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![]() Computer simulations, which are built on mathematical modeling, are used daily in scientific research of all types, for informing decision making in business and government, including national defense, and for designing and controlling complex systems such as those for transportation, utilities, and supply chains, and so on. Simulations are used to gain insight into the expected quality and operation of those systems and to carry out what-if evaluations of systems that may not yet exist or are not amenable to experimentation.Īs an example, one of the most important and spectacular events in the universe is the explosion of a star into a supernova. Such explosions seeded our own solar system with all of its heavier elements they also have taught us, indirectly, a great deal about the size, age, and composition of our universe. How can you study something that cannot be duplicated in a laboratory, would fry you if you got close to it, and rarely even occurs?īut within our galaxy, the Milky Way, supernovas are exceedingly rare. ![]() That is where mathematical sciences enter the story, via computer simulation. In scores of applications, from physics to biology to chemistry to engineering, scientists use computer models-whose construction requires the formulation of mathematical and statistical models, the development of algorithms, and the creation of software-to study phenomena that are too big, too small, too fast, too slow, too rare, or too dangerous to study in a laboratory. Now an additional rotation will be R, which leads to the projection u P R u. While scientists and engineers have long been able to write down equations to describe physical systems, before the computer age they could only solve the equations in certain highly simplified cases, literally using a pen and paper or chalk and a blackboard. Let u be a vector of the object, then its projection will be u P u. For example, they might assume the solutions were symmetric, or simplify a problem to two or three variables, or operate at only one size scale or time scale. Now, however, the scientific universe has changed. The study of supernovas is a perfect case in point. It is possible to create a rudimentary theory of supernovas by assuming that the star is perfectly symmetrical. Math formula for making a kaleidoscope image. Unfortunately, it doesn’t work: You can’t get a one-dimensional star to explode, and so simulations based on that simplified model cannot represent all of the important aspects of this complex system.Īstrophysicists call this a one-dimensional theory because all of the quantities depend on one parameter, the distance from the center of the star. Of course, real stars are not so symmetric they bulge at the equator, due to rotation. So astrophysicists began to simulate stars with a shape parameter as well as a size parameter, and they called these two-dimensional simulations. The basic function for one circle is a cone with the apex pointing down and compressed up. In order to do what you want, you need to go back to the 3D object, rotate that, and then project down to a 2D picture again.However, such simulations still cannot capture the behaviors of interest: Some fail to explode, while others explode but with less energy than a real supernova. Cybernetic Kaleidoscope 1 A superposition of many circles. ![]() Therefore there is no way to determine, from the picture, how some part of the object will move when you rotate it. But there is no consistent way of determining distance away from the viewer in a 2D picture (although modern software is getting quite good at calculations like this from video). This, again, means that the transformation you're asking for is not a function, and thus cannot be applied in any consistent manner to the 2D picture.Īnother way of looking at it is this: How something in the picture moves when you rotate the object depends on how far away from you it is. Tools to draw graphs or diagrams, and export to SVG or Tikz (Latex) format. That means that two things that had the same coordinates in the picture before the rotation will have different coordinates after the rotation. Access from anywhere via your web browser Very rich sets of symbols, layouts for your mathematics editing Quickly insert mathematic symbols with Suggestion Box (without knowing LATEX) By Name By Category By Drawing. However, once we rotate the 3D object, the two points will separate in the 2D picture. That means that in the 2D picture they have the same position. ![]() The easiest way to see this is to imagine two points of the 3D object such that in the 2D picture one is behind the other. At least not in the sense that you seem to be asking about.
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