December 08, 2003

One Hundred Interesting Mathematical Problems, Puzzles, Diversions, and Amusements: Number 21: Ancestors

One Hundred Interesting Mathematical Problems, Puzzles, Diversions, and Amusements: Number 21: Ancestors

"Dad?"

"Yes?"

"Were my ancestors... famous back in the Middle Ages?"

"Your ancestors? Which ancestors? You have a lot of them."

"Well, most of them."

"There is a tradition that your Ridgeway ancestors--part of the English wave of conquest of Ireland under the Angevin kings--were descended from an illegitimate daughter of William the Conquerer, and that they were also descended from Earl Leofric and his wife Godiva, of the well-known story about Coventry. This was possibly the first supply-side tax policy in history: she asked her husband to lower taxes, and he replied 'Sure, I'll lower taxes when you ride naked through the streets on a white horse'."

"Nobody knows that story, Dad."

"No?"

"I asked people in my class. They all agreed that Lady Godiva was the inventor of a kind of chocolate."

"Oh."

"But what was my typical ancestor like?"

"Well, first let's calculate how many ancestors you had. You were born in the 1990s. Let's suppose that there is a generation born every twenty-five years. That means that a century ago you had 2 x 2 x 2 x 2 ancestors born in the 1890s. How many is that?"

"16"

"In exponential notation: you're studying exponents in math this week."

"24"

"And a century before that? Another four generations, with your number of ancestors doubling as we go back each additional generation?"

"You'd multiply by two four more times."

"Which would give you?"

"28, two multiplied by itself eight times."

"That's 1790. How about 1690?"

"212."

"And what is 212, approximately?"

[Silence]

"Well, 212 is 210 times 22, and 210 is about 1,000--1024 exactly, but we'll say 1000."

"1000 times 4 equals 4000."

"Yep. About 4000 ancestors born in 1690. Now let's take it back to 1490--eight more generations. 212 times 2 eight more times is?"

"220"

"And back to 1240--another 2 1/2 centuries, another ten generations?"

"230"

"And that is? Remember, 230 is 210 x 210 x 210."

"1000 x 1000 x 1000... is a billion."

"Excellent! Now consider that there were only 400 million people alive on the earth in 1240. What does that mean?"

[Silence]

"Well, it means that you must be descended from a lot of people by a lot of different lines of inheritance--a lot of distant cousins marrying each other. I mean, there are a billion ancestral slots that have to be filled, and only 400 million people back in 1240 to fill them. So a lot of people have to be filling multiple slots on your ancestral family tree."

"So does that mean I'm descended from everybody alive in the world in 1240? That they are all my ancestors?"

"Not quite. There were a number of people alive in 1240 who have no descendants living today. And mixing between European populations and the Amerind, sub-Saharan African, Indian, southeast Asian, Chinese, and Japanese populations over the past millennium has been very slow. But if they (a) have any living descendants at all, (b) were alive in 1240, and (c) lived in Europe, on the south or eastern shore of the Mediterranean, or in Mongolia, odds are that you're probably descended from them."

"Mongolia?"

"Your mother's maternal grandfather had Tartar eyes. Mongolia. Chingis Khan. The Mongol Empire."

"So when I read about the history of Medieval Europe, I should think that everyone's my ancestor?"

"Yep. The Holy Roman Emperor Friedrich I Barbarossa of the House of Hohenstaufen, and also the peasant Ludovico il Pazzo from Caserta outside Napoli. Pope Innocent III, or at least one of his close relatives. William the Conqueror and a number of his knights, but also Aethelraed Illraed and Harald Hardrada. Lady Godiva, but also Brunhilde the washerwoman, and Thraxa the house slave back after the Roman and before the Saxon conquest of Britain. Rich and poor, noble and serf, pretty and ugly, smart and dumb--all of them, at least all of them who have any living descendants at all today."

"Kind of stupid that they spent so much time fighting each other, isn't it?"

"Yep."


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Posted by DeLong at December 8, 2003 07:52 PM | TrackBack

Comments

At a guess, there's more cousin-incest, and less social diversity than you think.

At least in my (Askenazic Jewish) family, we've occasionally FOUND ancestors on both (or several) sdies of the tree--it's a small genepool.

Not that I diagree with your ultimate conclusion--an brief affair in a haystack makes a lot of people suddenly related to each other.

Posted by: Dan on December 8, 2003 09:11 PM

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Wow. The funny thing is, it reminded me of this book called "Venus on the Half-Shell." It was written by Philip Josť Farmer, in reality, but it's under the pseudonym Kilgore Trout (whom you probably know as a Kurt Vonnegut character).

Anyway, it's a kind of picaresque sci-fi adventure story, and the main character comes to this society of people who share their minds with their ancestors on a rotational basis. There's this one guy who had a REALLY bad temper, from whom pretty much everyone is descended one way or another. When enough people happen to have him operating their brains (apparently his personality is embedded in each brain, rather than his "soul," as a united entity, having access to a bunch of brains. I think, at least.) on a given day, there are huge riots as a large number of people become violent. I think the guy's name was "shag," and such days are known as Shag Days, or something.

Uh... it's not really a good book or anything (though I enjoyed it), but I just felt the need to bring that up anyway.

Posted by: Julian Elson on December 8, 2003 09:23 PM

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Well, all I can say is, if you are descended from a Godiva branch, please do not discuss economics in your office on one of those hot un-airconditioned days like you had last fall.

Posted by: northernLights on December 8, 2003 09:52 PM

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We're coming up to christmas and you think it's odd that people should have wars with relatives?

Posted by: Andrew Brown on December 8, 2003 10:15 PM

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Masterful.

Posted by: bad Jim on December 9, 2003 12:03 AM

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>>At a guess, there's more cousin-incest, and less social diversity than you think.>> Especially so before industrialisation, when villages were small and cost of travel excessively expensive.

Especially so within the elites of agrarian societies, where marriage was a part of the political game.

Then consider the differences in birth-rates: low productivity populations/households have a strong tendency to have higher birth-rates. Famous people are mostly busy being famous and have little time left for family.

Hence we are probably all descendants of a bunch entirely dominated by competely unknown slaves serfs and poor farmers and fishermen. If you think this is sad, then consider what Darwin found out...

Posted by: Mats on December 9, 2003 12:55 AM

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The technical term for this is "pedigree collapse". It was first publicized by Lawrence Shoumatoff in his excellent book _The Mountain of Names_, but I don't know if it's original to him.

IMS, Shoumatoff cites the specific example of a 19th century King of Spain whose family tree was so tangled that he had only eight great-great-grandparents instead of the normal 16.


Doug M.


Posted by: Doug Muir on December 9, 2003 03:03 AM

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I don't know, Mats. In today's society, poorer people generally have larger families than the wealthier, more famous types, but back in the Malthusian days that comprised all but maybe the last ten generations of humanity? The nobility had access to a better nutrition, leading noble women to have less likelihood of miscarraige and more regular ovulation. I don't know who was more vulnerable to things like epidemics, but I hardly think it's a forgone conclusion, from how we see things in modern times, that the common people were multiplying faster than the elites.

Posted by: Julian Elson on December 9, 2003 03:25 AM

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Ah! I wasn't fast enough! I wanted to post the phrase "pedigree collapse" and I, too, loved Shoumatoff's book--a much racier, more fascinating read on kinship, geneaology and naming than anything I ever read in Anthropology.

Isn't it Alex Shoumatoff?

Posted by: Kate Gilbert on December 9, 2003 07:06 AM

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Some statistical physicists (that's actually a discipline, my discipline, in fact) study this. Here's a paper:

http://arxiv.org/abs/physics/0003016

(One of the authors is Derrida, who is a relative - nephew I think - of the philosopher). Among their findings: about 80% of the population is in the genetic tree, and about 20% belonged to lines that died out.

Posted by: Ben Vollmayr-Lee on December 9, 2003 07:07 AM

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Ah! I wasn't fast enough! I wanted to post the phrase "pedigree collapse" and I, too, loved Shoumatoff's book--a much racier, more fascinating read on kinship, geneaology and naming than anything I ever read in Anthropology.

Isn't it Alex Shoumatoff?

Posted by: Kate Gilbert on December 9, 2003 07:11 AM

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apologies for the double post. I swear I didn't do anything suspect!

OT: Brad, someone just nominated you for Secy of Treasury over at Calpundit (in a Dean administration).

Posted by: Kate Gilbert on December 9, 2003 08:29 AM

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My father dabbled in geneology. My ultimate patrilineal ancestor supposedly turned out to be a noble Norman supporter of Bad King John who was demoted to a commoner by Richard the Lionhearted. H opposed the Magna Charta and everything. Now that's something to be proud of!

Posted by: Zizka on December 9, 2003 08:29 AM

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>>The nobility had access to a better nutrition, leading noble women to have less likelihood of miscarraige and more regular ovulation. I don't know who was more vulnerable to things like epidemics, but I hardly think it's a forgone conclusion, from how we see things in modern times, that the common people were multiplying faster than the elites.>> Maybe so. However, considering that fragile genes according to this desription would survive in the elites, and considering that social mobility is downward mobility for those folks, noble ancestorship is still probably rare and entirely dominated by simple ancestorship. It would be interesting to see some fact here - should be facilitated by the detailed written records on noble genealogy.

Posted by: Mats on December 9, 2003 09:17 AM

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The demographic diamond is one of the cutest things in all of social science.

My ex-wife Susan Schmidt and I once drove onto the sidewalk in front of the mother church of Mormonism in Salt Lake, and sat there discussing whether to go in and get the microfilm on our ancestors.

We elected for a beer -- available when you join a "club" at the back of the restaurant -- instead.

Posted by: David Lloyd-Jones on December 9, 2003 09:43 AM

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Now where did I read that 1/7 of Asian males have Chingis Khan's Y-chromosome?

Posted by: Brad DeLong on December 9, 2003 10:11 AM

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Ben V-L,

Derrida's nephew, cousin and uncle, perhaps?

Posted by: K Harris on December 9, 2003 10:55 AM

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Consider how many were killed off also. Entire families were slain to purge a royal line so that there would be no heir cropping up later. Also, as some have mentioned, there was the issue of incest. Look at the Caesars and how many didn't bother looking past their sister when choosing a wife. That, among other things, lead (no pun intended) to insanity. To be perfectly honest, I'd be rather glad to know I'm not descended from nobility.

Posted by: Matthew Ross on December 9, 2003 11:03 AM

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About Genghis (Chingis) Khan: Here's a UPI story (http://www.upi.com/view.cfm?StoryID=20030205-100301-1566r) about the study The Genetic Legacy of the Mongols (http://www.journals.uchicago.edu/AJHG/journal/issues/v72n3/024530/brief/024530.abstract.html). I think Khan would have found the conclusion of this item "Kind of stupid that they spent so much time fighting each other, isn't it?" to be inexplicable, since his winning made sure he won the genetic contest.

Posted by: Ken Hirsch on December 9, 2003 11:54 AM

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In addition to the Derrida et al. pre-print mentioned, I think some people here might be interested in a recent American Scientist article on this:

Manrubia, S. C., Derrida, B. & Zanette, D. H. (2003) Genealogy in the era of genomics. _American_Scientist_91_, 158-165.

The related web page here is:

http://www.americanscientist.org/articles/03articles/manrubia.html

But you have to be a subscriber to read the article on-line. :-(

Posted by: Jonathan King on December 9, 2003 12:33 PM

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Brad asks >>Now where did I read that 1/7 of Asian males have Chingis Khan's Y-chromosome?>> and Ken replies with a reference saying that >>The Y chromosome of a single individual [Gengis Kahn]has spread rapidly and is now found in 8% of the males throughout a large part of Asia.>>

Now, rather than everyone in today's (local) population being the descendant of everyone in the ancient (local) population, only 8% is the descendant of the noble (!?) Gengis - despite him being indeed very sucessful in this game. So much that Ken's reference concludes >>that events of this magnitude have been rare>>.

Forget any good chance of lots of distant celebrity ancestorship!

Posted by: Mats on December 9, 2003 12:39 PM

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Zizka,

Your "ultimate patrilineal ancestor?" He didn't have a father?

Still, the fact that you have the Sheriff of Nottingham or one of his pals as an ancestor explains a lot

Posted by: Bernard Yomtov on December 9, 2003 12:41 PM

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Mats:

If 8% of Asian men have G. Khan's Y-chromosome, that means that 8% are descended from him BY STRAIGHT MALE DESCENT -- one slender strand of their family tree. I haven't gone through the math, but my statistical instincts tell me that this means that the vast majority of this population would be descended from him on one or another strand of the family tree over the 40 or so generations since he lived.

With regard to "cousins" marrying, I have been able to go back quite a ways in my family tree, and I only have to go back 6 or 7 generations before I start seeing the same names popping up multiple times around the tree (in colonial and post-revolutionary New England).

Posted by: Curt Wilson on December 9, 2003 01:09 PM

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Sorry, my first reading of Ken's references missed that the 8% of population being direct patrilinear descendants of Gengis should be compared to a 800'000 times smaller number. Nothing is said about anscestorship in general, it is only about patrilinearity. So if you are a male in this region, you actually have a good chance of having many noble ancestors, of whome the most ancient is Gengis. But only if you yourself or your close male anscestors were among the noblesse!

>>The conqueror established a social legacy that benefited his sons' sons unto the seventh generation and even beyond.
...
Genghis Khan's male-line descendents clung to power ... For instance, they were not driven from Russia until 1502.

Incredibly, as late as the early 20th century, three-quarters of a millennium after Genghis Khan's birth, the aristocracy of Mongolia, which was 6 percent of the population, consisted of his patrilineal descendants>>

The entire aristocracy! Weird! What will happen in the next 100 years or so if economic development gets a hold?

Posted by: Mats on December 9, 2003 01:13 PM

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Do you take in account that Genghis Khan had a father? (etc.) So how frequent was his Y chromosome in Mongolia before him?

DSW

Posted by: Antoni Jaume on December 9, 2003 02:14 PM

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Stuff any good dog breeder could tell ya.

Posted by: Kate on December 9, 2003 04:31 PM

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I forgot what this post was about by the time I made it down to the "Comments" field. I'm sure that means it's good. Brad's a smart guy.

Posted by: Gabe on December 9, 2003 09:44 PM

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How far did the average human travel over their perspective lifetimes? Early family trees didn't have a lot of forks hence the pasty skin color of many Europeans.

Posted by: Stan on December 10, 2003 07:25 AM

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Re: aristocratic bloodlines, we can tell from genealogies that the typical aristocratic direct bloodline would only last about 4 generations (in pre-scientific medicine times) before dying out and switching to a cadet line. Figure peasant bloodlines to be no better (or worse, pre-science medicine often doing more harm than good). And then you'd have occasional catastrophes like the Black Death or the Wars of the Roses to winnow things further and reset the diversity gauge.

Keep in mnd that during most of the Middle Ages in Western Europe nobles were prohibited from marrying anyone within 5 degrees of relationship. This promoted genetic diversity.

So it's not just a math problem.

Posted by: The Prop on December 12, 2003 01:55 PM

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Re: aristocratic bloodlines, we can tell from genealogies that the typical aristocratic direct bloodline would only last about 4 generations (in pre-scientific medicine times) before dying out and switching to a cadet line. Figure peasant bloodlines to be no better (or worse, pre-science medicine often doing more harm than good). And then you'd have occasional catastrophes like the Black Death or the Wars of the Roses to winnow things further and reset the diversity gauge.

Keep in mnd that during most of the Middle Ages in Western Europe nobles were prohibited from marrying anyone within 5 degrees of relationship. This promoted genetic diversity.

So it's not just a math problem.

Posted by: The Prop on December 12, 2003 01:57 PM

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VERY interesting and entertaining.

The Aboriginal people this side (of the pond)
have a REALLY CLEVER "take" on this:
If you have any "dark" (THEY can say "Black(s)", but us "white dogs" can't ;-) blood in you then IPSO FACTO (?) you are "black". Hmmmmm: I'm sure you know what I am getting at. It's politics; that is, a numbers game ;-). Funnily enough,
(I'm perverse) I may have a few "indigenous" (in our geo. locale the respectful term which I use is "murri") kids running around... ;-)
Fine by me. They're gonna run the country (of Oz)
in ten years or so. Maybe they do now... ;-)
I never realised being white was so unpopular.

(My ex girlfriend and her niece attacked me with a hatchet because I questioned her right to do a financial takedown on me and when she assaulted me with fists, I made the error of eventually hitting her back. Police forensics never showed.
Curiouser and curiouser, replied Alice...)

I TRY to believe in at least six impossible thing every day. Helps me cope with the PTSD ;-)

Great work, Mr DeLong. Shame I'm getting a bit slow...

I apologize for dragging "race" into your "thread". It's just that over here we ARE NOT ALLOWED to talk (about the race situation and the covert race war which is running here)...

Posted by: Don Wallace (I ain't THAT scared) on December 20, 2003 08:38 PM

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