The Mathematics of Procedural Generation in Video Games. OK-AR Section Meeting of the MAA
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1 The Mathematics of Procedural Generation in Video Games Demitri Plessas Department of Mathematics and Computer Science Northeastern State University, Tahlequah, OK OK-AR Section Meeting of the MAA April 12th, 2014 Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
2 Procedural Generation Procedural Generation is the process of creating content algorithmically, typically from a pseudo-random number generator. Since the 1980 s it s been used in movies, computer graphics, and video games. In 1997 Ken Perlin won an Academy Award for creating the algorithm used for procedural texture generation in movies such as Abyss, The Lion King, and Terminator 2. The Perlin Arch (this is not a 3D rendered model). Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
3 A Brief Visual History Elite - Acornsoft (1984) Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
4 A Brief Visual History Elite - Acornsoft (1984) By the Guinness Book of World Records, Elite is the first game with procedurally generated content. The developers procedurally generated galaxies to explore. They were worried that amount would seem artificial. So they cut it down to 8 galaxies. Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
5 A Brief Visual History Rogue - Epyx (originally made 1980, published in 1985) Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
6 A Brief Visual History Rogue - Epyx (originally made 1980, published in 1985) Dennis Ritchie (the computer scientist who created C) said Rogue wasted more CPU time than anything in history. Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
7 A Brief Visual History.kkrieger - Farbrausch (2004) Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
8 A Brief Visual History.kkrieger - Farbrausch (2004) The game is 96kb in size. The image above is 8 times as large as the game itself. This talk in.pdf form is 56 times as large as the game itself. Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
9 A Brief Visual History.kkrieger - Farbrausch (2004) The game is 96kb in size. The image above is 8 times as large as the game itself. This talk in.pdf form is 56 times as large as the game itself. Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
10 A Brief Visual History Dwarf Fortress - Bay 12 Games (2006) Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
11 A Brief Visual History Dwarf Fortress - Bay 12 Games (2006) Not only is the entire world s terrain, fauna, and weather procedurally generated, Dwarf Fortress also procedurally generates the entire history of civilization. This history includes the lineage of every dwarf the player controls. Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
12 A Brief Visual History Minecraft - Mojang (2011) Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
13 A Brief Visual History Minecraft - Mojang (2011) Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
14 A Brief Visual History Minecraft - Mojang (2011) Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
15 A Brief Visual History Minecraft - Mojang (2011) Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
16 Money is Also Fun While researching video games might seem like a cheap ploy to play video games at work, it is a huge business yielded a $14.8 billion revenue for video games (from the Entertainment Software Association). By February 2014, Minecraft has brought in over $400 million in revenue (from Mojang). Last year, a YouTube star BlueXephos made between $195, 000 and $2, 000, 000 making videos primarily about Minecraft (from Business Insider). Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
17 Cellular Automata One big draw to Minecraft is the ability to mine and explore natural caves. The mathematical tool we will need to generate caves is cellular automata. A cellular automation consists of a set of cells arranged in a graph, usually a grid, that have a finite number of states and a rule that determines what the state of a cell will be in the next discrete time frame. The rules for John Conway s game of life. Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
18 A Example of the Game of Life Short Keys Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
19 A Example of the Game of Life Short Keys Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
20 A Example of the Game of Life Short Keys Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
21 A Example of the Game of Life Short Keys Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
22 A Example of the Game of Life Short Keys Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
23 A Example of the Game of Life Short Keys Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
24 A Example of the Game of Life Short Keys Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
25 A Example of the Game of Life Short Keys Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
26 A Example of the Game of Life Short Keys Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
27 Generating Caves Using Cellular Automata We can think of John Conway s game of life as following the rule that a cell is born if there are 3 live cells next to it, and it survives if there are 2 or 3 live cells next to it. We can encode this as B : 3 S : 2, 3. Conway s Game of life will not work for a cave system. The live cells will die off quickly and we will be left with emptiness. However, we can modify the game of life to generate realistic cave formations with random seed patterns. We would like walls and stalagmites, but open floors inside the cave. So we will set birth to be fairly restrictive. A cell will be born with at least 6 neighbors. Survival should not be as restrictive. A cell will survive with at least 3 neighbors. We will increment the automation 20 times. B : 6, 7, 8 S : 3, 4, 5, 6, 7, 8. Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
28 An Example B : 6, 7, 8 S : 3, 4, 5, 6, 7, 8. B : 6, 7, 8 S : 4, 5, 6, 7, 8. The probability of each cell initially being alive is.5. Special thanks to Jeremy Kun for writing the algorithm I used. Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
29 Issues This process is heavily dependent on the resolution of the seed. B : 6, 7, 8 S : 3, 4, 5, 6, 7, 8 on an increased resolution. This process is also heavily dependent on the initial distribution of live cells. B : 6, 7, 8 S : 3, 4, 5, 6, 7, 8 with.55 probability each cell is initially alive. Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
30 Questions? You can play too!.kkrieger website: Dwarf Fortress website: Ken Perlin s Noise: John Conway s Game of Life: Special thanks to P. Poole for creating a giant dwarven city in Minecraft. Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
31 References C. Bays, A note on the discovery of a new game of three-dimensional life, Complex Systems 4.6 (1990), pp Conway, John H., R. Guy, and E. Berlekamp, Winning ways for your mathematical plays, Vol. 2. (1982). Entertainment Software Association, Essential facts about the computer and video game industry, (2013) web accessed on 4/1/2014, ef 2012.pdf. H. Jacobs, The richest YouTube stars, March 10, 2014, Business Insider, web accessed 4/1/2014, L. Johnson, J. Togelius and G. N. Yannakakis, Cellular automata for real-time generation of infinite cave levels, Proceedings of the 2010 ACM Workshop on Procedural Content Generation in Games (2010). Guinness Book of World Records, web accessed on 4/1/2014, Demitri Plessas (Northeastern State University) OK-AR MAA April 12th, / 17
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