Langton's Ant

Langton's ant is a two-dimensional Turing machine with a very simple set of rules but complicated emergent behavior. It was invented by Chris Langton in 1986 and runs on a square lattice of black and white cells.^[1] The universality of Langton's ant was proven in 2000.^[2] The idea has been generalized in several different ways, such as turmites which add more colors and more states.

Contents 1 Rules 2 Universality 3 Extension to multiple colors 4 Extension to multiple states 5 Extension to multiple ants 6 See also 7 References 8 External links

Rules

Animation of first 200 steps of Langton's ant

Squares on a plane are colored variously either black or white. We arbitrarily identify one square as the "ant". The ant can travel in any of the four cardinal directions at each step it takes. The ant moves according to the rules below:

• At a white square, turn 90° right, flip the color of the square, move forward one unit
• At a black square, turn 90° left, flip the color of the square, move forward one unit

These simple rules lead to surprisingly complex behavior: after an initial period of apparently chaotic behavior, that lasts for about 10,000 steps (in the simplest case), the ant starts building a recurrent "highway" pattern of 104 steps that repeat indefinitely. All finite initial configurations tested eventually converge to the same repetitive pattern suggesting that the "highway" is an attractor of Langton's ant, but no one has been able to prove that this is true for all such initial configurations. It is only known that the ant's trajectory is always unbounded regardless of the initial configuration^[3] - this is known as the Cohen-Kung theorem.^[4]

Langton's ant can also be described as a cellular automaton, where most of the grid is colored black or white, and the "ant" square has one of eight different colors assigned to encode the combination of black/white state and the current direction of motion of the ant.

Universality

In 2000, Gajardo et al. showed a construction that calculates any boolean circuit using the trajectory of a single instance of Langton's ant.^[2] Thus, it would be possible to simulate a Turing machine using the ant's trajectory for computation. This means that the ant is capable of universal computation.

Extension to multiple colors

Greg Turk and Jim Propp considered a simple extension to Langton's ant where instead of just two colors, more colors are used.^[5] The colors are modified in a cyclic fashion. A simple naming scheme is used: for each of the successive colors, a letter 'L' or 'R' is used to indicate whether a left or right turn should be taken. Langton's ant has the name 'RL' in this naming scheme.

Some of these extended Langton's ants produce patterns that become symmetric over and over again. One of the simplest examples is the ant 'RLLR'. One sufficient condition for this to happen is that the ant's name, seen as a cyclic list, consists of consecutive pairs of identical letters 'LL' or 'RR' (the term "cyclic list" indicates that the last letter may pair with the first one.) The proof involves Truchet tiles.

Some example patterns in the multiple-color extension of Langton's Ants:

RLR: grows chaotically. It is not known if this ant ever produces a highway.	LLRR: grows symmetrically.	LRRRRRLLR: fills space in a square around itself.	LLRRRLRLRLLR: creates a convoluted highway.
RRLLLRLLLRRR: creates a filled triangle shape that grows and moves.

Extension to multiple states

Main article: Turmite

A further extension of Langton's Ants is to consider multiple states of the Turing machine - as if the ant itself has a color that can change. These ants are called turmites, a contraction of "Turing machine termites". Common behaviours include the production of highways, chaotic growth and spiral growth.^[6]

Some example turmites:

Spiral growth.	Semi-chaotic growth.	Production of a highway after a period of chaotic growth.	Chaotic growth with a distinctive texture.
Growth with a distinctive texture inside an expanding frame.	Constructing a Fibonacci spiral.

Extension to multiple ants

Multiple Langton's ants can co-exist on the 2D plane, as long as there is a rule for what happens when they meet. Ed Pegg, Jr. considered turmites that can turn for example both left and right, splitting in two and annihilating each other when they meet.^[7]

See also

• Langton's loops
• Paterson's worms
• Turmites
• Conway's life

References

1. ^ Langton, Chris G. (1986). "Studying artificial life with cellular automata". Physica D: Nonlinear Phenomena 22 (1-3): 120–149. doi:10.1016/0167-2789(86)90237-X. http://hdl.handle.net/2027.42/26022.
2. ^ ^{a b} Gajardo, A.; A. Moreira, E. Goles (15 March 2002). "Complexity of Langton's ant". Discrete Applied Mathematics 117: 41–50. doi:10.1016/S0166-218X(00)00334-6. http://www.dim.uchile.cl/~anmoreir/oficial/langton_dam.pdf.
3. ^ Bunimovich, Leonid A.; Serge E. Troubetzkoy (1992). "Recurrence properties of Lorentz lattice gas cellular automata". Journal of Statistical Physics 67: 289–302. doi:10.1007/BF01049035.
4. ^ Weisstein, Eric W., "Cohen-Kung Theorem" from MathWorld.
5. ^ Gale, D.; J. Propp, S. Sutherland, S.Troubetzkoy (1995). "Further Travels with My Ant". Mathematical Entertainments column, Mathematical Intelligencer 17: 48–56. http://www.math.sunysb.edu/cgi-bin/preprint.pl?ims95-1.
6. ^ Pegg, Jr., Ed. Turmite. From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Turmite.html. Retrieved 15 October 2009 .
7. ^ Pegg, Jr., Ed. "Math Puzzle". http://www.mathpuzzle.com/26Mar03.html. Retrieved 15 October 2009 .

External links

Wikimedia Commons has media related to (category): Langton's ant or Turmite

• Weisstein, Eric W., "Langton's Ant" from MathWorld.
• Online demonstration of Langton's ant
• Generalized Ants
• Java applet with multiple colours and programmable ants
• Langton's ant in 3-D (examples and small demo program)
• Mathematical Recreations column by Ian Stewart using Langton's Ant as a metaphor for a Theory of everything. Contains the proof that Langton's ant is unbounded.
• Java applet on several grids and editable graphs, it shows how the ant can compute logical gates
• Programming Langton's ants in Python using Pygame.
• A video demo of different multiple-color Langton's Ants
• Golly script for generating rules in the multiple color extension of Langton's ant

Retrieved from "http://en.wikipedia.org/wiki/Langton%27s_ant"

Categories: Artificial life | Cellular automaton rules | Turing machine

from Wikipedia

The LHC's First Big Discovery: Stephen Colbert is Wonderfully Insane

Science has spoken. Stephen Colbert is wonderfully nuts. Here is your proof, of Stephen Colbert's testimony before the United States Congress, for real:

No, of course it has nothing to do with the LHC. My bad, huh?

Yes, but so what? Pay attention to what the man says, and if it is at all humanly possible, would someone please explain to me what "a Brazilian" is?

I thank you, in advance. Unless of course "the Brazilian" is a "naughty" thing, in which case do not comment.

But I am not immune to receiving private e-mails. I have my e-mail address in the header of this site. I ain't stopping you bros and sisses, if you wish to respond. I suffer untold spam from West Africa to do so, trust that. Please make said suffering so worthwhile by responding, and again I thank you in advance.

For those who are even nerdier than I am (very impressive to me if so) and thereby wish to know "New Physics" from the LHC, I offer up this link by Tommaso Dorigo.

But at the end of the day? I still want to know what "the Brazilian" means. ;-p

Geometry - A Well-Received New Book

I confess, I am a Geometer at heart. I am not so just for the most obvious reason, that a picture is worth a thousand words, because there is much more to Geometry than that (Algebraic Geometry and Differential Geometry, for starters).

But perhaps most of all, because it is the oldest of The Fields of Study, and I am totally into the "origins" of things.

Consider: Socrates taught Plato taught Aristotle, and Aristotle's great contribution was to give us: Logic. A.k.a.: Philosophy. There is way too much "Pop Philosophy" in the world, but REAL Philosophy should be called: Logic.

Logic almost immediately birthed two new fields of study (while maintaining itself as an end in itself, most recently advanced thanks to Computer Science): Natural Philosophy (Science) and Mathematics. We wouldn't have the world we live in today without them. And the oldest Science is Physics. All the other sciences are based on it.

Before Logic though, we had Geometry. The Ancient Egyptians for example didn't know Mathematics, but they did know Arithmetic (at which they were experts in spite of 10 different systems of measurement, and we complain about having two?), a Geometry-based field of study. Geometry is natural, all of us humans do it in our heads, even those so impoverished they never went to school. Indeed, it provides the base of virtually all thought that came after it.

Enter the Chinese-American Mathematician: Shing-Tung Yau. He has written a new book, pictured above. Incredibly, although it has only 12 reviews at Amazon, all 12 are 5-stars, the highest rating! Have you ever seen anything like that? It's written for the general public, so don't be shy to at least investigate it, if not buy it.

I haven't read it, but former Particle Physicist Peter Woit of Columbia Math has: Here is his review.

More on this interesting book later, after I purchase and read.

First things first though, I have to find a job. Rutgers is killing me via the 2 kids the Mrs. and I are sending there.

Ciao.

UPDATE (Oct. 2, 2010): Alan Boyle at Cosmic Log posted a good interview with Shing-Tung Yau, here.

The Sad Demise of Reggie Garrett, Great American

There aren't THAT many people who are both the star athlete in their high school and a straight-A student as well. They are not only genetically gifted, but combine those gifts with the proper attitude and work ethic such that they are primed to be tomorrow's leaders, and it is hoped will fight the good fight, that is to fight the bastards who seek to enslave us all, as least economically. Reggie Garrett, a master of Physics in every way it can be mastered, was one such person, but we've lost him, as the following CNN article explains. Rest in Peace, Reg, and our most heartfelt sympathies to his surviving family members and friends. Most of us will never know Reggie, but those of you who did know him were blessed that way to know a great potential future American leader during his all too brief walk in the sun. Keep his memory alive, and encourage the young among you to follow in his footsteps, is the best any of us can do. Special props to his mother, who obviously raised him well. There's another great American among us. Keep her warm.

High school player collapses, dies after throwing TD
From Rick Martin, CNN
September 18, 2010 5:45 a.m. EDT

(CNN) -- A Texas high school football quarterback died Friday after collapsing during a football game, a hospital official said.

Reginald Garrett was rushed to Baptist Orange Hospital of Southeast Texas, but he did not survive, said Susan Courtney, a hospital spokeswoman.

Garrett, a senior at West Orange Stark High School, collapsed shortly after throwing his second touchdown of the night, CNN affiliate KBMT reported.

Courtney said Garrett wasn't just a football player, but "the star football player" and a straight-A student.

Cornel Thompson, one of the coaches of the football team, told KBMT players were devastated.

"I've coached this game for 40 years and football really isn't important is it? When something like this happens," he said. "You talk about a great kid, friend and teammate. These kids all followed him, you know. It's a shocker."

Garrett had a history of seizures and coaches told KBMT they believe he may have had a seizure Friday night.

Fans poured into the hospital's lobby and waiting room, wailing and falling to their knees when they learned of Garrett's death, Courtney said.

"Most of the community was here at the hospital," she said. "There was hundreds of people in the parking lot, there was people in the waiting room, cheerleaders, drill team, band members, community support by the hundreds."

Lunar Colony - SPACE.com's Interesting Poll

It doesn't look like America will be going back to the moon any time soon, but that doesn't mean China won't! So for the future lucky taikonauts who will return, where should they land next?

I have long recommended going first to the caves, to free any future colonists from solar radiation and cosmic rays, first, and second to wherever the water is (and if there is, there won't be much of it). The consensus seems to be going to the polar regions.

In fact there are a lot of opinions, both pro and con, in the wonderful replies at this forum thread and poll at SPACE.com.

The usual arguments for going to Mars first (no) can usually refuted by the following little facts:

1) Mars is 40,000,000 miles away.
2) The Moon is 210,000 miles away.

Both are inhospitable before the application of human ingenuity, but guess which one will cost less?

I close with a picture of the little-seen far side of the Moon: