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Dynamics of live_cells_in_conway_game_of_life_

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an idea which could initiate a new direction in scientific research of transport in complex systems

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Dynamics of live_cells_in_conway_game_of_life_

  1. 1. Dynamics of live cells in Conway’s game of Life in the absence of basic physical force laws. A Pan* Department of Physics, Vivekananda College, Thakurpukur, Kolkata – 700063, India Abstract Dynamical properties of basic physics such as rotation and translation can be imitated using the rules followed by the Conway’s Game of life. This thinking will open up the possibility of tackling harder problems of transport and relaxation in complex systems and nanostructures, where the basic interaction could not be exactly defined due to the complexity of the system. PACS 02.50.Le – Decision theory and game theory PACS 89.75.-k – Complex systems *Corresponding author, arvindpan@yahoo.com
  2. 2. Introduction There has been a recent surge in interest from all quarters of scientists, which include physicists, economists, people of artificial intelligence [1 - 3] around the world to a particular mathematical concept proposed by a mathematician John Horton Conway – called the game of life (CGOL) [4, 5], which, seems quite natural in the era of the rise of virtual digital world [6] of games and social networking. This game of Life actually consist of an infinite two dimensional grid of square cells, each of which can be in binary states – alive (1 -state) or dead (0-state). As each cell can have eight nearest neighbor, it can interact with them only obeying the following simple rules: 1) Each live cell with two or three live neighbor cells survives for the next generation (instant of time/click of clock). 2) Each live cell with four live neighbor cells dies of overpopulation, while each live cell with less than two live neighbor cells dies of isolation for the next generation (instant of time/click of clock). 3) Each dead cell with exactly three (no more no less) live neighbor cells gives birth to a live cell for the next generation (instant of time/click of clock). Since its inception it has attracted a lot of attention in various fields like Cellular automaton [7], complex philosophical constructs such as consciousness and free will [8, 9, 10] in spite of being completely deterministic in nature. This deterministic nature had compelled many to compare it with the physical laws governing our universe [11]. RESULT AND DISCUSSION In the present work, it is asserted that we can explain the physical world even without the knowledge of the basic force of interaction. To achieve this, the Conway’s Game of Life is compared with the very basic concepts of Physics such as dynamics. The motto is - If the basic dynamical properties
  3. 3. such as rotation and translation can be imitated without the knowledge of the basic force of interaction (CGOL does not give the force of interaction but states few rules), then many complex phenomena such as transport properties, which include conductivity and relaxation in complex systems [12, 13, 14] and nanostructures [15, 16] can also be predicted to determine the design of future devices. Depending upon the initial condition, many different types of patterns exist for the Game of Life, which can be broadly classified as still life, oscillators and spaceships. With an objective to fulfill our present purpose, an oscillator of period 2, which is called a Blinker is considered in the figure 1. The live cells (1 – state) are shown in green, while the dead (0 – state) ones are shown in black. Let us assume that three live cells are placed horizontally initially ( time = 0). Now the central live cell survives as it has two live neighbors, while the end ones die at time =1. But the dead cells above and below the central one experience a birth obeying the rules of CGOL. The row turn to a column and this goes on repeatedly for the future time sequences. In the language of physics, we generated rotation without the knowledge of dynamics. Let us consider another pattern from the class spaceship, which is termed as a glider, shown in the figure 2. We can observe that, the pattern transforms in various ways for different times. For example from time =1 to time 2, the pattern suffers a clockwise rotation of 900 followed by a reflection about the vertical axis. Similarly the transformation from time = 1 to time =3 can be considered as a counter clockwise rotation of 900 followed by a reflection about the horizontal axis. All these transformations can form a group, which can lead to the emergence of certain kind of symmetry [17]. Again speaking in terms of the language of physics, we generated translation from time = 1 to time = 5. This translation is the resultant of one cell horizontal and one cell vertical translation. For a long duration of time they appear to crawl along the diagonal squares.
  4. 4. CONCLUSION In conclusion we note that the basic two types of motion in dynamics i. e. translation and rotation can be imitated using the Conway’s Game of Life without using the basic physical laws such as Newton’s law. This approach will open up new way of analyzing different complex physical situations even without the knowledge of exact physical laws of interaction (as the exact physical relationship are not always available due to the complexity of the system).
  5. 5. Figure Captions: Figure 1: Blinkers: Evolution of live cells with time generating rotation. Figure 2: Gliders: Evolution of live cells generating Translation.
  6. 6. Fig 1
  7. 7. Fig 2
  8. 8. Reference: [1] S. M. Reia, & O. Kinouchi, Phys. Rev. E 89 052123(2014). [2] D. Bleh, T. Calarco, & S. Montangero, Eur. Phys. Letts. 97 20012(2012). [3] A. R. Harnandez-Montoyaet. al., Phys. Rev. E 84 066104(2011). [4] M. Gardner, Sci. Am., 223 120(1970). [5] http://en.wikipedia.org/wiki/Conway's_Game_of_Life. [6] http://en.wikipedia.org/wiki/Virtual_world [7] A. Adamatzky, (Editor), Game of Life cellular Automata (London: Springer-Verlag)(2010). [8] D. C. Dennett, Consciousness Explained (Boston: Back Bay Books) (1991). [9] D. C. Dennett, Darwin’s Dangerous Idea: Evolution and the Meanings of Life. (New York: Simon &Schuster)( 1995). [10] D. C. Dennett, Freedom Evolves ( New York: Penguin Books) (2003). [11] S. Hawking & L. Mlodinow, The Grand Design (New York : Bantam Books)(2010). [12] A. K. Jonscher, Jour. Of Phys D. 32 R57(1999). [13] A.Ghosh & A. Pan, Phys. Rev. Letts. 84(2000)2188. [14] R. J. Sengwa & S. Choudhary, Indian J. Phys.88, 461(2014). [15] T. Shinada, S. Okamoto, T. Kobayashi, & I. Ohdomari, Nature 437, 1128(2005). [16] A. Pan, Y. L. Wang, C. S. Wu, C. D. Chen, N. W. Liu, J. Vac. Sci. & Tech B 23 2288(2005). [17] Pan, Z., http://arxiv.org/pdf/1003.3394.pdf.

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