Flexigation
A short story by Brian Howell

Sato-san sits to the left of Satomi. He is beyond sartorial criticism – smart black tie falling over the buttoned eye-holes of his shirt, dividing its crisply ironed shape in perfect symmetry.

They talk on a park bench within plain view of the school as the students make their way home. Sato-san is attempting to demonstrate to his prize pupil one of the many puzzles which absorb him. Satomi has only recently shown any inclination towards this side of her mathematical bent. Previously, she had worked ably and fulfilled her duties as the best in the class.

He is fascinated by patterns of all kinds. In nature, in abstract theory, in art. Morphic resonance is a favourite indulgence of his. This last is his next project with Satomi. He will often stare at the cream dissolving in his coffee, forming specific nebulae, sometimes planets. The contrail of an aeroplane will often stop him in his tracks as he tries to make out a certain figure.

While these concepts hold a genuine fascination for him, there is one which dominates him, and has grown in its compulsion throughout the years. His interest here is purely cerebral. It seems inferior compared to his other pursuits, but remains his secret passion.

He will scan the inside pages of the national newspapers in the staff-room, and curse the ever-increasing number of crimes against and, in many cases, by minors. There is an inner rage in him which does not find expression. He genuinely detests these men who abduct young girls and keep them locked up for years. If there is to be capture involved, it is rather the mind of this child he seeks to engage. His bait, his butterfly, goes by another name – the hexaflexagon.

The bewildering variety of uniform design, of which Satomi’s uniform is one example, often diverts him. It is well-known that many schools, including his, employ top designers to come up with ever more fashionable outfits. His young pupil Satomi sits attentively, tartan skirt splayed slightly, its pleats resting gently on each other, like inflated air pockets forming inside a parachute as it touches water. Sato-san deliberates on this image awhile, not too long, of course – long enough to glimpse dark folds resting on each other, patterned light just visible through the surface.

‘Now, Satomi, we are going to make a hexaflexagon. Have you heard of this phenomenon?’

‘No, sir. Is it a shape?’

‘Yes, very good. I’ve prepared all the material. In essence it’s all very simple. You take one strip of paper, like so.’

He takes out from his file a strip of paper about 4 cm wide. One side is already marked off in 19 equilateral triangles, numbered in an interlocking sequence of 1,2,3, 1,2,3, 1,2,3,1,2,3, 1,2,3, and 1,2,3, the last triangle being left blank.

The other side is divided in the same pattern but numbered 4,4,5, 5,6,6,4,4,5, 5, 6,6, 4,4,5, and 5,6,6, this time the first triangle being left blank.

Satomi observes in silence.

‘We have to get the same underside numbers to face each other, so that 4 is on 4, 5 on 5 and so on, and we do this like so.’

He folds the same-numbered triangles so that he soon has a strip about half as long as the original. The pattern on one side is now 1,2,2,3,3,1,1,2,2,3.

At this moment a damsel fly lands on the hem of Satomi’s skirt as it opens out slightly over the edge of the bench. Should he say anything? Sato-san thinks of folding wings. As a child he was terrified of these monstrous hovering creatures whose sudden shifts of direction you could not anticipate. Should he say something?

‘Next, you fold along this line.’

Satomi sees half a hexagon and suddenly there are more 2’s than any other number. She itches under the pleats of her skirt.

‘And one more time.’

Satomi sees a complete hexagon, of which every triangle is numbered 2 except the last, overlapping triangle, which Sato-san glues to the blank triangle on the other side. He turns the thing over to show on the reverse side all 1’s.

‘Now it’s time to flex our hexaflexagon.’

‘Flex?’ Satomi questions.

Sato-san rests the shape on his thigh, holding one side between thumb and index finger. He holds the opposite side with the same fingers of his other hand.

‘Now we pinch these two adjacent triangles and bend the paper a little.’

Immediately, another side appears, covered in 3’s.

‘Here, you try, ‘ he says, handing the shape over to her as if it is a delicate insect. At the same time, he has a strange feeling, as if he were indeed handing something incredibly precious over to her.

She copies his actions exactly and immediately another face opens up, this time revealing a side covered in 5’s.

She does not say anything. She does not say, How can this be? as he rather cruelly imagines her class mates reacting. He can see she is already working the mathematics through. Almost obsessively, she continues to flex the object, sometimes producing all the 3’s, then all the 5’s, and eventually all the 4’s and 6’s.

‘Can you see a pattern?’ he says eventually. He has noticed that she has realized if she concentrates on one corner until it refuses to open, then moves on to an adjacent corner, she gets into a cyclical sequence.

‘Yes,’ she almost shouts in her excitement. At this moment, she screams and lifts up her skirt, revealing a naked thigh as the damsel fly zooms out.

‘Yes, yes, ‘ she says, recovering quickly and barely noticing how shocked Sato-san had momentarily looked. ‘Numbers 1, 2, and 3 turn up three times as often as 4, 5, and 6, don’t they?’

‘Fantastic,’ he says, allowing himself a rather unusually demonstrative expression. ‘Well done, ‘ he enthuses again.

Satomi sits still – slowly, almost nonchalantly, flipping over the hem of her skirt. She seems suddenly deflated. ‘There is more, you know,’ he says. ‘There are many variations on this shape, with more complex numbers, many more faces.’

How long have they been at this? The sun has gone down considerably. It will be dark soon. There isn’t a teacher or a student in sight, not even the janitor. Then two schoolgirls whiz by on a bicycle, one girl riding pillion on the tiny hubs of the back axle.

Sato-san stares at the backs of the girl’s knees, her pristine white slouch socks, her browned thighs. My Satomi is not like that, he thinks. Then the mild-mannered, smartly-dressed teacher hears the impatient sound of his student flipping the hem of her skirt. He turns towards Satomi.

There is only her skirt. Her blouse flutters in the evening breeze a few feet away, as do her snow-white knickers and socks. He resists the urge to pick them up. Then he hears her voice coming, he thinks, from behind him.

‘Sato-san, Sato-san.’ He stands up.

‘Satomi, where are you?’

‘Here, Sato-san, here.’

‘Where?’

‘On the bench.’

On the bench, but there is only her skirt, neatly folded, in the shape of a hexagon, a perfect, kaleidoscopic tartan hexagon.

‘Satomi, stop playing games. You’re very clever but this is enough.’

‘But I am here, sir, in the hexaflexagon.’

He is surely going mad. But her breathy, almost asthmatic voice is coming from the very centre of her skirt. He touches it and realizes it has the texture of crinkly paper.

‘Unfold me, Sato-san.’

‘I am so sorry Satomi.’

He picks the skirt up and starts to flex, it is hard, as if starched, almost like cardboard. But the sides are impossible to tell apart. He flexes the shape, sweat beading his brow, for nearly an hour. Occasionally he sees a part of her face appear on one of the triangular sections. Then it disappears, only to appear again on another triangle, in an inverted position.

All the time, she seems to be cooing encouragement, wooing him. She is saying that she is happy where she is. But he has to go on flexing, he has to rescue her.

Sometimes the whole shape opens up into a giant aperture and he sees her in the centre, as if she is stuck in a black hole. Then he flexes again, and any hint of her disappears.

At some stage, he notices a pressure on him, a force of some kind. He looks straight down and almost laughs. His tie is caught, being pulled towards the centre.

‘Come, Sato-san, it is late. You have worked hard enough.’ He sees her hands, her arms pulling on the tie.

It is time, he is tired, he gives himself up. The pressure is too much.

On the bench a crisp, fluttering sheet of paper, lit only by a nearby street lamp, is suddenly lifted up by a gust of wind, and gives out an echoing sigh. A pile of clothes and a pretty girl’s uniform lie scattered, gently fluttering in the breeze.