3 April 1991
Brother Benoit licked his brush to get a really fine tip, dipped it into a pot of red ink, and added another stroke to the manuscript that he was illuminating. A remarkable multicoloured border was taking shape, a cross between a cat and a cactus, looping spiky tendrils in all directions. It contrasted rather well, he felt, with the Sierpinski gasket that formed the initial letter 'A' of the word 'axiom'. Many orders of monk, such as the Benedictines, specialise in the manufacture and blending of fine liqueurs. The relatively obscure order of Euclidean monks has developed a line of applied geometry, ranging from graphic design to the construction of mechanical toys. Brother Benoit had a special talent with fractals. He switched to a green brush to sketch in an array of seahorses...
Leather sandals slapped across the stone floor, and Benoit turned to see who it was. Oh dear, Brother Douglas. Benoit normally avoided him, because he had a rather shady reputation and was generally considered unsound. But on this occasion Brother Douglas bore an important message. "The Abbot requests your attendance," he said. Benoit's hand began to shake and he had to put the brush down before the seahorse became a sea-cucumber. A summons from the Abbot was always bad news. As he set off along the echoing corridors he tried desperately to think in what manner he might have been guilty of some lapse.
He stopped for a moment outside the Abbot's door. Then, stiffening his back and his resolution, he knocked on the worn oak panels. Adversity borne with fortitude strengthens the soul, he told himself. Even so, his heart sank as the thin reedy voice bade him enter.
"Brother Benoit, I am greatly displeased." Not a good start.
"My Lord Abbot, if I have offended in some way then I --- "
"Why no, Brother Benoit. I merely seek your excellent advice on a matter that displeases me. I hear you have considerable knowledge of mechanisms." As long as he doesn't ask me to fix his lap-top hymnal... The Abbot had a passion for modern innovations. Benoit relaxed, as much as one could in a Euclidean monastery. "You may recall," said the Abbot, "that recently the monastery was fitted out with new Venetian blinds. I am perplexed. With the old blinds, I could adjust them in mid-morning to admit partial light into this room, suitable for reading and writing, after which they required no further adjustment until the late afternoon. But now I find I must move the blinds almost hourly, to maintain a constant level of lighting. Why is that?"
Brother Benoit walked across to the window, and looked at the new blinds. Unlike the traditional design, with horizontal slats, these hung vertically. The Abbot had bought them from a salesperson who made great virtue of the fact that very little dust would adhere to a vertical slat. Benoit fiddled with the cords that adjusted the angle of the slats, turning them this way and that. The room sank into almost total darkness, then a bright beam of sunlight illuminated the far wall, then it fell dark again.
Metaphorical light dawned. How to put this in a way that wouldn't offend the Abbot? "I believe, my Lord Abbot, that it is a consequence of the vertical slats."
"Nonsense. Slats are slats, are they not? Turned one way, they keep the light out, turned another, they let it through."
"Indeed, my Lord Abbot. However, in the sun we have a moving source of light, and that has a profound effect on the --- er --- geometry." That was a good word to use: like any good Euclidean, the Abbot always approved of it. "The slats are closely spaced parallel lines. When the light source is aligned with those lines --- that is, when the rays of light that it emits are also parallel to them --- then the greatest proportion of the light that impinges upon the blind is admitted. Indeed, if the slats are infinitely thin, then 'almost all' of the light is let through --- the proportion blocked is vanishingly small." The Abbot nodded, permitting a degree of annoyance to cloud his features, and Brother Benoit made haste towards the point of his explanation. "If the incident light meets the slats at an angle, then much of it is blocked. Indeed, the greater the angle between light rays and slats, the less light there is admitted."
"Brother Benoit, this is admirably and concisely put, but I do not yet find that it illuminates the distinction between horizontal and vertical blinds."
"Lord Abbot, in order for the illumination to remain approximately constant, the angle between the light rays and the slat must change as little as possible. Here the source of light is the sun, which moves through the sky as the day progresses. However, because our monastery is placed in northern latitudes, the sun's angular variation in altitude is considerably less than its motion from east to west. Therefore a horizontal blind can be set in a position of compromise, a little dark in the early morning and late evening, a little bright at midday, but acceptable for the entire period. Whereas a vertical blind must be adjusted every few hours to follow the sun's movement." (See Fig.1)
1 Comparison of (a) horizontal and (b) vertical slats in a Venetian blind.
The Abbot nodded, but also frowned. He had ordered the blinds.
"If I may be permitted a personal observation, it is a small price to pay for the noticeable absence of dust on the blinds," Benoit added hastily. "And of course the more frequent adjustments actually help to dislodge any dust that might be attracted by static electricity."
"Well spoken, Brother Benoit. You may take the reading at matins tomorrow. And since you are in such fine fettle, perhaps you would offer your advice on a matter of greater importance. It concerns the monastery's sundial." This rusty and unreliable device, some five centuries old, had crashed to the ground the month before when a flock of pigeons had alighted upon it simultaneously.
"I could build a replacement, my Lord," Benoit offered eagerly. "The design is based upon the same geometrical principles of the motion of the sun, in combination with that of the Earth."
"Mmmm... I think not," said the Abbot. He pulled back the sleeve of his robe to reveal an expensive gold watch. "I have something more akin to this in mind."
"We could decorate the sundial with gold leaf, my Lord."
"No, Brother Benoit, I was not referring to the decorations. Observe the face."
"Ah. It is a digital watch."
"Precisely. I feel it would be more in keeping with the modern age if our monastery wase equipped with a digital sundial."
"Lord Abbot, it would be no trouble to incise upon the rim of the sundial the angular numerals favoured by designers of digital watches."
"No, Brother Benoit, you do not catch my drift. Let me demonstrate with this signet ring. See, when I hold it to the light, it casts a shadow on the desk?"
"Yes, my Lord."
"And in what form is that shadow?"
"A circle, my Lord. Like the ring."
"Indeed. But when I twist the ring on edge to the sunbeam, so?"
"A single line, my Lord."
"Precisely! And do those two shapes remind you of anything, Brother Benoit?"
"The numerals 0 and 1, Lord Abbot."
"Excellent! Yes, this ring casts a shadow which, when illuminated from one angle, resembles the digit 0; but from another, the digit 1. What I have in mind is an object whose shadow changes with the motion of the sun, and at each minute resembles the appropriate time, written out in digits." (See Fig.2)
2 The digital sundial.
"You mean that at twenty-three minutes past seven the shadow should look like the digits 7.23? And at twenty to eleven like 10.40?"
"Yes. And the shadow should correspond to the time at every minute of the day, while sunlight lasts."
"It is... a great challenge, my Lord Abbot."
"Indeed. But, I am sure, Brother Benoit, one that will not prove to be beyond your capacity. I certainly hope not, as I am sure so do you. Now, I must attend to more pressing business. You may go."
That evening, after vespers, Brother Benoit begged an audience with the Abbot. He had come up with one possible solution, a system of lenses and etched glass squares, which effectively ran a movie whose consecutive frames were 7.23, 7.24, 7.25, and so on, and projected it on to a wall. But this the Abbot rejected. "Digital devices do not contain moving parts," he said reprovingly.
Brother Benoit was sitting miserably in a cloister, at his wit's end, when Brother Douglas sat down beside him. Despite the monk's bad reputation, Benoit, who needed all the help he could get, poured out his troubles.
"Casts different shadows, eh? From different directions? Mmmph, reminds me of something... Back in a minute." Douglas reappeared with a book. Benoit looked at the cover, and dropped it in shock.
"But, Douglas, this book is on the Index! It's proscribed. I can't read this! Where did you --- "
"The Abbot's private library, brother. Old duffer keeps the titillating stuff for himself. I often borrow something juicy."
"You could get me in really serious trouble with this," muttered Benoit. He bent down and retrieved the copy of Gödel, Escher, Bach --- an Eternal Golden Braid. He thumbed its pages, found nothing salacious. "Why is it prohibited?"
"They say that the relationship between Achilles and the Tortoise is a bit unsavoury."
"Oh. And is it?"
"Don't know. Haven't read that far. But let me show you the bit that I was thinking of. Look at the cover."
The cover depicted a curiously shaped wooden object. It cast three shadows. To the left, the shadow was shaped like the letter G. Below, E. To the right, B. "I thought it might give you a clue," said Brother Douglas. "At least it shows how you can get three totally different shadows from the same object. Like the old puzzle of the cork that fits a square bottle, a round one, and a triangular one. Maybe there's a general principle."
"Hmmm. General principle... You know, brother, you may just have a point. I've been concentrating too much on details, like getting the shape of the digits right. What I should have done was to study a more general problem: what is the relationship between the shadows that an object can cast in different directions?"
"And do you know the answer?"
"I think so," said Benoit, turning the book in his hands. "This picture makes that much clear."
"So what is the relationship between the shadows that an object can cast in different directions?" asked Douglas.
"There isn't one," replied Benoit.
It took two weeks of non-stop effort in the workshops before the monk was satisfied. He slid the final strips of etched gold leaf into place, thanking every deity he could think of that the monastery was equipped with the very latest machinery for thin-film photographic etching. He carried the finished artefact carefully to the Abbot's quarters for a demonstration...
"My Lord, the scale model is finished," Brother Benoit said proudly. "If it meets your approval, I will order the construction of a full-sized version for the courtyard." The Abbot looked down at the table, where the faint but clear shadow 1.52 was visible. He looked at his watch, which read 4.17.
"My apologies, Lord Abbot," the monk replied, when the discrepancy was pointed out. "A small maladjustment." He fiddled with the sundial until it too read 4.17. The Abbot placed his watch on the table next to the sundial. As the watch clicked over to 4.18, there was an instant where the shadow looked a bit fuzzy; then it too read 4.18. For several minutes the shadow of the sundial tracked the digits on the watch face.
"Ingenious," said the Abbot. "Perhaps you will explain how it works."
"My Lord, I began to understand the nature of such a device when I asked myself a very general problem. What is the relationship between the shadows that an object casts in different directions? Now, for the common objects of daily life, which are relatively simple in structure, composed of flat planes or smooth surfaces, the answer is that the shadow must vary continuously, so that small changes in the angle of illumination produce small changes in the shadow. Indeed, that is how the human brain recognises objects from their outlines.
"However, my training as a monk of the Euclidean order has left me well versed in all aspects of geometry. As you are aware, my own speciality is fractals. Now fractals have fine structure on all scales of magnification. It occurred to me that if I could somehow 'amplify' that fine structure, I might create a fractal whose shadow varied discontinuously. I conceived the notion that, given any list of shadows whatsoever, there should exist a shape which, when illuminated from a fixed series of directions, exhibits those shadows. In a very strong sense, the possible changes in the shadows cast by a suitable fractal should be totally arbitrary.
"It remained to establish whether this notion was true. I found that in essence it was: a mathematical theorem to that effect was proved in 1986 by Kenneth Falconer of Bristol University in the United Kingdom. There is one technical modification, however: the shadows, instead of agreeing precisely with those prescribed on the list, may have to differ by sets of zero area. Of course, such sets are invisible to the eye, and therefore do not affect the operation of the sundial. The underlying idea, remarkably, is the Venetian blind principle, which is used in an iterative manner (Fig.3) to obtain a set whose shadow in some directions is large, but in others vanishingly small. By piecing together many such sets, and applying a limiting argument, the theorem follows. In our workshops I was able to carry out this procedure in sufficient detail to produce the digital sundial that now sits on your desk."
3 Fractal Venetian blinds have shadows that are large in some directions, small in others. These are the basic building blocks for sets whose shadows can be prescribed arbitrarily.
The Abbot, despite himself, was impressed. "Brother Benoit, your sundial is a marvel! Your reward will, of course, be in heaven, this being spiritually more uplifting than any earthly recompense." A thought struck him. "Perhaps you would care to assist in another small task?"
The monk had little choice but to agree.
"I hear that in Tibet they use machines in which a written prayer is wrapped around a wheel. When the wheel is spun, the prayer revolves, and is deemed to have been 'read'. That is a highly efficient innovation, which I believe has much to commend it. It would increase our prayer production quota a hundredfold. However, as a devout Euclidean you will be aware that only spoken prayers are meaningful. I wonder whether principles similar to those that underlie your digital sundial could be used to create a machine that, as it revolves in the wind, creates the appropriate sound vibrations..."
Oh my Lord, thought Brother Benoit. Now he wants an alternative energy prayer synthesizer.
Kenneth Falconer, Sets with prescribed projections and Nikodym sets, Proceedings of the London Mathematical Society 53 (1986) 48-64.
Kenneth Falconer, Digital sundials, paradoxical sets, and Vitushkin's Conjecture, Mathematical Intelligencer 9 (1987) 24-27.
Kenneth Falconer, Fractal Geometry --- Mathematical Foundations and Applications, Wiley, Chichester and New York 1990.