Part of my duties as a chemist was preparing serially diluted standards for analysis, and the typical approach was to make up a series of 100 ml flasks, create a master flask into which I would dissolve a known weight of a given standard, then dissolve the standard in water, filling the flask carefully to the 100ml line. Then, I would pipet a certain amount- typically 10ml, and transfer that into the next flask, fill to the line... lather, rinse, repeat. The nice thing about such a method is that if I did my job right, you could draw a line between the analytical results and you would get a near perfect correlation coefficient, typically in excess of three or four nines.
The thing about dilution is that if you do it enough times, that last pipet's worth of solution you transfer into the next flask might not contain any of your original substance. You may (or may not) remember Avogadro's number, which states that a "mole" of any molecular substance is it's atomic weight, in grams, and that a mole contains about 6.02 X 10^23 molecules. So if you have a mole or so of a given substance, say, sodium chloride, which weighs 57 grams- if you dissolve a gram of table salt into a 100ml beaker, there are about 10^22 molecules floating around. If you did the procedure above, the next solution would have 10^21 molecules floating around. Twenty one more times, and at last you only have only one molecule left in the flask, and chances are the next time you dilute, and in the next flask, poof! there's no more salt in your salt water.
Now, if you're inclined to believe in homeopathic remedies, you'd recognize that as an "X" dilution. An "M" dilution is one milliliter in 1000 (roman numerals, ftw), and goes even faster- 8 "M", and poof! all gone. Homeopaths believe that the water has a memory of the salt, particularly if the procedure is done by a specialist and shaken "just so". But, if you're to believe science, any time a particulate characteristic is "diluted", at some point the act of dilution fails to transfer the original characteristic, and it becomes "lost". This is a fairly difficult concept to grasp, and I've struggled a bit to find a good way to reconcile all of that in my head.
So, I spent some time thinking about the issue of "great-great-whatever-grandparents",
It was a lot quicker than I thought. The picture here shows the result- in about 30 or so generations, or about 1000 years. Of course, in this picture, some of these "place settings" in your family tree will be occupied by the same person. I have superimposed the approximate world population in each year going back, and this has to happen a lot quicker- back past the year 1150 or in about 850 years, there are more place settings than people on earth on my tree, alone. Of course, if on this tree someone marries their cousin, as must happen to us all, the dilution happens a bit slower, but at some point, we are all ultimately cut from the same cloth. In the end, there is "nothing left behind" of each of us, no matter how many generations we sire- poof!
I've been thinking a lot about generations and children, as I think having genealogy as a hobby is likely to cause you to do. I've often discussed with my spouse what I think about us not having children of our own. I think on reading that last paragraph, some people might feel a little saddened, and assuredly, it's not my intention to bring people down on new year's eve. I think what we leave in our genes is a very small part of the picture, and dwindles rapidly with each generation- but, too, we leave behind our thoughts, and our philosophies, and the impact that we've had on everyone else. If we've been really good, we leave the world a better place- safer, healthier, more easily understood. Leaving behind art, or science- these are things that can last for much more than 1000 years, if you care enough and work hard enough to make them count.
I think all this stuff about DNA and genealogy teaches me a lesson, too. Each and every one of us has a massive family. We are all, genetically speaking, cousins, even if the paper trail is really difficult to ferret out. It's pointless to be xenophobic or to hoard your relationships and your wealth when the inexorable act of dilution is constantly reshuffling the deck.
Be good to your family, and have a safe and prosperous new year.