That is the sequence of walking times for the particles.

It grows linearly, as expected for random walks, which grow as the square of the distance, which happens in two dimensions to be the square root of the particle number.

The pointiness in the lower left is because it starts in the corner and only goes up or to the right.

If we can go left, right, or up, then of course the bottom is flat, but the rest looks elliptical.

Here we can go left, left, up, or right.

The pointiness in the upper left corner, and adjacent straight knight-move edges, have no immediate explanation. (No, the blob has not reached the left boundary yet.)

Here we rotate: (up, right, right, left, up, down). There seem to be suspiciously flat parts from the NW side on up around to the E side.

Ellipse for (N,E,E,W,N,S,N,S). There are some suspiciously regular parts.

Ellipse for (NEEWNNS). What a deformity!

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Simultaneous Propagation

ghosts is how many particles have come and gone. ghosts+blob is how many have come.

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Guessing the odometer lump

Oops, a Gaussian was totally the wrong thing....

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Permuting the Stacks

The walking times are much less random for the sorted stacks.

Recall that the same graph for cyclic stacks yielded fuzz ranging roughly from y=0 to y=x.

y is walking time and x is particle number (walk number).

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Rotated Repetition

I don't see any clear pattern to the walk numbers which are a rotation of the previous walk.

Hmm, they barely overlap. Maybe that's why they succeed at matching?

There is quite a bit of rotational symmetry here.

The rotor directions are shown in hues that gradually change, so after 1/4 turn rotors which are also 1/4 turned appear the same color.

Converted by *Mathematica*
July 22, 2002