Computer Simulations What is the real purpose of computer simulations? a) To discover whether scientific theories really predict what they are supposed to predict. (Or to prove to people that their theories don't really predict what they are intended to.) b) And also to discover if each theory also predicts other observable or testable phenomena c) Also, to help design experiments to distinguish most decisively between alternative competing theories. Few scientists understand these three purposes. * They tend to believe inventors of a theory will automatically know what their own theory predicts. ** ...or that mathematical calculations can find out what any theory predicts. (more easily, and with more certainty) *** Most believe simulations are for producing visible illustrations of a hypothesis working, for teaching purposes.. ------------------------------------------------------------------------------ IN FACT: Inventors of a theory have no better idea than anyone else what results their theory predicts. Calculations require extreme over-simplifications (such as exactly linear proportionalities between variables) Animated drawings produce better visible demonstrations. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Examples of computer simulations, starting with the simplest. Cellular Automata * Each point in space has a numerical value. * These values often just 0 and 1 ("empty" & "filled") * These numerical values are repeatedly recalculated. (In simultaneous "generations") * (Based on the numerical values of a "neighborhood" of nearby points, often including a points own value. For example, diagonal neighbors might be ignored. One dimensional, two dimensional or 3-D automata are equally easy to write programs for. 1&2 are used most. John Conway's famous Game Of Life is a 2-dimensional C.A. Each square has 8 neighbors. (diagonals are counted) The number of "filled" neighbors can be 0,1,2,3,4,5,6,7, or 8. The specific rules of The Game Of Life are these: A square becomes filled if it has 3 filled neighbors. A square remains the same if it has 2 filled neighbors. Squares become empty if they have any other number Of "filled neighbors" i.e. 0,1, 4,5,6,7 or 8. You start the game by filling squares randomly, or in pattern. For several years, more that a third of all main-frame computer time in US and British corporations was used up by professional programmers doing experiments using these rules to generate spatial patterns. (It was a big scandal; cost millions of dollars) They weren't supposed to be doing any such thing. But company administrators had no clue how to stop them. Can you predict what will happen if I start with 3 filled points on a line? (And repeatedly apply the Game Of Life Rules.) What will happen if you start with 4 filled points in a square? What about 4 filled points in a straight line. 5 in a line? 6? 10? What about 6 filled squares in a hexagon? 8 in an octagon? What simple pattern will propagate sideways, as if moving? NOW DO YOU BELIEVE that nobody is quite smart enough to predict mentally the net results of even fairly simple rules? <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><> How many alternative sets of rules can there be? For a 2-D C.A. with 8 neighbors? (Such as Conway's Game of Life?) Is it 49, or 94, or 28, or maybe 49, or 94, or 28 ? Maybe 9! ? Don't worry if you can't figure that out! Just consider my point that such questions are not so easy. (But that definite answers do exist) The answer is > quarter million alternative sets of rules. Cellular automata can do lots of different things. Some are even used for military purposes. Some are "classified" secrets. That's nothing to do with why I am teaching you about them. They are the foundation of all simulations of spatial phenomena. {}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{}[]{} NEXT, imagine a cellular automaton in which the rules allow each point to have any of 10 alternative values: Maybe 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Or maybe A, B, C, D, E, F, G, H, I, J. What happens if the alternative values are 0, 1, 2,3,4,5,6,7,8,9 And if the rules of the game are: Add up the numerical values of each point's 8 closest neighbors Take the average of that sum. Substitute that average in place of the previous number at each point. Intuitively, you might be able to visualize what category of real physical phenomena this cellular automaton simulates. That set of rules is an example of a Finite Element Simulation. ############################################################################# We did not need to limit the numerical values to integers. Those values could just as well have been 1.2345, etc. Or 20.34 Continuous numbers (what they call "floating point") We could have had three different numbers at each point. Points (that have the numerical values) could have been allowed to move. How to simulate pulling or pushing forces distorting the shape of a sheet of cells, or any other object? Changing the rules obeyed by forces can cause an object to change shape spontaneously. This is directly relevant to gastrulation, neurulation, somite formation, shaping of arteries, glands, bones, etc. Measure pulling and pushing forces in organs, tissues, and individual cells. Write simulation programs to predict shape formation by combinations of forces. Demonstrate * That a certain combination of forces is sufficient to cause cells to form each specific anatomical shape. (by a simulation) ** That embryonic cells exert this same combination of forces. (the same as the computer simulation shows is sufficient) ()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()() The big mistake most embryologists have made is to expect that it will be obvious what shapes would be produced by any given combination of forces and signals. I meant for these computer simulations to prove to you that it isn't so obvious what shapes will result from a set of rules. And that computer simulations are the only way to find out.

 

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