All the humans who have ever lived on earth, number less than 10 billion!  The explanation for this contradiction is that our ancestors have married each other by the thousands and millions in the remote past (and in some cases, not very remote).  We may have had a trillion ancestors 40 generations ago, but not a trillion different ancestors.  

This phenomenon, the pruning of our family tree to look like a Lombardy poplar instead of a spreading oak, is called ‘pedigree collapse’ or ‘coalescence’.  We all share common ancestors.  "We are all related:  the question is how far back in time is the common ancestor."  As all family historians have seen, there are many cases where our ancestors married cousins;  thus reducing the actual number of our unique ancestors.

Scientists who study population genetics have done a great deal of research on this subject.  An article by Steven Olson in the May 2002 issue of The Atlantic Monthly describes some startling results of their studies:

	In all probability, you and I are descended from English Royalty,
	Everyone in the world is descended from Nefertiti and Confucius,
	Everyone in the Western world is descended from Charlemagne,
	Eighty percent of Charlemagne's contemporaries are also ancestors of us all.

Olson based these statements on the work of Joseph Chang, professor of statistics at Yale University.  Chang's paper, 'Recent Common Ancestors of All Present-Day Individuals'

The first part of Chang's article is quite readable;  later, when he presents the formal proofs of his conclusions, his mathematical formulation becomes quite complex.  But there is a fun experiment that you can do to bring home the essence of pedigree collapse.  You need a pencil, a sheet of lined paper, and a pair of dice. Imagine that many years ago there was an island on which six couples lived.  Down through the years, the generations remained distinct, and the population in each generation remained at six couples (see Pedigree Collapse Chart below).

Near the top of your paper put six dots in a row and number them 1 to 6.  This is Generation 1.  Seven lines below that, put six more dots but do not number them. This is Generation 2.  For each dot in Generation 2, throw the dice, and write the numbers from each die above the dot.

If the number is 1, put it one line above the dot;  if a 2, two lines above, etc.  For each couple in Generation 2, we are randomly choosing the husband's parent couple and the wife's parent couple from Generation 1.

Now repeat this process.  Put 6 more dots for Generation 3 seven lines below the previous dots.  Throw the dice for each couple in Generation 3.  This time, do not record the die face number, say 4, but use that number to find the 4th dot in Generation 2 and read the two numbers above that dot.   Record these two numbers above the Generation 3 dot, eliminating duplicates.  Do the same for the second die.

Now we have chosen the grandparent couples for all of the Generation 3 couples.   Probably by this time there will be some anomalies.  Each Generation 3 couple can have at most 4 numbers above it, but some may have fewer.  If there are fewer, cousins have married and the collapse has begun!   It could even happen that brother has married sister, not taboo on this make-believe island.

Keep repeating this process and watch for the first time that the same number appears above every one of the couples in the latest generation.  That will probably happen by about Generation 3 or 4.  If that repeated number is, say, 5, it means that the 5th couple in the original generation is the "most recent common ancestor" (MRCA) of all of the couples in the current generation.  Keep repeating the process and watch for the first time that all of the numbers above all of the couples in the current generation are the same.  When that point is reached, and it will be reached with certainty, then all the progenitor couples whose numbers are in that list are common ancestors of everyone in the current generation.  Some of the progenitor couples will probably be missing; their descendancy has winked out forever.

Now imagine that instead of a population of six we had a population of one million or one billion.  According to Professor Chang's model, the same thing will happen, but it will just take longer.  Not as long as you might think, though.  If the constant, randomly mating population is N, the number of generations back to the MRCA is the logarithm of N to the base 2.

For a population of one million, the MRCA generation is 20, or about 500 years.  For one billion population, it is 30 generations or 750 years.  The time when everyone was either an ancestor of all in the current generation of an ancestor of none is roughly twice that for the MRCA.

So Charlemagne, who lived 1200 years ago, is the ancestor of us all, and you are my Nth cousin, where N is probably less than 20.

George Anderson programmed Professor Chang's model on his Macintosh. Anderson’s simulations agreed with Chang’s.  Because the simulations involve random selections, multiple trials were conducted to get statistically valid results.  That's where it gets hairy!  To do 250 trials for the population=4,000 case, Anderson had to run his poor Mac overnight (you should use a PC George)!

For a population of 4,000, the MRCA appears at about 12 generations, and the all-or-nothing common ancestors appear in about 22 generations.  Of course, the mathematical model is idealized.  But the main conclusions are being accepted as true.  Humans really do become related to each other very quickly - in centuries, not in millennia, and not ever, as some believe.  Just think of the churning in England.   In succession there were the Picts, Celts, Romans, Saxons, Vikings and Normans.  If they did not interbreed, willingly or otherwise, it would defy all human nature.

Hello all you cousins.  Let's have a reunion!