Doubling Process
It is impossible for either ying or yang to exist without the other. To imagine death without birth is a logical impossibility. To imagine birth without death, quickly leads to unimaginable over population. For our first step in understand these powerful forces, we will imagine yang without ying.
When a quantity increases very rapidly, we say that it increases exponentially. A precise measurement of its rapidity is the period of time during which the quantity doubles. The smaller the doubling period the faster that quantity increases. Thus, the power of the exponential function, the thing we call yang, reduces to one number, the doubling period. We examine the incredible growth of a number when it is doubled repeatedly.
A commoner once performed a great deed for a king. When the king asked what the commoner wanted for a reward, the commoner replied, “give me one grain of wheat for the first square of the chess board, 2 for the second, 4 for the third, and for each subsequent square double the number of grains of the present square. That is he wanted:
1+2+4+8+16+ 25+26+ … + 263 = W, grains of wheat.
It would appear that in calculating W, one needs to perform the laborious task of adding up the 64 numbers,1, 2, 4, 8, … 263. There is, however, a clever short cut.
There was a person who had the remarkable ability to glance at a heard of sheep and know immediately their number. When asked about this ability, he explained that he counted the number of legs and divided by four. While you might not think this funny, you will certainly agree that it is silly. Let us modify this story to a person who had the same ability to count pigeons. He might reply that he counted their wings and subtracted all of the left (right) wings, depending on his political persuasions. We will apply this method to the calculations of W.
Assume that each grain of wheat as two wings. Count the wings and subtract all the right wings. That is, 2W-W=W. Our calculations look like this:
2W – W = [ 2 +4+ 8+16+ … 263]+ 264-1 -[2 +4+ 8+16+ … 263] = 264 - 1 = W!
2 multiplied by itself 64 times minus one, well 264. (No one could possibly notice that extra grain of wheat).
Now we have a new problem, imagining
264 =1. 8446744072 x 1019 = 18,446,744,072,000,000,000 grains of wheat. (263 = 9.223372036 x 1018 grains on the last square, rounded off to about 1019 )
It is estimated that the age of the universe is 13.8 billion years, that is 1.38 x 1010 years. To change this to seconds multiply by 365x24x60x60 to get:
30758400 x 1.38 x 1010 seconds = 42446592 x 1010 =
4.2446592 x 1017 seconds.
This means that if we were to count 44 grains of wheat per second (264/4.2446592 x 1017 = 43.45871648 grains of wheat) and started at the Bing bang, we would just be finishing.
Another way to imagine this enormous number, 264, is to consider the thickness of a stack of 264 papers.
Take a standard sheet of paper, 11 X 8 ½ inches. Fold it exactly in two. Fold the result in two, and continue folding it in two, alternating the fold on the vertical and horizontal. Each fold doubles the thickness of the paper and halves the area. Most people can do about 6 folds. The result has the thickness of 128 sheets of paper.
(One could do better if they took a larger and thinner sheet of paper to begin with. Try a full sheet of newspaper. It is 0.07 mm thick. One would do well to get seven folds, about the equivalent of 256 thickness of newspaper, which is 17.92 mm thick, 0.7055 inches. A Sunday newspaper of 1024 layers is 2.8 inches thick.)
Each sheet, of ordinary typing paper, is about 0.1 mm thick, so 24 sheets plus 2 reams, of 500 each, gives us a stack of 1024 sheets of paper about 1024 x 0.1 mm or 102.4 mm = 10.24 cm (4.0315 inches) thick.
How thick is a stack of 220 sheets of paper?
344.021 feet
# of sheets | thickness |
1 | 0.0039 inches |
128 | 0.35275 inches |
256 | 0.7055 inches |
210 = 1024 | 4.0315 inches |
220 | 344.021 feet |
230 | 66.7192 miles |
240 | 68320 miles |
250 | 70 million miles |
264 | 280 light years |
280 light years about 1.64567176 x 1015 = 1,645,671,760,000,000 miles. To put this in perspective, it is only 93,000,000 miles from the earth to the sun. So this thickness is 12,522,227.7 times the distance between the earth and the sun!
(The stack of 264 newspapers would only be 196 light years thick, a difference of 84 light years. Small differences become huge during the doubling process, a lesson useful when applied to IRA’s and Roth IRA’s).
When we consider the time interval for the doubling to take place, the power of pure yang reveals itself. We now apply yang (the power of birth) to the very beginning of human life.
The Beginnings of Life
After fertilization, the human egg divides into two cells. These two cells divide in to 4 cells. The next division yields 8 cells and each further division doubles the number of cells. This is like folding paper, or putting wheat on the chessboard, above. The time interval for this yang is somewhere between 10 and 30, let us say 24 hours. In 64 days, how many cells does the fetus have? We made the calculation above. The answer is the same as the number of grains of wheat on the last square of the chessboard, 263 = 1019 cells. This is outrageous! An average adult human has approximately 100 billion (1011) cells. If there were no cell death in the first 64 days, our two month old fetus would have same number of cells as 1019/1011 = 108 = 100,000,000 adults.
This does not happen because the power of ying restores a perfect balance. Ying, the power of death, kills off cells at an enormous rate. We can infer its power by the job that it must do in restoring the perfect balance against the very powerful yang. Clearly, many of the cells die from starvation because the mother cannot eat enough food. Ying must kill cells from the very start, otherwise yang will become too powerful. If one of the first two cells dies in the second day, without dividing, the production is cut in half. On the other hand, a cell dying on the 64 day makes no discernable effect.
How long do embryonic cells live? After viewing the exponential growth of yang, we know that they must have a short life, certainly much less than 64 days. I also expect that as the embryo ages, the rate of division increases. I have heard that the adult human replaces every cell in their body in 7 years. If this is true, adult cells have a much longer life and a much longer dividing time than embryonic cells.
Of course, we have only covered the first two months of a pregnancy. There are seven more months of this incredible battle between ying and yang.
So it is, the perfect balance between ying and yang bring forth the miracle of a baby.
I am a monkey’s uncle!
Another balancing of ying and yang might alter your feelings towards fellow humans. Everyone has two parents, 4 grandparents, 8 great grandparents, 16 great great grandparents and so on, or do they? If a generation is 20 years (the doubling time) where time is moving backwards into history, how many direct ancestors do you have 64 generations ago (64 x 20 = 1280 years ago or in the year, 725ad)?
From the calculations above, for 64 doubling periods we get the outrageous answer of 1019 ancestors! However, the world population in the year, 725, was about 210 million (2.1 x 108). Some of these people had no children, no grand children, etc. and hence were not ancestors of anyone alive today. To be generous, assume 2 x 108 of them have progeny. If every one of them were in your ancestral tree, then each of these people in the year, 725, would need, on the average, to play the roll of 1019/2 x 108 = 5 x 1010 = 50,000,000,000 of your ancestors. What is the controlling counterforce to this exponential madness? We cannot kill off our ancestors like embryonic cells, but each of them must perform duplicate rolls. This power of ying is not of death but of duplication.
If you marry your sibling, your children have only two grand parents, not 4. Your parents are both maternal and fraternal grand parents to your children. Marry your first cousin and your children have 6 great grand parents, not 8. Mary your second cousin and your children have 14 great great grandparents, not 16. (It is possible that your second cousin qualifies through two sets of great great parents and hence you have only 12). Clearly, there is a lot of marrying nth (n = 1, 2, 3…) cousins.
Just like in the death of embryonic cells, this ying must act quickly and often to reconcile the incredible discrepancies in the numbers, 1019 and the total number of possible ancestors, 2 x 108. In other words, n is much smaller than one might think (15 or 20 would be huge!). It inspires a new feeling that all people are much more closely related than they imagined. The problem of cousin closeness is incredibly complex. “n” is relatively large between the queen of England and an Australian aborigine and quite small among the residents of Iceland. Next time you see a homeless person or a person of a different color, remember he/ she is probably a close cousin, much closer that you might have imagined.
Too many people
The exponential is not only important in studying our ancestors, but is quite crucial in counting progeny. The yang of population growth is very complex. The doubling period changes over time and location. For many years, the knowledge of the power of this function had demographers predicting doom. However, ying comes to restore balance. It is in the form of disasters, pandemics, war, and even technology like the green revolution that postpone doomsday. Will aids save us as the black plague once did?
If one thought ying was a bad force, consider this. By preventing juvenile death and supplying family planning to developing countries, their birth rate is reduced. Ying’s power can be revealed in paradoxical ways.
In part, 3 we will derive the mathematical equations of yang. It requires a little calculus, but everyone will understand the basic assumptions.
DICK
