By Daniel Hubbard | May 3, 2010
I always loved the film Donald Duck in Mathmagic Land when I was a child. With a few vignettes, it showed how mathematics was everywhere—in music, architecture, nature, games, sports… Funny that they never mentioned genealogy. There is quiet a bit of math lurking within genealogy. Here are a few of my favorite bits of genealogical math.
So Many Grains of Rice
Since everyone has one biological mother and one biological father, it is pretty clear that every succeeding generation will contain twice as many ancestors. In fact, each generation in a pedigree chart has one more person in it that all the later generations put together. At first that is quite manageable. Later though our brains have a hard time grasping just how unmanageable it becomes.
There is an old Chinese story that goes something like this. An old man asks the Emperor for some rice. When the Emperor asks how much, the old man pulls out a chessboard and says he would like one grain for the first square, two grains for the second square, four for the third square and so on, doubling every time. The Emperor laughs, says of course you simpleton and thinks nothing of it until someone in his entourage starts to multiply. There are 64 squares on a chessboard and all those doublings start to add up (so to speak). How many grains does the old man get for that last square? It’s about 9,000,000,000,000,000,000 grains, or at 25 mg per rice grain about 200 billion tons of rice. Actually, it’s a bit more but when you’ve just bankrupted an empire, who’s counting? And that’s just the last square.
Pedigree charts work exactly the same way, doubling in size with every generation instead of every square. That’s a lot of ancestors. A totally unreasonable amount actually. We all have those 9,000,000,000,000,000,000 positions in our pedigree charts 63 generations back but we don’t have that many actual humans in our ancestry all those generations ago. Long before we get to that point, we will have run into what has become known as pedigree collapse.
Pedigree Collapse occurs when the same individual appears in a pedigree more than once. If you imagine yourself moving back in time, in one generation steps, pedigree collapse starts when an individual makes his or her second appearance in your pedigree. From that point on you can count on not having as many individuals in your generations as you have positions in your pedigree chart. That person’s whole pedigree will appear twice within your pedigree and, of course, the generations in that ancestor’s pedigree will be doubling in size as well, leading to more and more duplicate ancestors in each of your generations.
Here is where another small bit of genealogical math comes in handy—the average number of years in a generation. The rule of thumb I learned was 30 years. With that number we can calculate how far back in time we have traveled as we step through a pedigree. One generation back, two ancestors, 30 years. Two generations back, four ancestors, 60 years. Three generations back, eight ancestors, 90 years… Thirty generations back, 1,073,741,824 ancestors, 900 years. Now we should stop our time travel because something remarkable has happened. Nine hundred years in the past it is estimated that the world’s population was far less than one billion but that puts us about 30 generations into the past when we have over one billion positions in our pedigrees. Think about that for a moment. Most of us, I would think, do not descend equally from people who were evenly scattered around the world 900 years ago but instead any one person’s ancestors tend to be people who inhabited only a few regions of the Earth in 1100 A.D. Clearly, long before we trace our ancestry back that far, the same people begin to occupy our pedigrees not just twice but many, many times.
That kind of pedigree collapse is in everyone’s past. There are more extreme sorts of pedigree collapse that have real historical impact. Monarchs often have pedigrees that are twisted in intricate ways. In the English Wars of the Roses, some claimants to the throne laid their claims based on descent from the same king in multiple ways. Charles II, the last Hapsburg King of Spain, had very few individuals in the recent generations of his pedigree compared to the maximum number. So extremely few, in fact, that Spanish geneticists at the University of Santiago de Compostela have argued that his failure to produce an heir was due to his genetics.* That failure lead to the War of the Spanish Succession which involved not just Spain but France, Britain, the Dutch Republic, the Holy Roman Empire, Prussia, Portugal, Savoy and Bavaria. It lasted thirteen years.
So Few Genes, So Many Ancestors
The Human Genome Project currently estimates the number of human genes to be between twenty and twenty-five thousand. Fifteen generations back our pedigrees contain thirty-two thousand positions. Pedigree collapse will reduce the number of actual individuals in that generation. Nevertheless, that doesn’t seem to actually leave all that many genes for us to inherit. Aside from the purely male and purely female lines which have their own special properties, I have to wonder how far back one needs to go before actual biological ancestors stand a significant chance of not having passed down a single gene to any given descendant.
This is part one of two parts. The second part can be found at A Genealogist in Mathmagic Land, part 2.
*For example, Charles’s father was his mother’s uncle and she was the daughter of first cousins.Twitter It!