15 Minutes Past Sagittarius

The Mathematical Mean

The math professor, Miss Priest, was surely herself a Euclidean Solid – her form appeared sculpted and her features were so classical that when she moved it could be startling, as if seeing an ancient statue come alive. Indeed, movement seemed the only change she ever expressed – her face never showed any emotion, unless one counts such things as curiosity, contemplation, and the occasional flash of intellectual recognition.
Mark admired her from his desk midway in the classroom, but was distracted by the arrival of a new student. The girl was as light-haired as Miss Priest was dark, and her face betrayed the timidity which most people experience upon entering a new environment, especially when late. Mark made certain to draw her attention to the empty desk beside his, and smiled her a welcome when she took it.
The Professor continued her lecture without marking the new student. “Φ (pronounced Phi )is not the mathematical mean, but can be constructed from one.”
The girl whispered to Mark, “What does the ‘mathematical mean’ mean?”
‘What does ‘mean’ mean?’ – Mark toyed with the question in his mind. “It just means ‘average’,” he replied. But his usual playfulness fell under a growing shadow, a sense that this answer was somehow incomplete.
Miss Priest said, “Φ is called a ‘mean’ because it is a ratio, but it can’t be reduced to a simple fraction. In decimal terms it works out to one point six one eight dot dot dot, with the digits right of the decimal repeating infinitely, without a pattern. Thus Φ is an irrational ratio.”
The new girl stifled a giggle.
Mark remarked, “She means what she says quite literally – she has no sense of humor at all. But go ahead and laugh – she won’t be offended because the woman has no emotions whatsoever.”
He blanched, suddenly realizing that Miss Priest was now standing fairly close and looking at him – she very likely overheard his remark. But he dismissed his alarm by reasoning that if what he said was true then she couldn’t resent his characterization any more than the girl’s amusement.
The professor strolled to the chalkboard as if her entire demonstration was choreographed. Drawing the figure, she said, “Take a square ABCD, with sides of x. Now find the midpoint of the side AC – we’ll label it ‘E’. From E draw a line to an opposite vertex of the square, to B. The line EB is the hypotenuse of a right triangle, h, with one side x, and the other one half x. From the Pythagorean theorem we determine that h has length x times the square root of five divided by two. Now, swing the hypotenuse erect from E, above point A…”

“My my,” murmured Mark, “I hadn’t realized that our hypotenuse was limp.”
“..and label that point F. Complete a rectangle by extending BD up to point G, and connecting points F and G.”

Mark studied the movements of the professor’s body while she drew her figure. Amazing, really, that she never got chalk on her clothes during her lectures. As she stretched to draw FG her left side swept quite a pretty arc from heel to wrist, and every pause she made appeared posed.
Miss Priest did the math to show that when x equals one, the line FC has length Φ and the segment FA has length φ (phi, lower case Phi) because for any value of x the ratio of FC to AC is Φ.
Mark recited in his thoughts,”Fee fie fo fum – Reach for the vertex, my Pythagorean mum.”
The professor cocked her head as if trying to identify an unfamiliar noise while continuing,
“What we have constructed is the Golden Rectangle – so called because this is the most aesthetic proportion for a rectangle. From this, Φ is called the Golden Mean. Interestingly, when a square is deducted from this figure, the remaining smaller rectangle will have the same proportions – it will also be a Golden Rectangle.”
The professor marked off the discarded squares with arcs, reflecting “Perhaps Φ delimits the proportion of truth that Plato thought is lost with each successive mirror of Beauty on its downward spiral into our world.”
Miss Priest paused. “Having derived Φ by geometry, we can also find it algebraically. Remember that when x is one, the long side of the Golden Rectangle is Φ, and that the addition to the side of the original square is φ. But Φ is not only φ plus one – φ also happens to be directly related to the reciprocal of Φ. So we can set up an equation for Φ – ‘Find a number Φ which is equal to its reciprocal plus one.’” Developing the equation on the board, she concluded, “..and so by the Quadratic Theorem, Φ equals one plus the square root of five, divided by two, which you’ve seen is the value of Φ in our Golden Rectangle. The other Quadratic root is negative – meaningless as a line length and so cast out.”
“Pythagoras I know, and Euclid I know,” thought Mark, “But who is Quadrat?”
The professor turned, the chalk in her hand a pointer which probed among the students until marking Mark. She appeared puzzled by the humor shining in his eyes, but resumed, “We’ll examine the more exotic qualities of Φ in the next lecture. Now let’s take a break to go over your progress evaluations.”
It was unusual and certainly not required for a professor to provide these reviews, and it spoke well of her intention that each student do his best. She conducted the evaluations individually, letting half the students leave the room while she called the others in alphabetical turn to her desk.
During the initial shuffle Mark overheard Miss Priest telling the new student, “When I was a little girl in Ohio, I used to play along the Chattahoochee River, too.”
Mark protested, “Professor, the Chattahoochee is in Georgia, not Ohio.”
“I know what you mean,” she said with a warning look. “Nothing interesting ever came out of Ohio, so we have to borrow features from other states.”
Mark went outside to take his break with the first group, thinking about her odd remark. “What could that mean? What could she mean by that? What does ‘mean’ mean?”
He mused,”‘What does that mean?’ is asking what thought something is intended to convey. But ‘What did she mean by that?” is more like asking what motivates the thought that was actually expressed, or what effect the idea was intended to produce.
He recalled from logic class that a mean argument is a middle term in a syllogism, uninteresting in itself, but a necessary step between the premises and the conclusion. The hypotenuse that Miss Priest drew in deriving Φ from a square was like that – the hypotenuse itself was discarded once the Golden Rectangle was drawn, but it was needed in the middle stage of construction. “Sometimes ‘mean’ is the intention, and sometimes it’s the technique used to achieve it. Then the hypotenuse was a mean used to derive the Golden ‘Mean’.” Mark sensed that his conclusions were spiraling down toward non-being.
Again Mark panicked, realizing that because his last name belonged in the first half of the alphabet he should have remained with that group for the evaluations. Whatever the professor may think of it, he would regard himself guilty of a serious academic breach if absent when called. He rushed back into the classroom, but fortunately she was counseling the student just before him in the order. She showed no regard for privacy, conducting the interviews at her desk in full hearing of anyone else interested. Still, she meant well by her counsel. Mark took a seat in a quiet corner near the front to be at hand when called.
He noticed a pair of curvy parallel lines drawn on the blackboard. Maybe they were sine waves. “More aesthetic,” Mark thought, “for this professor of polygons to be drawing curves than straight lines.” Then he noticed that the figure could be a quick sketch of a river winding its way – had she reasserted her claim on the Chattahoochee? But it could equally well be the hastily drawn symbol for ‘approximately equal,’ or even the glyph for Aquarius – that was surely Miss Priest’s sign. He could feel the meaning in the mark like a verb in a sentence with subject hidden and object unexpressed.
Then a familiar rascal in his mind whispered that if the bottom wave was flipped, the curves would be become the sketch of a reclining female form. Mark imagined the Golden Rectangle as a horizontal picture frame around a portrait of Miss Priest lounging on a sofa, inadequately draped, of course. “And so our professor is the pi in Phi,” he thought. “Aesthetic indeed – Φ(π) is a sweet function.”
He leisurely painted the curve of her hips in his mind, his brush strokes making them blush pink, when he began to sense a cold scrutiny. The professor had finished the current evaluation, and was studying Mark prior to counseling him. Feeling her attention as a marked presence, he suddenly realized that she was a psychic, and was using that skill now as a guide in how to approach her next interview. Even worse – she was an empath; she hadn’t only been eavesdropping on his thoughts – she was actually experiencing them through him. Her face was becoming sweaty and blotched as it reflected the hormones in his blood. As she examined him, what had simply been his boyish fun in imagining something vaguely naughty looked quite beastly on her. Positively negative. Mark felt violated, nasty – and now she was calling his name. The logical inquiry of Professor Priest was quite mathematical, and mean.

Robert C Flanders
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