II

  Sofie gives a simpler proof. First, she replaces (x, y, z) with her own letters (u, v, s) so that the boundary surface equation is s = 0.

  Sofie had thought there was excitement on the boundary between two worlds. The path she walks is knife-edge, and in the right mood finding her balance is thrilling, but there are times when she finds the path is bland as buttermilk: slow and stifled, a nothing. She is used to feeling separate because of what she is, because of what she can do. She is not used to the separation resulting from what she cannot.

  “It will be alright,” friends assure her. “You’re a quick learner.” But until she can make herself understood, Sofie’s life is one of silence, of exclusion. This is not a deliberate thing. Most people are kind, but it is frustrating to have communication reduced to pleasant smiles and sympathetic glances, when Sofie knows that pleasantry is not her defining characteristic. If there were other monsters about it would be different, as communication between kind is always an easy thing, but Sofie must deal with more than monsters.

  Limited, she skates on the surface of Stockholm where society is carefully friendly and too polite. Her language is a stilted thing. With only a superficial Swedish tongue, “It’s like wearing a mask,” she says, but she has her mathematics and her students understand numbers as well as words so there is one place at least where she can make herself heard. Still, the mask is an ill-fitting thing: rigid about her bones, and with none of the softness of leather or fur or smooth tiny scales. It takes time to wear in, to smooth over her features until it’s snug against and comfortable, and if Sofie hadn’t had so much experience with skin-changing she would find the difficulties increased.

  The monster bites at its own mask when Sofie isn’t looking; sits at her feet and scratches. The mask has a hole to speak through, and though the monster does not speak it whines–a puzzled, quizzical sound that echoes in the space of snout. Whiskers poke through hollow cheeks, and the muzzle is wet with venom. It has always enjoyed the snow, always frolicked best in cold climates so the monster crunches behind Sofie, its paws dragged through snow, the coils all held up and tucked in. Wordless, it has not the problems of its mistress. Blood and death and poison are constants that anyone can understand.

  Sofie, with her skates on, does her best to look serene. To feel the ice underneath and keep her feet. To close her eyes and circle, with her arms outspread for flying. The monster is not always helpful. It’s not built for streaming–when it walks upon a thin surface the claws punch holes and the poison eats through and the ice is left unsteady beneath. This trips Sofie up sometimes, the unexpected potholes, the gouging in the surface of language, but the monster is part of her and Sofie would cut out her own heart before she would send it away.

  (There is no hiding under beds here, no leaping so as not to notice, and the way its eyes glow in the night has become a comfort to her. The way her own eyes glow, as well.)

  If it were a matter of prejudice, of mathematical fitness, she could kick enough to break the ice but her tongue stifles even her sex. Sofie thinks in Russian (feels in Russian) and the Swedes may be generous but they are not home. They lack bad memories and family and feeling. There’s no-one there to tell her nightmares, no-one but Strindberg and she’s already living his and doesn’t find it bad at all. She could scrawl over walls if she would, ink over the paper with Russian instead of mathematics–symbols from the past, if not the future–but her housekeeping is already under scrutiny and too much monster will tip the path a single way and Sofie has come, finally, to a place of balance.

  She’s always translating, always has to be careful that her substitute words are correct. That a letter change here, a syllable there, does not over-complicate or communicate wrongly. Her life is complicated enough, and there is virtue in simplicity.

  III

  Then, her own theory of partial derivatives says that Poisson’s equation–when using u, v, s–has a unique solution.

  One thing Sofie does understand–she’s heard it before in other tongues than Swedish and the accent gets more familiar with repetition–is the talk of her ability to mother. Of how she put a career before motherhood. No-one asks, in her hearing at least, of how Vlad put his career before fatherhood. (It seems so obvious to them that rocks should come before a daughter when mathematics should not.) Of how he put his life and the ending of it before his duties as a parent. No. When people speak of his suicide, it is in hushed tones that she can only hear with wolf ears, and the tones of their whispers are blame. It is not Vlad that they blame, or not often. Death has covered over any betrayal he might have committed. Sofie, alive, has no such protection.

  Little Fufa, when brought to her mother, to her home and care, has popular sympathy. The Swedish matrons that are her neighbours stroke the little head as if she were not the spawn of monsters. In this they see clearer than they know. Fufa is a bright child, and with such a mother that brightness could be expected to spark, at times, into something more. It does not. If Fufa has not inherited her father’s nightmares then she has not inherited her mother’s either. It is always a mixed thing, to be disinherited. But inheritance is not the same as proximity, and so Fufa is talked about and tutted over and the neighbour’s voices are dripping with pity and indulgence.

  The child has a slew of bad examples to learn from. Depending on who’s talking, those examples are the mother who walks as a monster walks, with a step so heavy that it reverberates all around; or the conformity, if kind, that exists all around. This last is Sofie’s fear. She has seen what expectation does to children, seen them fight and choke and shatter. Like Mishel, prodded into conformity at Sofie’s knee, if not at her direction. She thinks about him sometimes, wonders what he could have been if he were allowed to be as he was. Not a monster, perhaps, but not a misery either, pushed into putting away his paints, into forcing his fingers to trace lines of analysis instead of instinct. At the time, as a child, she had felt little for him but contempt that what came so easy to her came so hard to him. That he had allowed himself to be overtaken, to be moulded away from self. Now, looking back, she just feels sorry for him. So very sorry, that he was stunted alongside her while she did nothing, while her monster grew beside her.

  Sofie doesn’t want to raise another Mishel.

  It would please the matrons if she did, if she herded Fufa away from science, if she set her at her feet with paint brushes in her fists, with needles to practice her darning. These are things that Fufa might enjoy, given half a chance. But when Sofie comes with the monster to see her daughter in bed of a night, to tuck the blankets close and watch her sleep, she sees the long child-lashes on the sleeping cheek and super-imposed over all is the face of her little cousin. The child that Sofie helped to break, though she was innocent of intent and indifferent to the breaking.

  She will not be party to a repetition of that time. Not deliberately, and not through neglect. Fufa will have to find her own way, her own talents and interest. It is all the more important that Sofie allow her this freedom, because she can see with her witch-sight and sharp wolf eyes, that Fufa is not a monster. She will not force her way as Sofie did, and no dark creatures walk beside her.

  (There will be other nightmares, Sofie does not doubt, for everybody has them, but most nightmares disappear with the day, do not come along behind. They are not contagious. They are intangible.)

  Because her daughter is not a monster, Sofie doesn’t cover her shabby walls with calculus, stand Fufa before them and stupefy her into another obedience. But she doesn’t paint them over either, set before her child a blank and blandness that hides from view what lies beneath.

  “I’ll raise my daughter my own way,” she says.

  IV

  This solution is conditional, and analytic.

  Sofie hears her Nanny’s voice in the mutters of monstrosity, the murmurs of broken homes and unnatural mothers, in the rumours of women that are half a wolf, that have witch blood in their veins. These things are all true, as far as they go. Yet Sofie has long accepted her own monstrosity; made peace with it and nurtured, even, the monster in herself, the one that walks behind and ready to hamstring. She is not interested in excision, and she will not give up her mathematics, cut it scalpel out of her, no matter what she hears. Poison runs through her as well as witchery, and there is no rumour so toxic that she fears the venom.

  Yet the one that concerns her most is the one she has heard before, and often. It is centred about her daughter, about her own capacity as mother. Sofie is called “unnatural” in her maternity and this does not particularly bother her. She is well aware that she lives outside the norm, that her work shares space with her daughter and that this breeds comment if not always contempt. “Unnatural” she can ignore, she who has been unnatural since smallness, when stories made her more than what she was. It is “bad” that Sofie has trouble with. She sees little correlation: an unnatural mother is not a bad one.

  A bad mother would raise a Mishel, would clip and stunt and smother. A bad mother would tell bad stories. Sofie does none of these things.

  She does tell stories, however. She is a mother, and stories and cradle songs come from her as well as anyone. Here again, Sofie walks the middle path. There are no nightmares, not for Fufa. But there are no milk stories either, no twee little tales and soppy narratives enough to build a dream girl, one who gives up everything for duty, for expectation, and ruins her life thereby. Instead, Sofie’s stories are more substance than spectre: algae, infusoria, and not one word about the black death.

  (Nanny is shut in a box, smothered, and not before time. Sofie has heard more than enough about that dark death, about burning and blackness and buboes, about shutting up and nailing in. These are not stories for Fufa, but they are entrenched in Sofie’s body, engraved upon her bones, and so she locks up Nanny tight, hammers at the door of the little room she’s locked in and lights the match. “This is what monsters do,” she says, with the nails clamped between her teeth and a mouth that tastes of iron. “We fight back.”)

  Sofie is prepared for social compromise. This is not a tactic that comes easily to monsters, but Sofie has learned the hard way about prices, has learned what she is willing to pay and what is worth paying for. She has never regretted the cost of mathematics–the emigration that was close to exile, the broken home, the fractured mirrors that showed the monster for marvellous–but these were not compromises, not really. They were the price of extremes, they needed no balance.

  Stockholm has taught Sofie balance, has upset the inner ears of monsters well enough so that she must concentrate to keep her feet, so that she does not fall into knives or into conformity, as far as she can with the university behind her and her skin imprinted with algebra, with calculus. If there are accusations of badness, of monstrous motherhood, these are things that can be ameliorated. Sofie will never embrace normalcy, not entirely. Her mental wallpaper is too well stuck for that, but she can deflect criticism in other ways. She can blend the smaller pieces of herself, keep the monster close and take comfort in camouflage. There are expectations, and some at least she can fulfil. She will converge with the world on dress, on acquaintances, on everything that the most prim and pristine women of her circle have imprinted on their bones, on their hearts. She will shun most of what they could possibly object to–on everything but mathematics, on everything but motherhood. On everything but Fufa.

  This is her boundary, her point of change, and she will derive nothing from their methods.

  This is the price of monsters.

  V

  Every point on the surface S is transformed by invertible equations.

  December, 1888. Sofie wins the Prix Bordin. It is a day of celebration for monsters. Sofie has a doctorate, she has a handful of published papers and these things says that she is not a debutante, not a dilettante. They say that she has talent, and that she has applied it. This is the most important thing, the application. In another world, Sofie might have stayed at Palibino, or near around, and she would have married a man her family approved of, would have given up her mathematics for trotters. She would have been friends with Olya–not best friends, perhaps, but warm acquaintances and any talent she would have had would be reserved for helping her children with their homework when her social duties were done and over with.

  Her wallpaper would have come from Petersburg, and it would have been beautiful.

  It was a life that she might have enjoyed. Sofie is a monster, but she is not so lost to monstrosity that she cannot see the appeal of the expected, the ease of life with mirrors that don’t reflect atrocity or the source of trailing footprints.

  In another world, Sofie might have been another woman. But in this world, in her world, the world of exile and hard truths and symbols from a darker place than Stockholm she is a mathematician, and now a well-rewarded one. Not so much in the way of wealth, for though the Bordin is rewarding it will not keep her all her nights. It is in reputation that the prize is so valuable, and Sofie’s reputation can always use the shoring up.

  Placed against the whispers and hamstring colleagues and poisonous playwrights, the Bordin is a success that cements her place as a mathematician of note, defines her identity, underlines it. In the shadow of the Prix, Sofie does not need to say “I told you so.” She does not need to say “Look at what I can do. Look at what you would have taken from me, from all of us.” She no longer needs to justify or to explain, for her success is so marked that there is no mistake and never has been. Sofie has not wandered into mathematics as a whim, or a means to draw attention. Everything she has done, the false marriage and forced, the disobedience, the not knowing her place... It is all proved to be necessary, to be a gamble paid off, and when Sofie walks the streets of Stockholm now she notes that people look her monster in the eye, acknowledge its existence, when before they would let their eyes slide over it. As if it weren’t there; as if it were impolite to stare, to note how out of the ordinary it was. As if it weren’t dangerous, a mouse trailing along behind instead of a monster.

  The Bordin transforms the monster brain. For Sofie, so used to transformation, to skin change and sorcery and alteration, it is an almost familiar thing. Marriage changed her choices, widowhood her chances and Fufa her dreams, so this is the next step only. A monster is not a static thing. Even Strindberg is forced to recant, his two plus two making five and this is about as far as his mathematics can take him so Sofie does smile, this time, in thanks for the reminder of knives more than for the making-up, for the bare truce that comes between them.

  It seems that Strindberg sees the salt water now, him and most of those who opposed her. There are holdouts, of course. There always are, but the Bordin is acting as a lens and the perception around her is altering. “I could have told them so,” says Weierstrass, writing to Sofie from his German fastness, who wakes now with sand in his bed and the appreciation for marvels that he has always had, the man who saw the mermaid in the monster and took her in thereby.

  Sofie, sandwiched in her bed with the monster snoring on one side and Fufa on the other, smiles to read it. There are times when she can see the mermaid in the mirror now, only glimpses it’s true, but glimpses are all that she wants. “I am a monster at heart,” she writes back, in letters that will be burned. Writes with a hand that holds witches and wolves and iron, that holds porcelain and venom and hammers. That can hold salt as well, and scales of a different sort, for she is better now at stories. “But a mermaid is a monster to others than you. I don’t mind. One more won’t make any difference.”

  VI

  The Poisson equation is then also transformed: Sofie’s theorem is satisfied–(and Brun’s)–the coefficients are analytic.

  Sofie is returning to Stockholm from a trip away. She is looking forward to home–or what has become home, with her daughter and her position and the clear light that still feels unfamiliar, sometimes, on her skin. Journeys seems to her to get longer as she ages. She doesn’t feel the excitement that she once did, when she and Vlad and Aniuta had Europe all spread out before them.

  The journey home is longer than she expects, and more difficult. She is advised by a fellow traveller, who brings bad news in narrow carriages like coffins. “Stay away from Copenhagen,” he says, and Sofie can see the fear in his eyes, can smell it on his breath. The monster perks, waking from its doze beside her and the fur stiffens, scents itself with pox and contagion, little scabs forming on the dozen snake heads and drifting down like confetti. Sofie brushes them off her dress, off the monstrous body. The weather is so cold, there is no more need for snowflakes.

  Sofie hears the warnings repeat, from more travellers than the first. There are rumours of smallpox in the city, and if no-one has thought of the black death, of nailing people in and hammers like iron fists then there is a push, at least, to keep others out. To keep the pox from spreading, to contain it within the boundaries of Copenhagen.

  (The mermaid has sores on her tail and blood in her mouth. It is a sea city, a salt city, a place of mermaids and marvels but unhealthy now. It is not a safe place for Sofie, who is just enough mermaid to be infected, whose monster blood is not panacea enough.)

  There is only one decision. Sofie changes her plans, changes her travel. A monster never likes to do as told but that does not mean she must ignore sensible advice, and beneath all else are Nanny’s stories: how it was to be a child of the black death, locked in and locked up for burning, with the body betraying all around. Pox and plague may follow her as perfume, the sweet sticky scent of nightmares, but this is nightmare given form and not her own. It is an unwelcome apparition, and concerning. A monster always recognises another of its kind and Sofie feels the pull as magnets, feels the connection and attraction between them. She can feel the black death winging down on her, an alien monster drawing down upon her, alien because when did any part of her monster have feathers?

 

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