Daily Archives: April 17, 2012


Let’s start with a little combinatorics problem.

Say there are four friends gathered for lunch. Let’s call them Alana, Alex, James, and Trish. They are sitting in a restaurant booth, two on one side, two on the other. Now let’s say that the two men – Alex and James – are right-handed and the two women are left-handed. As they are about to start eating, it is observed that they have managed the most unfortunate arrangement of conflicting chirality: on each side, a left-hander and a right-hander are seated so that their active elbows are towards each other and thus will be in conflict throughout the meal. Now: assuming that all possible seating arrangements of two and two are equally probable, what are the odds of this state of affairs?

This is the question with which we challenged ourselves while waiting for our pancakes, eggs, bacon, et cetera on Sunday. I invite you to ponder it. I will give an answer at the end.

I will say that we decided not to rearrange ourselves, and we managed suitably well anyway. But, ah, the bedevilment of chirality.

We know chirality is bedevilling. How many times have you used, or heard, “The other left” when a direction to look or act left was responded to with a look or act to the right? Or vice-versa? We may know which hand we write with (except on certain mornings), but we still manage to confuse the sides from time to time.

Mirrors, those semiotic prostheses, highlight the issue. Indeed, the Oxford English Dictionary definition of chiral (from which we get chirality) is “not superimposable on its mirror image”. It is often thought that a mirror reverses the image. It does not; it just happens that when someone turns to face you they rotate on a vertical axis and so they reverse on that axis. You might as well think mirrors reverse front and back: if someone has their right on your right and their left on your left, their face is away from you, not towards you as the mirror would suggest. Indeed, if we turned by flipping so that our heads were down and our feet up, some people would say that mirrors reversed head and foot, not left and right.

But, although these thoughts are automatically prompted by today’s word, I am risking drifting away from the word at hand. And chirality is a word at hand, quite literally. Its more plain-spoken English equivalent is handedness. It comes from Greek χείρ cheir “hand”, which shows up in various other words, probably the best-known of which is chiropractor (they use their hands to work their medicine). You will also see chiropodist, which is not someone who uses their hands on your feet but someone who treats both hands and feet (compare otorhinolaryngologist, an ear-nose-and-throat doctor).

There are few other cute words that use this root. Among them are chiromancy, palm-reading; chirognomy, the form of the hand and the study of one’s character from that form; chirosophy, a synonym for either of the previous two; chirography, handwriting; chiromachy, hand-to-hand combat; and my favourite, chirapsy, touching or rubbing with the hand. (Ah, good hands are a rhapsody, and a good touch of a good hand can lead to rapture.)

In every case, the ch is pronounced /k/ in English; the original Greek has it as a velar (or postalveolar) fricative, but it came through Latin, which made a /k/ sound of it, and English has long since lost its velar fricatives anyway. So chirality starts with a hard stop at the back, and then moves to the tongue tip: a liquid, a liquid, a stop. The vowels gradually and with a slight hesitation proceed from low-central to high front. But it exhibits no right-left chirality.

Could a speech sound exhibit chirality in its articulation? Actually, yes. Ask a speaker of Xhosa or another language containing a lateral click which side they click on. If you’re not sure what I’m talking about, it’s the sound many Anglophones use to summon a horse. I do it on the right, but can do it on the left if I want. Other than that one, you will see little chirality in speech articulation. And even with that one it doesn’t have a meaningful effect on the sound.

But chirality can have a meaningful effect in other areas. And I don’t just mean the stats that show that left-handers don’t live as long as right-handers. There are many things in nature that have chirality, from seashells right down to certain molecules – DNA, for instance. A mirror image of a molecule (an enantiomer of it) is a different molecule. Sugars twist to the right; the mirror-image version of them tastes the same but isn’t digested. There would be sweeteners made of left-handed sugars, except no one has found a way to make them economically. (Naturally, such apparent tampering with nature would be seen by some as sinister. Fittingly, since sinister is Latin for “left-hand”.)

Similar issues of chirality appear in electromagnetism, particle physics, and mathematics – including geometry and anything else that has spatial implications. Such as some combinatoric problems.

Oh, yes. To return to our opening problem: one in six (22/4!). There are 24 ways to sit (4×3×2) and 4 ways to sit in the chirally pessimal arrangment (2×2). The optimal seating arrangement, from an elbow perspective, would have males on one side, females on the other, with elbows away from the wall. The odds of that are also one in six. That’s assuming that the four of us won’t by habit have a male and a female on each side, always a possible inclination for us dioecious humans.

coccygeal, coccyx

Oh, look at that chick! The backs of her shirt and pants have parted ways a bit! See? Oh, see? See?


There’s a tattoo there! She has a coccygeal tattoo!

A what?


No, no, no, I mean what do you mean by coccygeal? Are you going cuckoo?

Ha! Thereby hangs a tail!

A tale? I have a bone to pick with you.

I’d say you’re picking a tailbone. Specifically a beak-shaped one.

A beak? But a beak pecks. That’s why in England they call your nose your pecker. Which I am aware means something else in North America. Here, though, you’re talking about a tail. No, not that kind of tail.

I’m talking about the coccyx.

The cock six? What? This is really going downhill.

No, no, no, coccyx. That means “tailbone”. It just sounds like “cock six.” It comes by way of Latin from Greek κόκκυξ kokkux “cuckoo”. Apparently your tailbone is shaped like a cuckoo’s beak.

At least cuckoos’ beaks are still useful. A tailbone is just there for jarring on things. And what was that other word you used to refer to the sigil she has intagliated on her vestigial entailment?

Not really intagliated, just inked. The word is coccygeal. The adjective relating to the coccyx.

“Cock sidgy all.” It occurs to me that it sounds a bit congealed, concealed, or occluded.

Well, not that one. I won’t say you can see the tip of her tailbone, but…

Yes, you can see her butt, a bit. Perhaps a bit to excess.

But don’t look now.

Did you just say “coccyx” again?

No, that was the sound of her boyfriend cracking his knuckles. I said don’t look… he might cold-cock you one and clean your clock…

Say, what is that tattoo?

It’s some kind of cyclic form… an ouroboros?

Ah, yes, the snake eating its tail. You know who dreamed of those and unlocked the molecular structure of the benzene ring?

Yeah… Kekulé.