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We were asked to write about fishing. Goodness me, what a dull topic... I had recently been discussing pi (π) with a friend and the obvious pun (pie) was hovering round my mind. Then the phrase pilateral symmetry appeared from nowhere at all and suddenly I had my story. All that remained was to write it down. I hope it works -- I remain uncertain as to whether I explained enough (or if I explained too much) and it's a rather odd story even by my standards. But I quite like it anyway.


Gone Fishing

The things that swim in the Martian canals aren’t really fish, but what else can you call a creature that spends its whole life gliding around under the water? Like all Martian fauna, the fish exhibit trilateral symmetry – they have three eyes, three large fins and their tails are trifurcated. Trying to catch them, which I suppose, for want of a better word, you’d have to call fishing, is a very popular sport – though you really wouldn’t want to eat anything that you caught. You won’t die if you eat one, but you’ll definitely wish that you had!

One day I was sitting around staring blankly at nothing much at all when Xerxes popped his head round the door and said, "Want to go fishing?"

His name isn’t really Xerxes of course, but I can’t pronounce most of the syllables that make up his name. I suspect that some bits of it lie outside the range of sounds that the human ear responds to. The first time I met him he shook my hand vigorously with both his right hand and his middle hand, as he introduced himself. The dog next door started to howl. That’s what gave me the clue. So I just call him Xerxes. He doesn’t seem to mind. It works both ways, of course. He can’t pronounce my name in any understandable manner, so he compromises by never calling me anything at all. It seems to be the best way to solve the problem, and we get along quite well together.

"Fishing?" I asked. "Yes, OK."

The nearest canal is about a twenty minute journey away from where I live. Xerxes and I piled into his brand new four wheel drive beach buggy, and he set off across the dunes. Usually this would be a very bumpy ride, but the fat, round tyres on the new buggy made the trip very smooth indeed. "Why did you suddenly decide you wanted to go fishing today?" I asked.

"Oh," he said, "I’m really quite excited. There are rumours that a pi-fish has been seen swimming in the canal. They are incredibly rare. I’ve never actually seen one in the wild."

"Pie fish?" I asked in bewilderment. "Are you going to make a fish pie?"

"No, no," said Xerxes impatiently. "Pi-fish. Pi, not pie. The ratio of the circumference of a circle to its diameter. 3.14159 followed by an infinite number of other digits that you can look up on the internet if you are so inclined. Or you can calculate them for yourself if you have a lifetime or two to spare for the task. Pi-fish."

"What’s a pi-fish?" I asked, honestly puzzled.

"Oh, it’s a really strange creature," said Xerxes. "It doesn’t have bilateral symmetry like you do, and it doesn’t have trilateral symmetry like I do. It has pilateral symmetry. It’s the only Martian animal that does."

"What is pilateral symmetry?" I asked. "Do you mean that it has 3.14159 eyes and 3.14159 fins and 3.14159 sticky out bits on its tail?"

"Don’t be silly," said Xerxes. "How can you have fractional bodily organs? What would 0.14159 of an eye or 0.14159 of a fin actually look like? That makes no sense at all! No – the thing about the pi-fish is that the ratio of its length to its breadth is always exactly 3.14159… And a tiny little bit, of course… Weird, eh?"

I thought about that for a time, then I said, "So that means the pi-fish is proportioned so as to be able to bend itself around its own body and put its head by its tail in an absolutely perfect circle?"

"That’s right," said Xerxes, "though actually it becomes a solid cylinder. That’s because the breadth and depth of its body are both the same size, and of course either one can be thought of as the diameter of the circle that is formed by the length of its body. So therefore it turns into a rather squat cylinder when it rolls itself up. That’s what it tends to do when it’s attacked. I’m told that it’s quite amazing to see it happen!" He parked the buggy carefully by the canal. We unloaded our gear and set off for our favourite fishing spot where the angle of the sun cast shadows that kept us cool while we fished.

The canal itself ran straight as an arrow across the rust-red Martian desert. The banks were lined with ancient concrete. Occasional hieroglyphic daubings interrupted the smoothness of the concrete slabs. It says a lot for the skill of the Martian builders, and the durability of their materials, that the graffiti are still legible all these thousands of years after they were first written down. Once I had asked Xerxes what the writing said but he only shook his head and told me that they were just complaints about the quality of the food, the harshness of the working conditions and the ugliness of the overseer’s wives. Workers are all the same no matter what planet they are working on.

It doesn’t matter whether you are fishing on Mars or on Earth, the procedure is always exactly the same. You impale something that wriggles and is enticingly yummy on your hook, you cast your line and hook out into the water and then you sit back and wait for something to take a nibble while you eat and drink and talk about life, the universe and everything. It’s a very relaxing way of passing the time. But today Xerxes wasn’t at all relaxed. He was tense, and twitchy, and he kept jiggling his rod as if trying to check whether there was something on the other end of the line yet. Eventually his impatience was rewarded. The end of his rod bent down towards the surface of the water and he started to reel his line in.

"Do you think you’ve caught a pi-fish?" I asked.

"I don’t know," he said. "I hope so." The line came up out of the water with something blue and wriggly on the end of it. "It is a pi-fish!" yelled Xerxes. "The blue colour is a dead giveaway. Nothing else on Mars is that colour."

He swung the pi-fish over on to the bank of the canal and released it from the hook. The pi-fish looked around in panic for a moment and then, just as Xerxes had promised it would, it rolled itself up into a tight blue cylinder. Then, as I watched in fascination, the cylinder began to shrink, becoming smaller and smaller until it was just a thin blue line on the sand. Then, a second or so later, the blue line vanished and there was nothing left to indicate that a pi-fish had ever been there at all.

"What just happened?" I asked.

"That’s why pi-fish are so rare," said Xerxes gloomily. "They keep disappearing. Once they roll themselves up into a cylinder, the mouth starts trying to ingest the body so the cylinder gets smaller as the fish eats itself up. All the other bodily dimensions start to shrink in sympathy because, of course, they have to keep the proper ratio between the fish’s circumference and its diameter. A pi-fish always maintains its pilateral symmetry even under the most stressful of conditions. That’s what pilateral symmetry is all about. So the pi-fish shrinks and shrinks as it continues to ingest itself. It shrinks itself down to atomic and possibly even sub-atomic dimensions, vanishing from view while the ratio of its ever decreasing length to its ever decreasing breadth continues to be exactly 3.14159..."

"And just a little bit," I said.

As I said that, my own rod began to vibrate and Xerxes got excited all over again. "Reel it in," he ordered. "Maybe you’ve caught a pi-fish as well."

I reeled it in, and sure enough there was another blue fish twisting and twirling on the end of my line. I swung it on to the bank and pulled out the hook. Just like the first one, it immediately rolled itself into a cylinder and started to shrink. But this time I interrupted the process. I jabbed a fish hook into each end of its body and maintained a constant pressure that grossly distorted the pi-fish as it shrank.

"Stop it!" cried Xerxes in horror. "What do you think you’re doing?"

"I’m destroying the fish’s pilateral symmetry," I said. "I want to see what happens. Think of it as a scientific experiment."

Xerxes rummaged in his tackle bag and produced a ruler. He measured the rapidly shrinking pi-fish and then punched some buttons on his pocket calculator. He frowned and shook his head at the result. Then he did some counting on his fingers. "That’s a bit odd..." he said.

Eventually the distorted, and by now completely unsymmetrical, pi-fish vanished from view just like its predecessor had done. Or, the thought suddenly occurred to me, perhaps this pi-fish actually was its predecessor, vanishing back into itself all over again deep beneath the waters of the canal, ready to begin life again, just as it did every time the thin blue line disappeared. After all, it had to go somewhere when it vanished from view. Indeed, maybe there was only one pi-fish in the whole universe. Let’s face it, pi only needs to be defined once. I wondered if anyone had ever seen a shoal of pi-fish.

Xerxes put his calculator and ruler away again, and we re-baited our hooks. But the day’s excitement appeared to be over. Neither of us caught any more fish for the rest of the afternoon and eventually boredom set in. "Let’s pack up and go home," I said. "I’ll buy you a beer."

"OK," agreed Xerxes.

We carried our rods and tackle bags back to the beach buggy.

"You really did get a magnificent vehicle when you bought this buggy," I said to him.

"Yes," agreed Xerxes. "I had to pay a lot extra for the square wheels, but I think that they give a much smoother ride than the triangular wheels most people settle for."


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