There are many hidden patterns in the world … mysteries to be revealed and conundrums to be solved. Names like Mandelbrot, Zipf, Paredo and the great Pyramid of Giza challenge our imagination. Let us explore a few of these challenges.
WHAT’S HIDING INSIDE THE MENDLEBROT FORMULA?
Did you know that deep within the Mandelbrot Fractal equation there lies a secret. Recall that the formula for the fractal is Z squared + C (where the result is fed back into Z with each step.) Along the horizontal axis, we know that no value greater than two can produce a fractal and that any starting C greater than ¼ will tap out with an eventual value greater than 2. So what if we add just a tiny bit of a number to ¼ (we can call this epsilon). The smaller the value of epsilon, the greater the number of steps it takes to tap out. If epsilon = 1, we tap out in only 2 steps, If epsilon = .1, we tap out in 3 steps, if epsilon = .01, we tap out in 30 steps and so on. Here is the table that shows how many steps it takes to tap out when there are various values of epsilon.
epsilon number of steps
What 3.14159? Really? Pi is hiding inside a Mandelbrot fractal? Yes indeed, we have a really cool way to calculate the value of pi. Just make epsilon smaller and smaller and count the number of steps it takes to tap out (Wolfram Alpha is useful here) and viola pi is revealed in ever greater glory with each incremental value down. Pretty cool huh? For more detail, watch the video below:
THE GREAT PYRAMID OF GIZA AND THE HIDDEN SECRET OF IT’S ARCHITECTURE
The Egyptians built a pyramid so big and well constructed that it was once said that “Man fears time, but time fears the pyramids.” On the point of ancient genius and will, there can be no dispute. What was never understood, however, is why ratios of pi appear throughout the measurements of the great monuments. Yes even though there is no evidence that pi was known to the Egyptians, things such as the ratio of the height to the width or even just the measurements of the base all have ratios of pi built in. Did the ancients know of the secrets of the Mandelbrot fractual? Uh No. Did they share secrets with long lost alien civilizations? Uh no. The great hidden secret was entirely accidental. It turns out that the cubit measurements were not done with a stick or a rope or strips of metal in a box … no the entire structure was laid out with … wheels. This technique, still in use today (to measure for marathons and the like), meant that pi was built in because a wheel was instrumental to the construction technigues of the time. Hidden patterns … wonderful mysteries. This one not solved for thousands of years. But there is more.
ZIPF’S LAW AND THE PATTERNS OF LANGUAGE
In the late 1940’s, a Harvard linguist named Zipf discovered that if you listed all the words in the English language by frequency of use, you will get a most incredible pattern. The pattern is this … the second most common word in the English language appears ½ as often as the first, the third most common word appears 1/3 as often as the first, the fourth most common word appears ¼ as often as the first and so on all the way down. Studies of encyclopedias and compendiums of fiction, the great works of Shakespeare and even the Bible produce the exact same pattern. The 5000th most common word appears 1/4985 as often as the first most common word. A quirk of English you ask?
No, it is not just a quirk of English. The same pattern appears in French, Spanish and German. It also appears in Arabic, Farsi, Chinese, Russian and Japanese. It even appears in dead languages for which we have no translation. Zipf’s mysterious pattern even appears with randomly typed words. Like pi being cloaked in the secret world of Mendlebrot, the zipf pattern lies deep in the bowels of language … all languages.
PAREDO’S PRINCIPLE IS AN EVEN BROADER VERSION OF ZIPF
AND digging even deeper, it turns out that ZIPS is a specific version of a more general construct that 80% of results are explained by 20% of causes. It happens again and again. 20% of your customers will account for 80% of your business. 20% of software bugs will account for 80% of all software complaints. It goes on and on like this. The 80/20 rule is simply everywhere. Below is a great video of Michael at Vsauce explaining how zipf works and why Paredo keeps happening over and over.
CAPITALISM IS USING PAREDO TO CREATE WEALTH
About 80% of all wealth is owned by 20% of people. The top 20% of income earners make about 80% of all income and about 20% of taxpayers pay about 80% of taxes. It seems that in most systems there is an order that is sought out … even super complicated systems like macroeconomics. Some people are just not very good at earning money and others excel to incredible levels. This is true for sports talent, musical talent and even size.
CAPITALISM AND TAX POLICY
One possible way to use Paredo and Zipf, when designing tax policy, is simply to work exactly in harmony with the great invisible hand of Paredo’s law. If you design a system that matches known percentages from the world of science and art, you should have the easiest time collecting revenue. Just set the top 20% so they pay about 80% in taxes and so forth and you will be working with not against the great patterns of our world.
Interestingly, it may be that no matter what you use for tax policy, Paredos patterns will emerge. For example, if you have many graduated brackets and even tax shelters for income to be manipulated, you might still see Paredo emerge. Likewise if you establish a flat tax, where everyone pays the same percentage, you might see Paredo as people who make more would pay more. Serious econometric analysis would be needed to verify this, but the principle that Paredo’s law will tend to manifest itself no matter what the system, is an interesting concept to contemplate. The key principle here is that while different econometric strategies might create different size pies, Paredo will decide who gets the biggest piece based on known and predictable patterns. At least that is the Wizards working hypothesis.
OK the Wizard has strayed into economics and this is very dry territory … tediously dry territory, so I leave you with this thought on the number of economists in the world …
If you were to stack all the economists in the world … end to end … it would be a good thing.
On the first day, God created the sun. In response, the Devil created sunburn. On the second day, God created sex. In response, the Devil created marriage. On the third day God created an economist. This was a tough one for the Devil, but, in the end and after a lot of thought, he created a second economist.
I could go on and on with economist jokes, but apparently 20% of all jokes about economists contain 80% of the humour (and since there are only 10 jokes about economists in the entire world, it is time to stop.)
NEXT: WHAT HAPPENED BEFORE THE BIG BANG? “OK, like the Wizard knows the answer to this one?” the dark elves exclaim. “Bruhaha,” shrieks the Wizard. “All shall be revealed.”