WHAT DOES RANDOMNESS LOOK LIKE?
RANDOM WALK asks this question and presents experiments in mathematics and physics, showing the mysterious interaction of chaos and order in randomness.
The project RANDOM WALK simulates randomness in visualizations, which are easy to understand. In this way, it delivers insight into a phenomenon, which has so far remained unexplained.
 
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There is one remarkable thing about randomness: Its existence is neither proved nor disproved it even appears everyday in science and in our everyday lives. Random walk is interesting for people who want to know more about the mystic character of this invisible companion.

This diploma thesis was created at the University of Applied Sciences in Mainz, Germany by Daniel A. Becker in 2009 and was supervised by Prof. Johannes Bergerhausen. 
  The project consists of 14 double-sided A2 posters contained in a transparent plastic sleeve. Ten sheets explain the phenomena of randomness in mathematics and physics - four focus on all-day randomness and the quality of pseudo random number generators. Without title or specific order the sleeve contains the folded sheets at random like a pack of cards. It has no thematic order, unlike the playing cards in a pack. The layouts of the poster backs are visually influenced by randomness. Each reverse side is unique as is the reverse of the same topic in different sets. Visualizations and random layouts are made with the program "proce55ing".
 
 Wikipedia: Randomness        
All visualizations of physical mathematical phenomena are not just illustrated representations; they are actually simulated with the help of "proce55ing". True chaos and order within randomness has been demonstrated by the use of pure numerical data and software simulations. Therefore, the software not only presents the visualizations but it also uses the graphics to prove the phenomenon of randomness.        
 
The constant number pi has an infinite number of decimal places with no recognizable system within the sequence. However, the distribution of the 10 possible digits is quite uniformly balanced – at least within the displayed range from 1 to 1,000,000 positions after the comma. Each digit is represented by a direction from 0° to 360°. For example, each time the 0 arises, a line with a certain fixed length is displayed with the value of 0°. The end of each line is at the same time the start of the line for the following digit; the length of each line remaining constant. The result of the lines is a path; the so called random walk.

  The colored areas represent the distribution of the decimal of pi. These always start with 0, but with each succeeding step the values are increased by 10,000. These areas are laid around the most extreme points of the random walk. It can be observe that the lager the displayed range becomes the more round the areas are.
 
 Wikipedia: PI        
 
Let us assume the following experiment: Grains of rice trickle onto a horizontal dartboard. Then the number of rice grains in each particular area of the board are counted. If this experiment is repeated several times, a regularity within the sizes of the rice grain piles will be observed. This regularity is characterized by the so called Poisson distribution. In applying this theory, one can easily calculate the distribution of piles of a given number of rice grains on the board. Of course the precise areas on which piles containing a given number of rice grains cannot be predicted.

  The areas of the dartboard in the middle of the visualization are all of the same size. On it, there is a random distribution of points, and the number if hits on each point are counted. These pile sizes are symbolized by the colored petals, whose number equals the number of different pile sizes. Example: The more areas there are with the pile size "two pieces" the bigger the respective petal. The grey line shows the idealized calculated results using the Poisson formula. Since the number of random points is limited, it is not surprising that the petals will never match the grey line exactly, but the Poisson effect is nonetheless clearly shown.  
 Wikipedia: Poisson distribution        
 
Physical experiments are also subject to mathematical probabilities. The bean machine, also known as the Galton Box, clearly shows normal distribution in action. The machine consists of a vertical board with interleaved rows of pins. A ball is dropped from the top and on its way down it bounces left and right until it leaves the machine at the bottom. The path of any single ball cannot be predicted; however, the number of balls in each row when all rows are taken together, approximates a bell curve.

  The most heavily colored areas within the visualization depict those parts of the machine which were passed through more often: Mainly the direct way down.  
 Wikipedia: Bean machine        
 
A prime number is a natural number which has exactly two distinct natural divisors: 1 and itself. Within the natural numbers from 0 to 1,000,000 there are exactly 78,498 prime numbers. Their distribution is chaotic – it cannot be predicted whether any subsequent number will be prime or not. Each number has to be checked precisely. However, the density of prime numbers decreases in the higher number ranges. In the range from 1 to 1,000 there are 168 prime numbers, whereas between 999,000 and 1,000,000 there are just 53.

  The visualization shows lines in a circle each representing 400 natural numbers. The more prime numbers there are within each package of 400 numbers, the longer the line grows towards the center of the circle. There is no regularity within the different lengths of the lines – the number of primes is randomly distributed in each package. However, in the long term a spiral is generated suggesting a decrease of the density of prime numbers in higher number ranges.  
 Wikipedia: Prime numbers        
 
Certain data, the values of which are not determined by intentional manipulation (e.g. the sizes of the world´s countries) have an interesting property: Observing only the first digits of the size, it can be observed, that the possible numbers 1, 2, 3 and so on until 9 lie in a certain proportion to each other. This proportion can also be observed with completely different data, such as DAX index values. This phenomenon concerning many real-life sources of data can often be observed by Benford´s law, also called the first-digit law. Perhaps, because the data is not intentionally influenced by human hands, it satisfies this rule. If the data is manipulated, this can be shown with some clarity with the help of this law.

  All the world's countries with their area sizes are depicted on the right hand scale of the visualization. Taking only the first digit of its's size, a line connects each country to it's respective group on the left hand scale. Each of the groups on the left hand scale grows with every new line joining it. Finally, the distribution to each digit is shown for each group. Remarkably, even with such a comparably small dataset, Benford´s law can be verified quite clearly.  
 Wikipedia: Benford's law        
 
The size of simple forms such as rectangles, circles or triangles can be calculated simply by the use of defined formulae. But how is the size of an irregular colored shape determined? The first possibility would be to divide the area into many small rectangles and then determining the sum of their sizes. A fundamentally different approach is described by the Monte Carlo method, named after the randomness to be found in a casino. In a circular area, there are two irregular forms, the size of which must be determined.


  For that purpose, points are randomly thrown on the circular area, based on the assumption that they will be uniformly distributed. Each individual point will take on a color depending on which shape it hits. Each colored point is connected to that part of the circumference relating to its respective color. The ratio of the two numbers of points to each other allows the calculation of the percentage of the circle which is captured by each shape.  
 Wikipedia: Monte Carlo method        
 
The frequency of lottery numbers basically satisfies the law of large numbers. Given that each of the 49 balls is picked with the same probability, the distribution of the 49 possible results will be all the more balanced, the more often the experiment is repeated. Ball no.1 will eventually be picked nearly a soften as ball no. 49, because all balls have the same chance. The next ball to be picked is nonetheless not predictable, even if a certain ball number has fallen behind.
  The flower-shaped pictures show the average drawing frequency of all balls in randomly selected years. The more often a ball is picked during each period, the longer the petal grows from the center outwards. On the top left hand side of the visualization, it can be seen that the range of different petals within one year varies greatly – obviously the numbers were not picked with the same frequency. Moving to the bottom of the visualization, the flower pictures summarize the average of an increasing number of years; the result shows that the drawing sequence is more and more balanced. At the bottom right hand side, the average drawing frequency of all 19,026 Saturday lottery numbers between 1955 and 2008 are shown. The frequencies are clearly more balanced.  
 Wikipedia: Law of large numbers        
 
Each atom and molecule in gases or fluids is subject to perpetual movement, resulting from the impulse of other particles. However, this chaotic process satisfies the rule that the distribution of the direction of each particle will be the same with the result that any one observed particle will not be biased in a particular direction.
  Starting in the middle, the visualization shows a random walk equal to the movement of a molecule. In this process, each change of direction is observed. The frequency of movement changes is depicted by the corners of the “net” which is formed. The rounder the resulting form of the “net”, the more balanced the distribution of the directions taken by the molecule. By comparison, a circle would always show a completely balanced distribution. The law of large numbers can again be observed; the longer the walk of the particle, the greater the balance of the “net”.
 
 Wikipedia: Brownian motion        
 
The atoms of each substance decay over a specific period of time. This phenomenon is known as radioactive decay. The process determines exactly, at which time in the future, half of a quantity of atoms will be decayed. Please note, that the substance itself will not be disappear, it will only be transformed into another substance. The resulting period of time is called the half-life. Remarkably, it is not clear at which time a single atom “decides” to decay. It may decay within the next millisecond or – as with uranium – not until several billion years have elapsed.

  This process can be compared to the bisection of a piece of paper, where it is decided at random, which of the resulting two halves will be divided next. The visualization shows this process and in doing so, the cut always takes another direction. The constant is that the newly created area is reduced to half of the previous area.  
 Wikipedia: Half-life        
 
The double pendulums is a simple chaotic, physical experiment. It is a system of two pendulums connected to each other; one under the other. Each moves independently of the other. If this construction is set in motion, it can be observed that the lower pendulum will move in a chaotic manner. This is shown within the visualization.
  The colored points measure the time the lower pendulum is found each side of a vertical axis. Counting the points clearly shows that they occur just as often on the left side of the axis as on the right side; leading to the conclusion, that the distribution of time units in which the pendulum spends on the left side of the scale is the same as on the right, despite moving chaotically.  
 Wikipedia: Double pendulum        
 
There are many mathematical models with the help of which a computer can produce random numbers. A computer cannot produce real random numbers, because it works completely deterministically. Therefore, the computer generated random numbers are called pseudo random numbers. Each chart here shows 30,000 pseudo random numbers between 0.0 and 1.0, which were produced using different random generators.
  In each chart, these 30,000 numbers are placed into a three-dimensional cube in the following way: Three sequenced numbers form the X-,Y-, and Z-coordinates. Then all 30,000 numbers are placed within the cube. An observation of all sides of the cube can produce a discernable pattern. These regularities are a sign of the random generator producing some numbers more often than others or even leaving some out altogether. This is an indication of a bad generator. The most modern random generator is called the “Mersenne Twister”, which was developed in 1997: It does not show any patterns in this visualization. Its distribution is shown at the bottom right of the visualization.  
 Wikipedia: PRNG        
 
 

Special thanks to: Anne Schnabel, Dr. rer. nat. Rudolf Schnabel and Prof. Johannes Bergerhausen and John Townsend.
Thank you for supporting my project: Dr. Hans Schnabel, Florian Jenet, Dr. Peter Hellekalek, Martin Anderle, Achim Krebs, Björn Knauf, Michael Sonnek, Jana Mattes, Jenny Lettow, Stephan Nuber, Nadine Roßa, Franziska Noack, Dr. Markus Trahe, Nina Pree, Oliver Hudec, Carmen Hudec, Sarah Primosigh, Stefanie Becker, Johanna Becker-Jenniches, the W.B. Druckerei GmbH and the Deutschen Wetterdienst.

 
© Daniel A. Becker 2009.
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Random Walk – The Visualization of Randomness by Daniel A. Becker is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License.
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