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Chapter 1.2:

The Two Images that Shook the World

             The math problem floated around for a few decades after that. The Collatz Conjecture was often enjoyed as a small enigma, but little progress was made on it. Not too many people were working on it.... that is, until something amazing happened. 

             In the 1970's computer technology reached a whole new plateau of functionality. With the advent of modern computers, math researchers were now able to perform billions of computations at lightning speed. This launched the field of Dynamical Systems wildly! With computers performing the calculations, it was now possible for mathematicians to explore vast amount of data and graph the results. In a certain sense we could finally "see" what was going on inside of these dynamical systems. Thus was born some now-famous visual graphs like the Logistic Mapping and the Mandelbrot Set. 

             The Collatz System, being a dynamical system, suddenly came back into mainstream focus. Some of these Dynamical Systems researchers, armed with the latest generation computers, decided to take a look at the Collatz System.... and were absolutely startled by what they found!

             We're about to take a look at the two "Classical" Collatz Images. They are both very famous. Consider the orbit of n=7 through the Collatz System. 

                                       7 >> 22 >> 11 >> 34 >> 17 >> 52 >> 26 >> 13 >> 40 >> 20 >> 10 >> 5 >> 16 >> 8 >>

                                                                                                                                                 4 >> 2 >> 1 >>

                                                                                                                                                 4 >> 2 >> 1 >>

                                                                                                                                                 4 >> 2 >> 1 >>


             And let us now keep track of two things:

                          (1.) How many steps did it take for 7 to reach 1? We'll call that the Stopping Time for n=7. 

                                                                                    In this case, n=7 has a Stopping Time of 16. 

                          (2.) What was the highest value reached along the number's descent down to 1? 

                                                                                    We'll call that the Max Value for n=7. 

                                                                                    In this case, n=7 has a Max Value of 52. 

             This lets us produce the two most famous images out in Collatz Space,

                                                                  the Stopping Times Graph and the Max Value Graph. 

                                                                  These are wild! Scroll down to take a look...

             Wow! Look at all those patterns! When you play with the Collatz system, the orbits feel very random and chaotic, and yet these two images go to show that there is obviously a lot of hidden structure going on. These are nothing at all like the random "static" we might have expected. Somehow, beyond human understanding, there are a great number of patterns occurring. 

             For the Stopping Times Graph, we see that the image is composed of a whole matrix of mysterious little horizontal "bands" arranged into some kind of curved structure. The bands are not "solid", so to speak, but are composed of a dense little line of seemingly random dots placed unpredictably next to one another. This can be seen in the zoom-in to the left. The question of why these little horizontal bands exist is another of the most famous mysteries out in Collatz Space. 

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             As regards n=27, take a look at this zoom-in. Do you see how the Stopping Times chart begins with this funny sort of fish-tail shape? In the beginning, most of the numbers have stopping times which are less than 30, but a small sprinkling of them have stopping times which are much higher, up in the 100's. For the first 100 or so positive integers, each one very distinctly has either a low stopping time or a high stopping time. It's fascinating and very inexplicable. 

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             For the Max Value Graph, we see that the image is composed of an array of mysterious horizontal and diagonal lines. What are they and why are they there? Adding more to the mystery, do you see that dark bold line cutting directly across the image? That line is precisely the maximum value 9232, the one that n=27 launched all the way up to! What is going on here? For it to be such a bold dense line, apparently many other input_n's share that same maximum value. Why is 9232 so popular?

             These two images resonated around the world, startling the math community and showcasing what a gorgeous, strange, and mysterious math problem the Collatz Conjecture really is. Who would have believed that such a conundrum could be started all by asking such a simple question! 


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