#joseph fourier

[isomorphismes]

Test functions and [tempered] distributions require the notion of topological vector space … distributions can be traced back to Green's functions in the 1830’s to solve ordinary differential equations … the 1936 work of Sergei Sobolev on hyperbolic PDE’s.

Laurent Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on.

tempered distribution

topological vector space

(^ if there is a dipole, there must be a notion of subtraction, hence the need for a vector, and to speak of this very conceptually, use a TVS)

[religion-is-a-mental-illness]

"There cannot be a language more universal and more simple, more free from errors and obscurities… more worthy to express the invariable relations of all natural things [than mathematics]. [It interprets] all phenomena by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes."

-- Joseph Fourier, "The Analytical Theory of Heat"

[teachersource]

Joseph Fourier was born on March 21, 1768. A French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier’s law of conduction are also named in his honor. Fourier is also generally credited with the discovery of the greenhouse effect.

[diagnosed-anxiety-disorder]

Joseph Fourier info for my history presentation due today

so I have a presentation on Jean-Baptiste Joseph Fourier due in a couple hours and instead of making flashcards I'm typing up my presentation notes on tumblr okay ANYWAYS

Born a year and a few months before Napoleon Bonaparte, Joseph Fourier was a French mathematician and physicist whose theories formed the basis and discovery of the greenhouse effect. Studying the Earth's temperature from a mathematical perspective, he was able to develop theories on heat transfer that were inspired by Newton's Law of Cooling.

Newton's Law of Cooling states the proportional relationship between the rate of heat loss from an object and the difference in temperature between said object and its environment. This is often shown as an exponential decay graph where both axis are proportional. This proportionality stated in his Analytical Theory on Heat influenced his discoveries on temperature in the Earth's atmosphere.

Random fact to make my history teacher impressed: he accompanied Napoleon on Egyptian campaign as a scientific adviser in 1798.

Fourier discovered that the light and energy from the sun would heat up the Earth but the atmosphere (ozone layer I think) would act as an insulator for the long-wave infrared radiation the heated Earth would emit. This heat and radiation is trapped underneath the atmosphere, therefore heating up the earth.

The Sun is exponentially heating up the Earth and therefore exponentially decreasing the difference in heat between the Earth and its atmosphere.

This then influences the formulation of the Greenhouse Effect due to the similar idea of the atmosphere trapping things. The Greenhouse Effect traps greenhouse gasses in the atmosphere while Fourier's theories say how heat is trapped under the atmosphere.

This may not be entirely accurate since my source is the wikipedia link my teacher provided me with. I'm not aiming for total historical accuracy, I just want to sound like I know what I'm doing haha. But if I get anything wrong feel free to correct me!

1 like = 1 wishing me good luck on my presentation /j

[jurph]

I am a mechanical engineer who mostly works on computer security, and occasionally bumps into ML. And yet somehow despite never being formally taught the Fourier Transform, I have continued to run into it throughout my career. Piano practice? Fourier. Rocket propulsion vibration testing? Fourier. 300-baud modem handshakes? Fourier. Ethernet, electronic warfare, inferring traffic states from localized noise? Fourier. Van Eck eavesdropping? FOURIER. Reading Neal Stephenson books? Gotta know your Fourier Transforms to really appreciate the humor there! 

Rewatched the 3 Blue 1 Brown video on Fouriers yesterday, got it pounded back into my head, took a break and went and read some ML papers. Sure enough someone figured out how to make Neural Radiance Fields that are actually composable polynomials. I am not sure how to express the weird creeping dread when I saw those words. “Oh good, composable polynomials, bob’s your uncle. Now all we need is a method for--” 

[jurph]

"If you want to be a physicist, you must do three things -first , study mathematics, second, study more mathematics, and third, do the same."

Arnold Sommerfeld

[aklingolgesi]

ISI VE SICAKLIĞIN DOĞASI: BÖLÜM 1 | TERMODİNAMİK YASALARI NASIL GELİŞTİRİLDİ?

Isı ve sıcaklık nedir? Birer madde mi? Enerji mi? Yoksa bazı olguları açıklayabilmek için varsaydığımız kavramlar mı? Fiziğin önemli alt disiplinlerinden birisi olan termodinamik, bu iki kavramı merkeze alarak maddeyi inceliyor. Peki, termodinamik ilmi zaman içinde gelişerek ısı ve sıcaklığı nasıl açıkladı? Termodinamik Kanunları olarak bilinen 4 yasa nasıl ortaya çıktı? Günümüzde ısı ve sıcaklık…

[canmom]

tempted to start pretending Charles Fourier and Joseph Fourier were the same person and telling people the Fourier transform was invented by a utopian socialist who believed we should all live in communes which sometimes fight each other by having massive orgies. because wouldn't it be cool if that was true

[scotianostra]

26th June 1824 saw the birth of William Thomson, Lord Kelvin, eminent physicist.

Most people remember William Thomson, or Lord Kelvin as he is universally known nowadays as the man who gave us the Kelvin scale of temperatures but as with most clever people there is a lot more to know about him besides.

Although born in Northern Ireland, William Thomson spent most of his life in Scotland after his father became Professor of Mathematics at Glasgow University. Kelvin attended university classes from the age of 10. He wrote his first scientific paper, under the pseudonym ‘PQR’, aged only 16.

In his teens he learned French well enough to read the work of eminent French mathematician Jean Baptiste Joseph Fourier. Philip Kelland, Professor of Mathematics at Edinburgh University, had criticised Fourier’s work on the theory of heat. Kelvin boldly stated that Kelland was wrong – and later scientists agreed with him.

From 1841 to 1845 Kelvin attended Cambridge University. He was an excellent student – although he shocked his father by spending his allowance on a boat. After graduating, Kelvin worked in a Paris laboratory with physics professor Victor Regnault.

Kelvin was influenced by the mathematical ideas of George Green, a self-taught miller’s son from Nottingham. He later brought an essay of Green’s to the attention of fellow mathematicians and physicists by republishing it in a respected scientific journal. Until then the essay on mathematical analysis, electricity and magnetism had gone largely unnoticed.

In 1846, aged only 22, Kelvin returned to Glasgow to become Professor of Natural Philosophy. He remained in this post for an astonishing 53 years. Keen to see students involved in practical experiment, he established what became the first university physics laboratory. Kelvin continued to study the nature of heat. He realised that it would be useful to be able to define extremely low temperatures precisely. In 1848, he proposed an absolute temperature scale, now called ‘the Kelvin scale’.

After further research, Kelvin formulated the second law of thermodynamics. This states that heat will not flow from a colder to a hotter body.

His interest in the transmission of electricity prompted Kelvin’s appointment as Director of the Atlantic Telegraph Company in 1856. The company was to install a telegraph cable under the Atlantic Ocean. Atlantic Cable Expeditions in 1857, 1858 and 1865 failed, but communication by cable finally succeeded in 1866.

From 1870 onwards, Kelvin spent many summers on his yacht 'Lalla Rookh’, and invented several marine instruments to improve navigation and safety.

He introduced a mariner’s compass more accurate than any other in existence, and a machine to predict tide levels worldwide.

In 1884 Kelvin travelled to the United States to give a series of lectures. These were enthusiastically received, and were published in 1904 as the 'Baltimore Lectures’.

Kelvin became a Lord in 1892 and took the name Kelvin because of his Glasgow connections. By the time of his death in 1907, he was an international celebrity, widely respected and honoured.

The statue in the pics is in Kelvingrove Park, Glasgow.

[hero-israel]

Hi, I'm the anon who was asking for pro-peace initiatives.

I really appreciate the suggestions you made but man it's depressing that there's so little out there.

When I've gone looking for voices from the other side, I have to admit that I've been rather dismayed at the dearth of non-Jewish anti-Zionists who are willing to approach this issue with sensitivity, nuance, a willingness to accept history as it is, who see Jews as humans with the same rights as everyone else. I'm open to hearing other perspectives, but I am completely unwilling to listen to an argument that would treat Jews as subhuman. Like, the bar is on the floor here, folks.

But then of course you say something like that amongst mainstream leftists and everyone freaks out that you're a horrible human for wanting a solution that doesn't involve a lot of dead Jews and completely decimated Jewish communities and subcultures. It honestly makes me feel a bit like I'm losing my mind. These are people who care about literally everyone and everything else, it seems, except us.

Sorry for the vent; I'm just trying to learn and also frustrated.

Palestinians are entitled to be anti-Zionist.

When I encounter anti-Zionism from any other goyim, I have long since learned to assume they will have the dumbest, most hypocritical, most punishment / ostracism / killing-focused opinions on the Jews, and be the most fragile and spiky upon receiving any correction. They can quite often be MORE punishment / ostracism / killing-focused than Palestinians, while of course having absolutely nothing at stake, not the slightest shred of personal or familial excuse behind which they can hide their vengeance fanfic.

Antisemitism is older than political Leftism and so was incorporated into it as it developed, just as surely as with the Right:

Karl Marx: "The true god of the Jews is money, the true religion of the Jews is swindling"

Pierre Leroux, inventor of the word "socialism": "The Jewish spirit is the banker spirit... their individualistic and egotistical industry is destined to reign, for a time, over the ruins of any true social organization"

Pierre-Joseph Proudhon, inventor of the word "anarchism": "The Jew is the enemy of the human race. One must send this race back to Asia or exterminate it."

Charles Fourier, inventor of the word "feminism": "Jews are a parasitic sect... the source of all evil."

[aedesluminis]

26 for the ask game （╹◡╹）

26. Forgotten hero everyone should know about

This could have been a perfect occasion to talk about Claude-Antoine Prieur again, but given that I plan to devote him many future posts on my blog, I thought it would have been more appropriate to use this ask to share my knowledge about an important and unfortunately still rather unknown STEM personality, who truly inspired me when I was a young student. I'm referring to Sophie Germain.

Born in Paris in 1776, Sophie was one of the rare mathematiciennes of the 18th-19th century. She had her first approach with mathematics during the days of the storming of the Bastille, when it was too dangerous for a young 13 years old girl to go outside. To pass the time, she turned to her father's library and a book named "Histoire des mathématiques" by Jean-Étienne Montucla captured her interest. The story of Archimedes narrated in the book fascinated her deeply, eventually leading her to start studying mathematics on her own through the works by famous mathematicians like Euler, Newton, Cousin. Her interest and dedication to the discipline was so strong, that during winter, when her parents denied her warm clothes and a fire in her bedroom to prevent her from studying she kept doing it anyway despite the cold; at the time maths wasn't considered appropriate as a studying discipline for a woman.

When the Polytechnic school opened in 1794, women couldn't attend, but the policy of the school allowed to everyone, who asked for them, notes of the lectures. She requested them under the pseudonym of Antoine-Auguste Le Blanc, a former student who had dropped out. Given that, as a student of the Polytechnic school, one was expected to send written observations about the lectures - a sort of homework - Germain wrote and sent hers to Joseph-Louis Lagrange, one of the teachers and renowned mathematician. The latter was so positively impressed by her essays that requested a meeting with the brilliant student LeBlanc, who unexpectedly had improved so much. She was then forced to reveal her identity. Lagrange was pleasantly surprised to realize Monsieur Le Blanc was in reality a young and talented woman and decided to support her, becoming her mentor.

One of her most noteworthy contribution to mathematics was in number theory, where she proved a special case of the so-called Last Fermat's Theorem (1), which has remained one of the hardest mathematical theorems to prove for more than three centuries and whose final proof was actually found only in 1994 by Andrew Wiles. Other important works of hers include treatises on elastic surfaces, one of which, Recherches sur la théorie des surfaces élastiques, awarded her a prize from the Paris Academy of Science in 1816.

Although she often faced prejudice for being a woman, Germain was praised and also supported by various well-known mathematicians of the time. Some of them include the aforementioned Lagrange, Legendre, who thanks to her work on the Fermat's theorem, was able to prove it for another special case; Cousin himself, Fourier, who managed to grant her the permission to follow the sittings held at the Paris Academy of science and last, but obviously not least, the great Gauss, who after Germain's death advocated for giving her an honorary degree in mathematics.

Notes

(1) In short, the Last Fermat's Theorem asserts that for n > 2 there are no integer solutions to the following equation:

with a, b, c being positive integers. Sophie Germain proved the theorem for all numbers n equal to a prime p, so that 2p + 1 is also prime. The whole thing is much more complex that how I explained it, my aim was to write down a simple intoduction. If you want to read more about that I recommend you this link.

[okamiz36]

PIXELS ARE NOT SQUARES.

'You may well ask, If a pixel is not a little square, then what is it? And those tiny colored squares on the screens we stare at all day—if they are not pixels, what are they?

To answer these questions, Smith takes us on a grand tour, starting in Napoleonic France, where Jean Baptiste Joseph Fourier proclaimed that the world is made of waves. This idea is easy to grasp when it’s applied to a one- dimensional signal such as music, where the sound produced by an entire symphony orchestra can be decomposed into a set of pure sinusoidal waves of various frequencies. Fourier showed that the same process also works in two or three dimensions. For example, patterns of light and dark in a photograph can be represented by waves extending across the width and height of the image.

But waves are not pixels. Sine waves are smooth and continuous; pixels, whether or not they are little squares, are discrete objects. To get from waves to pixels, Smith leaps ahead a century, from the aftermath of the French Revolution to that of the Russian one, when Vladimir Kotelnikov invented the sampling theorem. Kotelnikov showed that you can capture all the information in a waveform without tracing out the details of its undulations. It’s enough to take samples at discrete points, as long as those points are spaced closely enough that the highest frequency component of the wave is sampled twice in every cycle. Those sample sites, Smith informs us, are the true pixels. They are not little squares but dimensionless points where the wave amplitude is measured. From a set of such point-like samples, the complete image (or sound wave, or any other signal) can be fully reconstructed. And the reconstruction is not just a good approximation; it’s exact. No information is lost in the sampling process.

If pixels, properly understood, are sample points, we are left with the question of what to call all those little squares of color that light up the screen of your phone or computer or television. Smith doesn’t have an inspired answer. He suggests the term display element, but for the most part he grudgingly calls them “spread pixels.”'

Thank you copy and paste, also link here, https://www.americanscientist.org/article/a-pixel-is-not-a-little-square

[yourdailyanticistamines]

Yves Saint Laurent: Transparencies - Part 2

Near the end of the exhibition, we got to see the working space of Yves Saint Laurent, which had mostly been preserved since his retirement and death. It was one thing to see his works, which were polished and complete, but another to see his office that was cluttered with books, sketches, and fabrics. It was like being inside his brain. As I stood inside the place where Saint Laurent made history, I felt admiration and awe.

I realized soon after that the emotions I felt there were the exact same as what I felt when I visited the graves of famous mathematicians earlier in the morning at the Cimetière du Père-Lachaise. Math is a lot like fashion, in that the specifics are highly technical, but the value of the finished work is almost entirely dependent on subjective notions of beauty, creativity, and novelty, which are heavily determined by current trends. At least, that is how it is in a short timeframe, but only the works that are truly beautiful stand the test of time. Joseph Fourier died in 1830; his name frequently appears in three different math and physics classes I’m taking this semester, and I will most likely be doing things that are heavily based on his work for the rest of my life. It’s the same with fashion - people will probably still remember Christian Dior’s New Look or Coco Chanel’s little black dress hundreds of years from now.

Looking for graves of significant people, especially mathematicians, physicists, artists, and composers, is a regular hobby of mine. I wasn’t exactly sure myself why I like grave hunting until I visited Saint Laurent’s office and Fourier’s grave - I think I have a desire to leave something beautiful and timeless in the world too.

[dreams-of-mutiny]

MORTIMER ADLER’S READING LIST (PART 2)

Reading list from “How To Read a Book” by Mortimer Adler (1972 edition).

Alexander Pope: Essay on Criticism; Rape of the Lock; Essay on Man

Charles de Secondat, baron de Montesquieu: Persian Letters; Spirit of Laws

Voltaire: Letters on the English; Candide; Philosophical Dictionary

Henry Fielding: Joseph Andrews; Tom Jones

Samuel Johnson: The Vanity of Human Wishes; Dictionary; Rasselas; The Lives of the Poets

David Hume: Treatise on Human Nature; Essays Moral and Political; An Enquiry Concerning Human Understanding

Jean-Jacques Rousseau: On the Origin of Inequality; On the Political Economy; Emile, The Social Contract

Laurence Sterne: Tristram Shandy; A Sentimental Journey through France and Italy

Adam Smith: The Theory of Moral Sentiments; The Wealth of Nations

Immanuel Kant: Critique of Pure Reason; Fundamental Principles of the Metaphysics of Morals; Critique of Practical Reason; The Science of Right; Critique of Judgment; Perpetual Peace

Edward Gibbon: The Decline and Fall of the Roman Empire; Autobiography

James Boswell: Journal; Life of Samuel Johnson, Ll.D.

Antoine Laurent Lavoisier: Traité Élémentaire de Chimie (Elements of Chemistry)

Alexander Hamilton, John Jay, and James Madison: Federalist Papers

Jeremy Bentham: Introduction to the Principles of Morals and Legislation; Theory of Fictions

Johann Wolfgang von Goethe: Faust; Poetry and Truth

Jean Baptiste Joseph Fourier: Analytical Theory of Heat

Georg Wilhelm Friedrich Hegel: Phenomenology of Spirit; Philosophy of Right; Lectures on the Philosophy of History

William Wordsworth: Poems

Samuel Taylor Coleridge: Poems; Biographia Literaria

Jane Austen: Pride and Prejudice; Emma

Carl von Clausewitz: On War

Stendhal: The Red and the Black; The Charterhouse of Parma; On Love

Lord Byron: Don Juan

Arthur Schopenhauer: Studies in Pessimism

Michael Faraday: Chemical History of a Candle; Experimental Researches in Electricity

Charles Lyell: Principles of Geology

Auguste Comte: The Positive Philosophy

Honore de Balzac: Père Goriot; Eugenie Grandet

Ralph Waldo Emerson: Representative Men; Essays; Journal

Nathaniel Hawthorne: The Scarlet Letter

Alexis de Tocqueville: Democracy in America

John Stuart Mill: A System of Logic; On Liberty; Representative Government; Utilitarianism; The Subjection of Women; Autobiography

Charles Darwin: The Origin of Species; The Descent of Man; Autobiography

Charles Dickens: Pickwick Papers; David Copperfield; Hard Times

Claude Bernard: Introduction to the Study of Experimental Medicine

Henry David Thoreau: Civil Disobedience; Walden

Karl Marx: Capital; Communist Manifesto

George Eliot: Adam Bede; Middlemarch

Herman Melville: Moby-Dick; Billy Budd

Fyodor Dostoevsky: Crime and Punishment; The Idiot; The Brothers Karamazov

Gustave Flaubert: Madame Bovary; Three Stories

Henrik Ibsen: Plays

Leo Tolstoy: War and Peace; Anna Karenina; What is Art?; Twenty-Three Tales

Mark Twain: The Adventures of Huckleberry Finn; The Mysterious Stranger

William James: The Principles of Psychology; The Varieties of Religious Experience; Pragmatism; Essays in Radical Empiricism

Henry James: The American; ‘The Ambassadors

Friedrich Wilhelm Nietzsche: Thus Spoke Zarathustra; Beyond Good and Evil; The Genealogy of Morals; The Will to Power

Jules Henri Poincare: Science and Hypothesis; Science and Method

Sigmund Freud: The Interpretation of Dreams; Introductory Lectures on Psychoanalysis; Civilization and Its Discontents; New Introductory Lectures on Psychoanalysis

George Bernard Shaw: Plays and Prefaces

Max Planck: Origin and Development of the Quantum Theory; Where Is Science Going?; Scientific Autobiography

Henri Bergson: Time and Free Will; Matter and Memory; Creative Evolution; The Two Sources of Morality and Religion

John Dewey: How We Think; Democracy and Education; Experience and Nature; Logic; the Theory of Inquiry

Alfred North Whitehead: An Introduction to Mathematics; Science and the Modern World; The Aims of Education and Other Essays; Adventures of Ideas

George Santayana: The Life of Reason; Skepticism and Animal Faith; Persons and Places

Lenin: The State and Revolution

Marcel Proust: Remembrance of Things Past

Bertrand Russell: The Problems of Philosophy; The Analysis of Mind; An Inquiry into Meaning and Truth; Human Knowledge, Its Scope and Limits

Thomas Mann: The Magic Mountain; Joseph and His Brothers

Albert Einstein: The Meaning of Relativity; On the Method of Theoretical Physics; The Evolution of Physics

James Joyce: ‘The Dead’ in Dubliners; A Portrait of the Artist as a Young Man; Ulysses

Jacques Maritain: Art and Scholasticism; The Degrees of Knowledge; The Rights of Man and Natural Law; True Humanism

Franz Kafka: The Trial; The Castle

Arnold J. Toynbee: A Study of History; Civilization on Trial

Jean Paul Sartre: Nausea; No Exit; Being and Nothingness

Aleksandr Solzhenitsyn: The First Circle; The Cancer Ward

Source: mortimer-adlers-reading-list

[xxlet]

Them: ok but you Better not get weird about 17th century mathematicians again

Me: dwdw

Me a drink later: i need to open the grave of joseph fourier and fuck his corpse nasty

[strings0fcontrol]

Hannigram – Post-Fall (1)

He traversed the deathly passages of his mind palace in a slow-motion pace, where sounds dulled to a muted whisper. Yet, amid the eerie silence, an ever-escalating rush of water steadily approached. The once-luxurious carpets beneath his feet were already soaked, yet nothing deterred his advance. As challenging as it was, he pressed on through the eerily vacant corridors, where lights that had once shone brilliantly now flickered uncertainly.

Intermittently, his mind stuttered, and the distant sound of water enveloped him entirely.

It was drawing nearer.

Yet, he continued his frantic search, dashing through rooms in an environment now rendered unrecognizable. His gaze scoured every corner, never relenting. But despite his efforts, the answer remained elusive, forever out of reach.

Then, the water arrived, a force that seized him in its merciless undercurrent. His ears were overwhelmed by the gurgling roar of the ocean, which seemed to rush in and out of him, and the images blurred, never coalescing into distinct shapes. His mind fractured, desperately clinging to the fragile precipice of consciousness.

The encroaching darkness stretched its icy tendrils, almost enveloping him completely.

Yet, he was no ordinary mortal.

He was the MORNINGSTAR, blazing with celestial brilliance.

Lecter stoked the fires within, his dark eyes flaring to life, their radiance surpassing the dark's futile attempts to extinguish him. The distant light beckoned, a mere glimmer on the horizon, akin to a remote, unforgiving moon. He might have welcomed the embrace of death if not for a faint scream in the recesses of his mind—a realization that he had struck the water before anything else, and if he had endured, then it was highly probable that Will was still clinging to life as well. And Hannibal knew, with certainty, that he would not succumb to death until Graham did.

Will had slipped from his grasp, severed in that fleeting moment of darkness. However, if he could just hold on, if Hannibal could remain afloat a while longer in this unending abyss, he might yet discover Graham.

His oxygen supply dwindled, and his body began to entertain the perilous notion of inhaling the liquid, a desperate gambit to defy the odds.

With the utmost restraint, he allowed his head to turn gradually, as if he could peer through the encompassing void. Lecter made the barest of movements, conserving every ounce of energy to prolong his endurance for as long as humanly possible. They found themselves submerged approximately six or seven meters below the water's surface. The impact had been absorbed largely by Hannibal's back, the icy sting of the water's lash still fresh in his senses. Will had been positioned atop him, causing Hannibal to tilt his gaze slightly upward. However, Graham's weight leaned more toward Hannibal's right side, suggesting that Lecter's impact had tilted Will's body, presenting a smaller target area. This implied that Will might have possessed a higher velocity upon impact. As Hannibal's chin dipped in contemplation, he calculated, taking into account the water's density and the relatively subdued movements at this depth, that Will would most likely be situated beneath him.

Within the gradually submerging chamber of mathematics, he pursued the lingering echoes of luminaries like Johannes Kepler, Isaac Newton, Joseph Fourier, and Carl Friedrich Gauss. A portion of his focus remained firmly fixed on what he believed to be the moon, its bright radiance a guiding beacon from the heavens above. This celestial sentinel served as a steadfast reference point to chart his own gradual descent since the moment of impact.

Lecter minimized his movements, intervening with the natural drift as sparingly as possible. This meticulous approach allowed him to discern and retrace the patterns, erupting in a calculation aimed at pinpointing the precise coordinates.

Then, with a resolute surge forward, he sensed it—a faint, trembling presence existing alongside him in the frigid depths. Like a whisper of life, it called, a shadow even darker than the abyss itself. And there, he felt it, the distinct warmth of another body under his touch. It was unmistakable: Will.

Graham hadn't made a single attempt to break the surface, already entangled in a fierce battle with death, his body wracked by convulsions. Lecter pondered whether Will, like himself, had been conscious until just a brief moment ago and deliberately hid himself. However, Will was no swimmer, unlike Hannibal. The doctor had honed his lung capacity for such dire circumstances. It was rather adorable that he attempted to end Lecter with the very thing that had become almost second nature to him. What a deviously clever boy Will could be. Cunning. To annihilate himself meant to obliterate Hannibal in the process. If Will perished, Hannibal would have no purpose to persist, even if he somehow managed to survive. In that move, Graham had effectively checkmated Hannibal, regardless of whether Lecter lived or succumbed.

Regrettably, it had all been rather predictable.

'I don't know if I can save myself. Maybe that's just fine.'

If Will believed that duping him a second time would be a straightforward affair, he was gravely mistaken.

Hannibal could sense it, a feeling akin to butter sizzling in a pan, creeping gradually through his being.

If the fickle hand of time wasn't conspiring against him, he would have gladly remained within this suspended moment indefinitely, contemplating the possibility of cradling Will close and waiting until the enveloping darkness claimed them both.

Yet, a simmering fury churned deep within, a bitterness that bordered on tasting like betrayal. Will had cast them both over the precipice before his wings could fully unfurl and bear him to the heavens. Hannibal had been ensnared by beauty and would have almost welcomed such an end, but an insatiable curiosity lingered, a desire to witness the expression on Will's face—the very revelation denied by the encroaching darkness.

He yearned to glimpse what lay behind those eyes, curious whether it would reveal relief or sheer horror if he were to behold him once more. Hannibal stood at the precipice, with the choice to perish in the embrace of his dream's illusion or awaken to the stark truth.

And that spark of curiosity, that ember of spite, burgeoned into an enraged blaze, becoming the fuel that propelled his ultimate choice.

Approaching the light, a cruel irony indeed, yet Lecter pressed forward with every last ounce of his remaining strength. His body, guided by the residual authority it still possessed, propelled them onward, but the darkness eventually closed in.

Existence melded into an indistinct blur, time unfurled and wavered, and once more, there reigned a profound stillness.

Everything surrounding him appeared dim and remote, save for a single word that echoed relentlessly in his consciousness: 'Hannibal.' Nothing seemed coherent, nothing felt recognizable. The only constants were the persistent sound of the ocean's roar in his ears and the pearls of saltwater cascading down his ebony locks.

Shivering, he sat huddled beneath a colorless blanket, his eyes vacant, mirroring the emptiness that now enveloped his mind.

In a curious way, Will had achieved his goal. No longer was he truly alive; only his body remained, a hollow vessel. As for Will, he was adrift at sea, lost to its fathomless depths.

Beneath it all, the steady hum of a motor provided the backdrop, while the gentle breeze whispered its soft song.

That soothing purr, an oddly familiar sound, as if Will had once known it in a previous life…

Then his eyes gently closed, and along with it, the hum faded into nothingness. Everything ceased to exist.

In this stillness, a fragile tranquility prevailed, yet it lacked the essence of reality.

He knew that, at some point, his eyes would have to reopen, forcing him to confront the unvarnished truth before him.

It felt as though an eternity drifted past before he summoned the courage to unveil what lay hidden on the other side of his closed lids.

And as he cautiously parted his eyelids, what he beheld was…