorously stated)
*)
• This fails when Cut and Reglue II is selected and used to remove the two squares directly above where the ant starts. As below. It also fails if the two squares directly below where the ant starts used to Cut and Reglue II. However interestingly, if both deformations are done at the same time, the Ant draws as usual, only upside down. Returning infinitely often with symmetry is still satisified by these examples.

Here is a good reference on Langton's Ant:
D. Gale, J. Propp, S. Sutherland, S. Troubetzkoy
Further travels with my ant.
Mathematical Intelligencer 17 (1995), no. 3, 48--56.

Among other things, they demonstrate this reflective symmetry property of some Ants. A while back I wrote a similar proof of this fact. It is available here.

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