A Genealogist in Mathmagic Land Part 2

A couple weeks ago I posted the first part of these thoughts about genealogy and mathematics. Thinking that too much math might not feel so magic to everyone, I decided to break what I wanted to write in two. So, once again, with apologies to Donald Duck and his journey through Mathmagic Land, I thought I would finish my current journey with some binary numbers and a little crochet.

Ahnentafel Numbers

Anyone familiar with the binary system of numbers (the system of 1s and 0s computers use) may have noticed something about ahnentafel numbers, those numbers commonly assigned to people in our pedigrees. I first noticed this years ago while stuck in bed with the flu. I offer no excuse for thinking about such things while in bed with the flu except that I was doing lots of programming at the time.

Binary numbers consist of only two digits, 1 and 0. Instead of 11 being one unit of ten plus one, in other words eleven, in binary it is one unit of two plus one, so that 11 binary is three, despite appearances. Pedigree charts are very binary entities because we all have two biological parents and when we assign ahnentafel numbers to the people in them we follow a very binary pattern. Assuming you start your pedigree chart with yourself, you number yourself 1, your father 2, mother 3, then father’s father 4, father’s mother 5, mother’s father 6, mother’s mother 7, father’s father’s father 8 and so on. Funny thing about those numbers, if you ignore your own gender, males are always given even numbers, females odd numbers.

If we use binary ahnentafel numbers, more of a pattern starts to become clear. You are 1, your father 10, your mother 11, your father’s father is 100, father’s mother 101, mother’s father 110, mother’s mother 111, father’s father’s father 1000 and so on. The binary ahnentafel number of every female ends with a “1” making the number odd, just as it should be. The number of every male (except you, if you are male) ends with a 0, making the number even, just as it should be. Every generation needs one more binary digit for it to be specified and the path through the pedigree chart is given by the sequence of digits. The number always starts with the 1 you assign yourself. Every time you go back a generation to a male, put a 0 on the end of the number. Every time you go back a generation through a female, put a 1 at the end. So can also figure out an ahnentafel number as you go back through the generations. For example, as you move backward might move like this to your (1) father’s (0) mother’s (1) father’s (0) mother (1), a binary ahnentafel number of 10101, which is 1x16+0x8+1x4+0x2+1x1 or 21.

If you imagine yourself walking back through the generations on a giant pedigree chart, you can follow the digits of a binary ahnentafel number as if it was a list of directions for following a map. “0” means go to the left as you move back a generation where  you will reach a male. For a “1” move to the right as you go back a generation where you will reach a female. Then follow the next digit where it leads you.

What Does Crochet Have to Do with Genealogy?

When I was a kid, I would try to figure out how to best fit a pedigree chart onto paper. Anyone who has ever dealt with a chart that stretches page after page, with the line you are trying to follow skipping pages along the way, might sympathize with the desire to get it all to fit better. The problem is that as you add generations you might be willing to add some width to the paper to fit in the next generation. You also might keep your sheet of paper roughly square by adding the same amount in the other direction as well, but the number of ancestors in each generation just doesn’t cooperate.

Say you have a four generation chart on a square of paper. If you want to add four more generations to your chart, you would double the length and width of your paper as you double the number of generations. That gives you four times the area to write on. However, it doesn’t work out so well. Going from a square of paper big enough for four generations with 15 people on it, to your new “eight generation size” piece of paper might give you space for four times as many ancestors but that means only 60 people and those eight generations will have 255 total people in them.

It turns out that the problem comes from trying to fit a pedigree chart on a flat piece of paper. Flat things don’t grow fast enough in area as you stretch their sides. What you need is a thing called a hyperbolic plane. They have areas that grow faster than flat surfaces. They may sound strange and too mathematical but you see them or things very close to them all the time—the wavy edge of a leaf or that funny wavy look that you can see at the edge of a roll of toilet paper that was put down on a damp bathroom counter. Even the way the edge of a horse’s saddle goes up, then down, then up, then down again is like a tiny bit of a hyperbolic plane.

So, I implied that I would mention crochet somewhere in this. It turns out that about a dozen years ago, a mathematician named Daina Taimina came up with the first practical way to make models of large hyperbolic planes. You can crochet them. If you search for “hyperbolic plane” and crochet you can find instructions and pictures. I found some here that seem appropriately binary (even to someone who does not crochet), a paper by Daina herself that also explains how to do it and you can also see some color pictures and the story of how she came to crochet hyperbolic planes. If you have the crochet stamina for it, you can actually crochet your pedigree chart in one piece with as many generations as you like. Just remember that each generation you add will be as much crocheting as everything you’ve already done. Finally, beware, your creation won’t lie flat!