rantonse.no - June 7, 2016

Fixed Point Fascination | Roger Antonsen

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Fixed Point Fascination | Roger Antonsen Roger Antonsen homeabouttalksteachingresearchbookspressblogartENNO Magical PatternsJune 7, 2016Fixed Point Fascination (Published in Aftenposten Tuesday June 7, 2016, in Norwegian.) I am infinitely fascinated by so tabbed stock-still points. My fascination does not stem from any particular mathematical result or fact, but rather from the eyeful of the concept itself. Even though it is an extremely useful concept both in mathematics and computer science (it is for example used in the wringer of programming languages), it has an intrinsic value and weft that I am tightly infatuated with.It is a good example of a mathematical concept from "higher" or increasingly "advanced" mathematics that is utopian and powerful, yet not particularly nonflexible to understand. In the video above, I try to explain in simple terms what a stock-still point is. Have a look!What is aStock-stillPoint really?Imagine that you have a hammer, a nail and a wall. We can view the act of hammering the nail into the wall as the using of a mathematical function. Eventually, the nail is all the way in, and then no remoter hammering has any effect. Or you can hush students in a classroom; sooner everybody (hopefully) will be quiet, and the hushing has no effect anymore. Then, you have reached the so-called stock-still point. Photo: COLOURBOXOr imagine that you have two identical sheets of graph paper. Take one of the sheets and crumble it up into a ball. If you now place this wittiness on top of the other sheet, there must be a point that is placed directly whilom itself. This ways that there is a point on the crumbled up sheet that has a respective point on the sheet below. This point is moreover tabbed a stock-still point. No matter how you crumble up the paper, there will unchangingly exist at least one stock-still point like this. Photo: COLOURBOXDo stock-still points like this unchangingly exist? In the specimen of the nail and the graph papers, the wordplay is yes. This is moreover the specimen in many other similar situations.Fixed Point TheoremsIn mathematics and computer science, there are many heady results well-nigh stock-still points, and these are usually tabbed stock-still point theorems. In general, a theorem is a true mathematical interjection for which one has found a mathematical proof.Stock-stillpoint theorems usually say that a particular mathematical function has a stock-still point under unrepealable conditions.One of the most famous stock-still point theorems is Brouwer's fixed-point theorem, named without the Dutch mathematician and philosopher L.E.J. Brouwer (1881–1966). In simplified terms, this theorem states that if you stir virtually in a cup of coffee, there will unchangingly be at least one point that has not moved relative to its starting point. Foto: COLOURBOXA Visual Proof for the Existence ofStock-stillPointsLet us do an experiment with a rectangle like this. We now make a reprinting of the rectangle that is slightly smaller. Below, we have made a rectangle that is exactly 61.80339% (because why not?) as large as the original, and we have placed this on top of the original. We now have a situation where there has to be a stock-still point! This is the point that is "on top of itself" and that has not moved relative to the starting point. If we imagine that the original rectangle consists of points or coordinates, and that all of these are moreover in the smaller rectangle, then at least one of these points must be exactly whilom its respective point on the rectangle below.We can requite a visual proof of the existence of the stock-still point by standing this process of taking smaller and smaller copies in the same manner. The stock-still point is located exactly at the place where all the pictures disappear inwards. Can you see where the stock-still point is located? We can moreover do the same with a screenshot from the video. First, we pinpoint a transformation of the image in the same way as above: By repeating this transformation many times, we get the pursuit picture! Concepts and Understanding in MathematicsMathematics is fundamentally well-nigh ideas, patterns and concepts – not only formulas, computations, and rules, like so many mistakenly believe. One such concept is the stock-still point of a mathematical function.A related concept is that of infinity. You may have noticed that in order to unquestionably reach the stock-still point in the transformations above, we would have to repeat the process infinitely many times? This is a very worldwide miracle in mathematics, and it has lead to notions such like convergence, limit and many others.It is worthwhile to ponder concepts like these, and the concept stock-still point is one of the most trappy in all of mathematics. What is your favorite stock-still point?by Roger Antonsen Roger Antonsen 2018 go to Norwegian site / gå til norsk side