I
Consider the ring: it has cylindrical coordinates. Consider the ring, and find its potential.
Sofie is older now, a woman grown, or nearly. She can never truly be adult in her father’s house, ringed round as she is with restriction, with limitation. In one sense there is safety there, the safety of a closed-up house with bars over the door and the windows nailed shut from the outside. There is nothing that can hurt her there, nothing that can sneak in from the outside unobserved.
Sofie wants to be able to be hurt. She doesn’t want to be tucked up, wrapped up in wool and confined against contagion, albeit on the wrong side. The atmosphere is already contagious, and although she doesn’t recognise the source of infection in her sister’s former suitor it has infected her nonetheless, her immunity undermined and Sofie wide open to a different type of disease–one more subversive than bubonic–and to a poison that will not surface for years. She thinks herself both healthy and half-mad with health, with boredom and insulation and exile from the world she wishes to enter, the world of wishes and wallpaper that is foreign and familiar at once.
“No,” says her father. “I will not send you to Petersburg. What good would it do? Girls are not permitted to enter university in Russia. You know that. You should not be so selfish; those places are reserved for men.” For Mishel, who is long gone and little brighter. For Fedia, who is so unexceptional that Sofie and Aniuta often forget that he even exists.
“No,” her father says again. “I don’t care what those foreign colleges permit. I’ll not allow any daughter of mine to go gallivanting about the continent.” And Sofie, kept safely at home where she belongs, without the necessary paternal signature for flight, spits and groans and does not settle, turns restless as snakes. Aniuta is more thoughtful, counsels patience and makes her own plans–but when she rests her hand on Sofie’s the younger is too angry to feel commiseration.
She has to get out of the house–away from the wallpaper she’s outgrown and the walls behind it. Her father’s house, with gates like threaded finger-bones and calculating thorns that anchor to a nursery of lectures she’s either outgrown, or doesn’t want to hear. “There’s more than one way to skin a cat,” says Aniuta, and Sofie should know in her own wolf pelt, her own changing form, that permission can be drawn from more sources than one.
“I can’t forge his signature,” she says. “My fingers don’t have the knack.”
“But you do have fingers,” says Aniuta, cryptic, and when she goes away to her correspondence, still as secret as before if not so underground, Sofie holds her hands before her and stares. She does have fingers–and in one of those fingers a bone that, when bound about with gold can force its way into a keyhole without breaking and trip the lock.
“I know some people,” says Aniuta. “Some men.” Young, like them, and raised in the shadow of the Crystal Palace. Young men who wish to be educated, who are willing to take wives in name only so that their wives can be educated too, and the marriage annulled when the learning is over. “It is not only women who care for rights,” says Aniuta. “And a husband can supersede a father.”
“You want us to marry?” says Sofie. It is not an avenue for freedom that she has considered, and the monster beside and inside growls in scepticism.
“One of us has to,” says Aniuta, and they are walking in the garden together, conscious of poison fruit and too subtle now to eat where they can be seen doing so. “A wife can be a chaperone. Father would have to trust us. He knows we would not allow the other to come to harm.”
Why would they? Their teeth are the same, if different shapes, and if Sofie drags nightmares behind her then she has Aniuta to balance her out.
II
The Fourier series takes jagged, ragged functions with saw-blade teeth and serrated edges, strips them down to a sum of simple sines. The potential is a Fourier.
Aniuta goes visiting with Sofie, with a friend. The three of them together are come to interview, almost. Come to search out a comrade rather than a husband: a man who will not share their bed but who values freedom for others as much as he values it for himself. This is a rare thing, and Sofie expects nothing but hopes for a doll that will not shatter. That is what she searches for: a puppet with the strings cut, who is willing to share his scissors. Who will marry her in expectation of a broken home and make the breaking a friendly thing, and not the stuff of Nannies and nightmare. She does not want to gain her freedom at the cost of stuffing all pulled out and spider cracks in porcelain.
It seems a lot to ask. “Is it really?” says Aniuta. “Would you not suffer minor inconvenience to save another from slavery? It is a question of principles, Sofie. We are not the only ones who have them.”
Easy for her to say, Sofie thinks. One can buy–or bypass–a lot of principles with a face like Aniuta’s. She sits quietly offside, feels like a pig brought to market. It is only when he comes to sit by her that she hears something to drown out the sound of trotters: his voice, and the monster’s. They speak in concert and in opposites.
His name is Vladimir, and he is many things. Interested in geology, in palaeontology, in women’s rights and revolution. He is as desperate to study abroad as Sofie. It would be nice for him to have a friend. Someone from home, someone to talk with. Someone to be a brother to. Sofie, who has a brother already when she bothers to remember him, is charmed. This brother would come to science as she does, would not circumvent her. He has the eyes of someone who has seen the Crystal Palace and Sofie forgets, has long forgotten, the time when she would have given it up to have another man talk of marriage to her.
Under her skirts, the monster growls. It does not stop, and constant echoes come up her legs, settle between her thighs and reverberate in a warning that Sofie has no interest in hearing. Vladimir is her ticket out. He means well. He is kind and clever.
(He prefers Sofie to her sister.)
He is also prone to depression and woeful with money. Sofie does not see these things–only eighteen, the sines of her paper nursery are more apparent to her than the signs of character, and Vladimir has so many good points that she forgets he must also have bad. None of these show on his surface. There is no warning that her freedom may be bought with black tides and empty pockets. His skin is pink and smooth and optimistic and it covers all the weakness beneath. There is nothing about him that Sofie can scent as wrong, but she gets so little preference that she mistakes scent for sensibility and her nose is all confused. The monster who stays with her, the exterior mirror of herself, is less involved. It snaps at Vladimir when he comes close: snaps with teeth like nails and the unconsecrated breath of witches. Sofie has to smack it on the snout to stop the growling, and feels the blow echo through her own flesh.
“Stop that,” she snaps in turn, and the monster retreats under her skirts, hissing from all its mouths and sulky. “This is my way out, the best it’s going to get.” Vladimir may be many things, but to Sofie he is in sum a husband.
She smiles when Vladimir shakes her hand, when they shake on their agreement. It is a pleasant feeling to be revolutionary, to twist expected behaviour to other purposes. To be able to upend tradition for the greater good, cast off the shackles and usher in a new world. In the silence of her skull, Sofie thinks she hears the sound of nails being dragged from wood, the sound of hammers imploding and peaceful sleep. It is the sound of wedding marches.
III
Laplace thought a solid ring orbited in an ellipse. Sofie came to think it was an oval.
Strange, to be married first. To be married at all. Sofie’s model is her parents: a father with the force of his daughters and a mother that faded thereby.
Sofie does not feel like fading.
There is a sense of excitement, or embarkation. Of freedom within reach, and mathematics spread before her. The chance is so close she is not willing to give it up, not for anything, so when her father disapproves of the engagement, wants to have it dissolved in silence and privately, Sofie holds with both hands and manipulates. Makes herself public, risks a loss of reputation and unless her father wants to see the entire family shamed, he has to cover up quietly and go along, and he does.
(This is how it is, she thinks. This is how it happens. Action, reaction, then being left to get on with it. It is a lesson taught her by her father and Aniuta, and one she has never forgotten. It is burnt into her blood like plague, like the buboes rising under.)
The marriage takes place and it is a fiction at first, a way to rope themselves together and ring the changes for women’s rights, for revolution. A way to get Sofie an education, to let her learn to fulfil herself, to bring her out past the borders of her country and into daylight, into lands where the native nightmares are not hers, and ineffectual.
That it is a fiction doesn’t matter, for Sofie and Vladimir are friends and young enough to believe it would be alright. They have a dual focus, after all–have matched themselves on matched assumptions and find themselves in Germany with the roads spread out before them. Sofie paves hers with symbols, with equations, and the borders of Russia lie behind her. For Vladimir, pavestones are made from rocks and fossils, and the rooms they share are scattered with books and untidy. There is no time for cleaning, and Sofie’s monster leaves paw-prints in the dust, leaves snake trails and the shadows of hob-nail boots. It disapproves, still growls at Vladimir when he knocks on her door of a morning, but as house-mates they are well-suited and Sofie, who would tolerate a lot for the education she is getting, has to tolerate very little and most of that is poverty.
“How many rich students do you know?” says Vladimir, and Sofie laughs. All their friends are poor. It is worth it to learn: she would rather have black bread and beer in Germany than the cakes and cognacs of her father’s house. Aniuta has left them to go to Paris. Fluent in the underground post, she sends her letters home via Sofie, so that their father thinks they are still together. She has even less desire to return home than Sofie does, and they spend their freedom in different ways, friendly shadows of each other.
Time passes, and passes again and for Sofie freedom and fiction become intertwined, and their previous habits are underwritten by familiarity. “There is more to us than friendship now,” she says, and she is not the only one to say it. The marriage is no longer a paper thing: it turns from fakery and into love, and Sofie, let out of the castle where she would have wasted her life away, gets a happy ending.
For a while.
Then the marriage that had become more becomes less and the relations between them distort. They split and shift under the weight of expectation. It is one thing to be comrades and students together, where separate interests are prioritised and expected to be so. Another when the last barrier is removed. Sofie’s life begins to be one of balances, of compromise and conciliation–but she cannot balance and compromise and reconcile by herself, and Vladimir is set as off-centre as she is and the bond between them, the golden band, starts to stretch itself out of shape.
“Told you so,” Sofie hears at night, with Vladimir beside her and so deeply asleep he does not wake to growling. The monster is back under the bed, banished there when the marriage changed so it wouldn’t get in the way of love-making, but Sofie is conscious of it where-ever it is, conscious that it sees Vlad as an intruder, as a weight and anchor.
“Told you so,” she hears in mirrors, when her reflection faces her and her complexion, pale with strain, is less witchy than before, her teeth neat and even and filing down on themselves to nothing. Even the air about her is cleared and bracing, the pox and the rattle of bones banished to corners and rattling with the distant peal of pig trotters.
IV
To determine the shape of the ring, the sum of energies, potential and kinetic, must be constant for every particle.
Her wedding ring unlocks the first door, then another. Permission, emigration, enrolment. Sofie skips through them, delighted, and gets a doctorate before a daughter. These things are bittersweet. It is hard to enjoy her own accomplishments when her worlds are far apart and fracturing. She severs her life down the middle: one half for marriage, the other for mathematics and she does not know for which she is more the mistress.
Vladimir is not awarded his doctorate. It’s years of effort and wasted work and Sofie feels her heart break inside her at his face, the misery and shame of it, and Is this your reward? she thinks. You who were so kind to me? He is better than his assessment, she knows, but for Vladimir his future is shattered and he can’t seem to see how to put it back together. Fufa helps, the new little baby who lies so plump and sweet in her cradle and that is a future there, albeit one not of science and unhoped for.
It is because of Fufa that Sofie moves home, ends her exile, returns to Russia. Because of Fufa–and of Vladimir. Sofie’s schooling is over but it is hard for women to find work and she has other obligations now, and one of those is to the man who she currently out-shadows. It is a disappointment to her teacher–to the man she has as counterbalance to her husband, the second father and the one more suited for her–but Sofie cuts him off to concentrate on the fragment she can still hold together, the family she created, and if the monster whines and cries at enforced absence then Sofie only mirrors it when she is alone. Her tears crack on the floors about her feet as crystals.
Back to Russia, bound by rings, all movement stops. Sofie closes her books for society and the nursery–she might as well, she can’t get a job unless it’s teaching schoolgirls arithmetic. This is an absolute waste of her talents, she knows, an almost calculated offense. “I am sorry,” she says, sardonic, when the job is offered to her. “I don’t think that I would be suitable. You see, I am too weak at multiplication for that,” and in her mouth is the taste of nails, of rejection and contagion and confinement. She’d go home and smash all the mirrors, but mirrors are expensive to replace so she kicks the monster instead, the one that growled and warned and whined, kicks with the heat and betrayal of witches and there is venom in her sleep.
(She dreams of Dostoevsky, of giving up the Crystal Palace, of the way his ring would have sounded like trotters. Of how she didn’t need his help for that.)
“I was such a stupid little girl,” she says, and if Fufa smiles at her with nothing of the witch about her there is no wallpaper Sofie can put in her nursery to change that. Even so, she would not give Fufa up for anything, would not upset her conception if she could. And yet, and yet... has it been so worth it, all of it?
It’s stagnation on the mathematical front, and degeneration follows at home. The ring is smashed all to hell. The ring she chose, and shaped to fit. Vladimir finds it worse than she does. It’s humiliating for him, to be married to a women seen as so much more fit than he. It is one thing to be generous, but the potential between them is no longer equal, no longer constant, and generosity fails him as scholarship. He tries to remake himself, reshape himself within the marriage, but he is much less fit for provision than he is for geology. His talents lie in rocks and not with credits. The home that was to have been dissolved when the study was over is less revolutionary than before and more of a battleground for all that. The sound of slamming doors echo in Sofie’s ears–the sounds of screams and regrets and inconstancy.
Sofie’s mirrors are unsympathetic. “There is more to life than this,” they say, and speak of fragments and reconciliation and a return to the life that suited best. “You are stuck here. Stuck, stuck,” and the sound of paws is heavy behind her, stony and depressed and anchored by inertia.
“I was stuck once before,” says Sofie, thinking of thorns and finger bones and action, reaction. “I got out of it then too.” If she goes far enough away from home and hearth, energy can return, and maths, and movement.
V
The ring could only be solid, be stable (said Maxwell, in 1859) if its density is so skewed that the ring becomes more satellite than circle.
Sofie knows what it is to be a satellite. The wallpapered room of her father’s house spins about her always–not the wallpaper from Petersburg, expensive and never to be touched, but the old pages of symbols, of mathematics, and she sees them glowing about her whenever she closes her eyes. And she herself revolves in turn, rotates as if around the black hole of nightmares that are her Nanny’s legacy. This too is symbolic in its way. It is no coincidence that her skin changes with the moon, that she has a monster who changes that way as well. The moon is round and malleable and she finds its curves in her own body, in the shape of her belly and the sharp crescent colour of her teeth. But it is not only the moon that tethers her to orbit, that underlines her nature. She knows what it is to be a satellite because she was raised to be one. Daughter, wife, mother–always a life revolving around someone else.
This, she is told, is what makes a stable society. It is the relationship that holds the family together. And Sofie, present in contagion and yet not fully inoculated to it, sees in her ring the presence of hammers, sees it shaped and shined and shimmery: a short length of metal, no longer than nails, made into a form that will set her as satellite. She gives it her best shot: for a brief time she tries orbiting her husband–the expected role, the sanctioned satellite–but try as she might the circle degrades. Gravity is too much for her, too much for them both, the relationship between objects complicated by mass and pull and movement. Vladimir has many things to weigh him down, but if he has monsters behind to anchor him Sofie does not see them, not even hints as she does with Aniuta who keeps her monsters to herself and all silent but for hissing. With the weight of second self–the heavy paws, the coils thick as thighs with a dozen heads like diamonds, the iron hammers and heaviness of houses shut up, broken homes and pieces of porcelain shattered into too many pieces to pick up again–Sofie is more massive than Vladimir.
More famous, more successful, more everything.
Their orbits shift and decay, swap places in the sky. Vladimir finds himself spinning around his wife, watches as she grows enormous in his sights and blots out the stars of his better nature. He gorges on the view of her, eats it up in lumps and they no longer taste of the colour of roses as they did when their marriage came out of story and into fact. In the new cold light, she tastes of thorns. And because he revolves around her, is tethered to her, Vladimir is always looking inwards. In her face he sees his failings reflected, but because his disposition is melancholy he sees those faults as worse than they are, and is transfixed by them. Transfixed by the thought of how Sofie must see him, as if she saw through his eyes and not her own. He sees her looking outwards, looking past him and into the universe, into the stars beyond, and because he is a satellite and small, he takes up less place in her universe than she does in his. He is fixed in a little corner of it, and the stars outshine him.
He cannot stay this way, of course. Vladimir struggles with his fixing as Sofie struggled with hers, wanting respect, wanting significance. He struggles so hard that his orbit is resentment and arguments and failed business ventures and sinking–but the struggles are not for nothing. His depression is a black hole, and what he sucks in to sustain himself makes him massive again, brings him back into balance and they are opposites again: the presence of sustaining nightmares and the gravitational suction of light, and the ring that binds them is broken beyond all repair.
VI
It is doubtful, said Sofie, that Laplace was correct. The solid ring structure is no longer acceptable.
Her husband swallows chloroform. An entire bottle. He dies.
Sofie no longer loves him but this is a nightmare come to life. The mirrors have blame written on them in wolf’s teeth and the memory of each argument between them, each kind word, stabs like the needle teeth of snakes, like iron nails caught under the heaviness of hammers. The monster is sympathetic: it licks her hand and whines, rubs its thick bony snout against wet cheeks, but Sofie can hear the hiss behind the whine and that hiss says I told you so. The witch voice, the second sight, and it is so easy now to see how wrong everything went between them. How little everything else matters.
Sofie bars the door to keep herself apart in grief. She is a monster now in truth, for she feels the blood on her hands that comes with the inability to save one she felt a responsibility for, an obligation to. He had helped her once, so much. There had been happiness too, for a while, until the marriage turned constricting for both of them and metamorphed into mistake. For that alone Vladimir deserves her grief.
With the door barred against well-wishers Sofie starves herself into coma dreams: a broken doll in a broken house (twelve headed snakes and witches and werewolves, the black death). The dreams are heavy; they suck her down and into darkness. She wakes in them as if the waking were a true one, but the dream world Sofie wakes into is that of the house at Palibino and the noises are back again, the noises under the bed and she lies once more in her child’s bed, the bed that is too small for her now, and listens. There is scraping and rustling and whispering beneath, a whine toned too high for growls, and when Sofie leans over, blanket-clutching, she sees what is under the bed and this time it isn’t the monster. This time it’s her Nanny, and she is chattering now not of monsters but of little girls who can’t be lady-like, who disappoint their friends and disgrace their families. Sofie is transfixed, frozen into place as if nailed, as if sent venom-stabbed into rigor mortis and Nanny reaches one liver-spotted hand out from under the bed and grabs for her flesh, grabs with fingers like sudden iron and painted fingertips the colour of teeth. Grabs and drags to bring Sofie under, to settle her in a place of endings and privacy, of nibbling at dead bones and regrets.
She almost goes, and it is almost willing. (There should be a price, she thinks, and the responsibility for payment cannot all be given to Vlad.)
Two things stop her. The first is the monster, sinking its teeth into torn clothes, the same clothes Sofie has worn for days now–since she heard, since she saw her face in mirrors. It growls and fights and bites until blood runs down Sofie’s legs, until she feels the smell thick in the back of her throat, the hot-iron-salt taste of it. The taste of Olya, of walking away. The taste of other people’s stories. The monster pulls her out from under the bed, and if she has Nanny’s lessons written in her skin now, punctures from those clingy fingers, then Nanny had embedded more and earlier, for skin has not the strength of bone and Sofie is bone-deep a monster.
The monster pulls her back but it cannot feed her anything other than itself, and Sofie is not a cannibal. Instead there is a doctor, nameless, who works with pox at his coat-tails and deep-furred movement about his feet, who is supervised by strange shadows and a jaw that could, slavering, crack him to pieces. The doctor feels himself watched all down his spine, and stills his shaking hands long enough to stuff a tube down Sofie’s throat, to force-feed her back to consciousness–to regret, to relief, to the thick coiled hug of snakes and the smoothing over of porcelain, the broken doll in the broken home filling in her own cracks and stringing herself back together.
“I could have been a better wife,” she says to the monster, as it shares her blankets and warms her feet, curled all about like kindling.
(Vladimir had said he wasted fourteen years with her.)
“But he could have been a better husband.”
(He said she was too good for him.)