While snowed in at his home, a geometer writes to his former lover about his students, his discoveries and how much he misses her.
This is one of those literary art pieces by an author for whom mathematics is nothing but meaningless poetry and the stereotype of the insane, antisocial mathematician. I am sure there are those who would praise such a work, but it is difficult for me to do so. Imagine reading a story written in your native tongue by a person who does not really know the language well, and that moreover the plot seems to be based on the worst misconceptions about your country, with characters that are stereotypes you've seen too many times before. For me, as a mathematician who reads a great deal of mathematical fiction, that's what this story is like.
Towards the beginning, some of the mathematical discussion is sufficiently poetic that I had hopes the story might contain some interesting ideas:
(quoted from The Song of the Geometry Instructor)
I was still working on the projection problem then, remember? The one with the helix. I felt certain that form this equation my students and I could extrapolate a whole new family of formulae and constructions. But progress was slow. I could get as far as the bending helix, the helix which twisted up to form another helix, but after that ste I would become confused. To even conceive a figure, Hazel, not only of infinite extention itself but one which created by its extension a second figure also of infinite extension which in turn created another such figure infinitely extended and to continue in this fashion so that ultimately not only is extension infinite but generation as well...to conceive such a thing is to drive thought to its knees....
dnv(f_{1},f_{2},f_{3}...) W *=C
dnv(f_{1})=4X_{1}
dnv(f_{1},f_{2})=17x_{2}^{2}+(?)X_{2}X_{1}11X (or (X_{2} . X_{1})^{2}/(6X))
dnv(f_{1},f_{2},f_{3})=(?)
(converge to polynomial in X_{1},X_{2},...?) ...
dnv(f_{1},f_{2},f_{3},....,f_{n})=does not compute.

This is essentially meaningless and does not sound at all like real math research, but it is sort of interesting without being offensive.
However, by the middle of the story where the character's insanity is apparent (see the quote below describing his hallucinations as he builds models of his "discoveries" out of junk in his basement), it became clear that this was not going to be the sort of mathematical fiction I enjoy:
(quoted from The Song of the Geometry Instructor)
But then one day after I had been working for several hours on a pair of adjacent, mirrorimage pentadecahedrons  I say several hours, but in reality, Hazel, it may have only been minutes or it may have been weeks  my fingers, grown quite stiff with the cold, slipped from one of the vertices and broke through two horizontal parallels. Before I realized what was happening, I was on my feet, trembling, waving the broken figure before me in the air. My students were shocked. I wanted to reassure them but somehow could not speak. And how strange they looked. Harelips, turgid pustules on cheeks and chins, snot encrusted septums, glabrous bellies that oozed over brass buckles, vermin infested pudenda, great strawberry marks over foreheads and eyes, slewed feet, scaley arms, brown teeth, denim crotches fat with priapic bloat. I Ieaned forward and took an obese student's nose between two fingers. It collapsed with a faint pop. They were so fragile. I bent down and pushed my hands into the soft surface of the mattress. And not beautiful at all. I ripped out two fistfuls of stuffing and threw the polyfoam particles up into the air. They drifted down over the bodies of my students. No, my students were clearly not beautiful. I picked up a can of nails and hurled it at them, but they did not move. And were they crying?

Published as part of the collection Plane Geometries and Other Affairs of the Heart, all by the same author, which together won the 1984 Illinois State University Fictive Collection Award. The story A Decisive Refutation of Herbert Dingle's Objection to Einstein's Twin Paradox, or, Gravitas which also appears in the book might arguably be considered mathematical fiction. However, I am not giving it a separate entry. (I would consider it to be "physical" rather than "mathematical" fiction.)
Contributed by
Vijay Fafat
A lyrical musing written in the form of a plaintive missive or diary of a geometry teacher to his girlfriend, Hazel, who has left him. The main subject is the geometer’s obsession with an infinitely convoluted geometric figure which he thinks must exist but whose equations he is unable to formulate completely. Over time, through a couple of inspired thoughts and sudden mathematical epiphanies, he starts grasping more and more of the shape of his curve, and while he still remembers his girlfriend, the new geometry supplants her.
The story has a strange background setting – of a very anomalous winter which blankets most of north Florida in multiple feet of snow for months, trapping the mathematician inside his house, where he finishes his research. But as you read the story, you suspect that perhaps the snow and the entrapment is not in the geography but in the mathematician’s mind – that he is slowly going insane. The author never says this is so but it works wonderfully in that context, if it were to be implied in the end. Berry’s style is quite poetic in this regard.
Some beautiful, extended extracts about his expression of his work and inspiration:
(quoted from The Song of the Geometry Instructor)
“To even conceive a figure, Hazel, not only of infinite extension itself but one which created by its extension a second figure also of infinite extension which in turn created another such figure infinitely extended, and to continue in this fashion so that ultimately not only is extension infinite but generation as well.. .to conceive such a thing is to drive thought to its knees. I could almost glimpse the near edge of the problem some mornings, like a huge shining scaffold against the blackness of space, fluorescent lines falling away to reveal curves, curves forming crystalline figures which then flared like great white novas to reveal thin lines again, a phantasmagoria of shapes. But when I would try to put it on paper, it would fade. Not shatter or crash or burst. Just fade”

(quoted from The Song of the Geometry Instructor)
“Then one day an extraordinary thing happened. I was sitting at my workbench worktracing a series of interpenetrating dodecahedrons, prisms, ellipsoids, and cones, when my eyes began to ache. I sat back in my chair and rubbed them. The light was not very bright, and I assumed they were tired. The stereo was playing Bartok's second piano concerto, and in the adagio movement where the piano plays pianissimo in the treble and the strings hardly seem to move at all, I closed my eyes and let myself float out onto the sound. All at once the scaffold appeared before me. Hazel, you can imagine my surprise. It rose up immense, even boundless, yet entireno longer truncated as beforeand flashing, spinning, silver as a thistle of ice. I watched as its network of lines dissolved into explosions of light revealing crystalline figures, trapezoidahedrons, pentadecagons, shining conic trisections, all slipping smoothly in and out of one another and spinning off into
coils within coils within coils. I leaned into my eyelids. How had I thought to tether such a figure to laws and signs? I saw my equations breaking loose as the curves twisted past, watched as divisors, exponents, coefficients, and radicands snapped off and tumbled into indigo depthless space.”

(quoted from The Song of the Geometry Instructor)
“Then, for the first time, Hazel, I understood what space could become for me. I realized what my figures had always been trying to mean, and that now when you saw them you would understand, too. I began to limber my fingers in the warm air. I would build again, only now I would not restrict myself to discrete and familiar forms. I would let possibility devour space. I envisioned twodimensional cones, thirteensided dodecahedrons, involuted ellipsoids, vanishing helices, spirals that shrunk and swelled, great networks of planar trisections, and impossible hyperboloids. I would fling thought into the realm of the scaffold itself.”


