Thursday, May 10, 2007

Michelle's sister

I have a distant relative from Hong Kong who can be briefly described as my grandmother’s cousin’s wife. Due to cumulative age differences, my grandmother’s cousin’s wife (who is obviously two generations above me) is only 12 years older than me. We will call her Michelle for convenience.

At one of the parties last week, Michelle was seated with a few cousins around her. “Yat Wai,” she called my name in Cantonese, a strange glint in her eyes, “come over here for a moment.” The cousins eyed me, a strange mischief seeming to dance on their faces.

I plodded over and Michelle asked, “Do you have a girlfriend?”

Time stops. Cue pan shot of motionless scene à la The Matrix.

A cluster of neurons discharge, triggering a cascade of electrical discharges in neighbouring cells. I articulate a careful response.

“Ah, good,” she clapped her hands together softly, a smile appearing spontaneously, “I have a sister 12 years younger than me. Just the same age as you!”
“Err, is that necessary?” I had no other ways to respond.
“She’s a smart girl, pretty, studying medicine.”
Out of nowhere, my mother materialises and interject, “yes, look at how pretty and smart aunt Michelle is.”

For the rest of the week, I get teased occasionally by the aunts about this sister of Michelle.


Which brings us to the main topic of today’s discussion: the family tree as a mathematical object.

Consider the typical family tree, neatly arranged so that offspring are positioned below the parents and siblings are located side by side.

As a consequence of the above rule, members of the same generation are always on the same row. Members on separate rows are from different generations.

Now construct a function called generation, which is simply a measure of which generation a particular member is in. The generation number increases as one looks at earlier generations. Arbitrarily setting my generation to generation 0, my parents would be labelled generation 1; grandparents, generation 2. And so on.

Now to apply the concept of a path integral to this generation function. The path integral of the generation function is the difference in generation between the member at the start of the path and the member at the end of the path.

Note that this path is completely arbitrary and need not follow the sequence in which members are arranged on the family tree.

Consider this short path, starting from myself and ending at my brother:
Myself, my grandmother, my aunt, my cousin, my father, my brother.

Now, take the path integral by taking the change in generation at each step:
Myself to my grandmother. Grandmother is 2 generations above me; hence the change for this step is 2.

My grandmother to my aunt. My aunt is my grandmother’s daughter, and hence my aunt is one generation below my grandmother. The change for this step is -1.

My aunt to my cousin. -1

My cousin to my father. My father is my cousin’s uncle, and my father is one generation above my cousin. The change for this step is 1.

My father to my brother. -1

The path integral is the sum of these integrals, which is 2 - 1 - 1 + 1 – 1 = 0. Which makes sense, as my brother is in the same generation as me and hence there is no generation gap.

Now, one can see that in a typical family tree, this generation function is conservative: the path integral of the generation function does not depend on the path; it only depends on the two end points.

It follows that any integral taken along a closed loop would result in zero. After all, this is a conservative function and the closed loop means that the start and end points are the same; hence there is no difference between the start and end points.

Having introduced the path integral of the generation function, now consider a hypothetical question.

What if I marry this Michelle’s sister (whom we shall call Φ for convenience)?

Marrying Φ results in a deformation of the family tree topology. As a marriage creates a new connection element in the family tree, it creates a paradox of sorts. If I look at the marriage between myself and Φ, then the marriage would mean that we are of the same generation. However, if I trace backwards to my grandmother, her cousin, his wife and her sister, it turns out that we are not of the same generation.

If we take the path integral of a closed loop in this scenario, starting at myself to Φ, then back towards Φ’s sister, her husband, my grandmother and back to myself, we would find that the path integral is not zero but +2 (dependent on the direction of the loop).

Hence, by choosing the appropriate path integral to loop around this abnormality, I can be my own grandfather. Taking two orbits around the loop, I become my own great-great-grandfather.

Due to slackness on my part, I’ll not attempt to produce drawings on a sample family tree, the associated path integrals and the deformed topology resulting from the hypothetical marriage (althought this last one would be enlightening).

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