Night passes, the Cray running deep toward pi. It seems that the Cray has hung up the phone, and may have crashed.

Once again, pi has demonstrated its ability to give a supercomputer a heart attack. It sometimes strikes young women and older women. I wonder if it is some kind of sluggish virus. It was a cold afternoon, and rain pelted the windows; the shades were drawn, as always. He lay against a heap of pillows, with his legs folded under him. They were two-tone socks, wrinkled and floppy, hand-sewn from pieces of dark-blue and pale-blue cloth, and they looked comfortable. They were the work of Malka Benjaminovna, his mother. Lines of computer code flickered on the screen beside his bed.

This was an apartment built for long voyages. The paper in the room was jammed into the bookshelves, from floor to ceiling. The brothers had wedged the paper, sheet by sheet, into manila folders, until the folders had grown as fat as melons. The paper was piled in three-foot stacks on the chairs. It guarded his bed like the flanking towers of a fortress, and his bed sat at the center of the keep. His happiness, it occurred to me later, sprang from the delicious melancholy of a life chained to a bed in a disordered world that breaks open through the portals of mathematics into vistas beyond time or decay.

This slice is Over here is some What you see in this room is our working papers, as well as the papers we used as references for them. Some of the references we pull out once in a while to look at, and then we leave them somewhere else, in another pile. Once, we had to make a Xerox copy of a book three times, and we put it in three different places in the piles, so we would be sure to find it when we needed it.

Unfortunately, once we put a book into one of these piles we almost never go back to look for it. There are books in there by Kipling and Macaulay. Eh, this place is a mess. Eventually, these papers or my wife will turn me out of the house. His holograph is dense and careful, a flawless minuscule written with a red felt-tip pen—a mixture of theorems, calculations, proofs, and conjectures concerning numbers. Some say he has published most of his work, while others wonder if his bedroom holds unpublished discoveries.

He cautiously refers to his steamer trunks as valises. They are filled to the lids with compressed paper. They wonder whether the digits contain a hidden rule, an as yet unseen architecture, close to the mind of God. A subtle and fantastic order may appear in the digits of pi way out there somewhere; no one knows.

No one has ever proved, for example, that pi does not turn into nothing but nines and zeros, spattered to infinity in some peculiar arrangement. It might be a small but interesting message from God, hidden in the crypt of the circle, awaiting notice by a mathematician. On the other hand, the digits of pi may ramble forever in a hideous cacophony, which is a kind of absolute perfection to a mathematician like Gregory Chudnovsky.

We have a sequence of digits that looks like gibberish. Pi is a transcendental number. For example, if you try to express pi as the solution to an equation you will find that the equation goes on forever. Expressed in digits, pi extends into the distance as far as the eye can see, and the digits never repeat periodically, as do the digits of a rational number. Pi slips away from all rational methods used to locate it. Pi is a transcendental number because it transcends the power of algebra to display it in its totality. Pi possibly first entered human consciousness in Egypt.

The earliest known reference to pi occurs in a Middle Kingdom papyrus scroll, written around B. Around B. There knowledge of pi bogged down until the seventeenth century, when new formulas for approximating pi were discovered. The Germans still call pi the Ludolphian number. In the eighteenth century, Leonhard Euler, mathematician to Catherine the Great, called it p or c.

Physicists have noted the ubiquity of pi in nature. Pi is obvious in the disks of the moon and the sun. The double helix of DNA revolves around pi. Pi hides in the rainbow, and sits in the pupil of the eye, and when a raindrop falls into water pi emerges in the spreading rings. Pi can be found in waves and ripples and spectra of all kinds, and therefore pi occurs in colors and music. Pi has lately turned up in superstrings, the hypothetical loops of energy vibrating inside subatomic particles. It is one of the great mysteries why nature seems to know mathematics. No one can suggest why this necessarily has to be so.

Fox learned while he was recovering in a hospital from a wound sustained in the American Civil War. Having nothing better to do with his time than lie in bed and derive pi, Captain Fox spent a few weeks tossing pieces of fine steel wire onto a wooden board ruled with parallel lines. The wires fell randomly across the lines in such a way that pi emerged in the statistics. After throwing his wires eleven hundred times, Captain Fox was able to derive pi to two places after the decimal point, to 3. If he had had a thousand years to recover from his wound, he might have derived pi to perhaps another decimal place.

To go deeper into pi, you need a powerful machine. The race toward pi happens in cyberspace, inside supercomputers. In , George Reitwiesner, at the Ballistic Research Laboratory, in Maryland, derived pi to two thousand and thirty-seven decimal places with the eniac , the first general-purpose electronic digital computer. Working at the same laboratory, John von Neumann one of the inventors of the eniac searched those digits for signs of order, but found nothing he could put his finger on.

A decade later, Daniel Shanks and John W. Wrench, Jr. The race continued desultorily, through hundreds of thousands of digits, until , when Yasumasa Kanada, the head of a team of computer scientists at Tokyo University, used a nec supercomputer, a Japanese machine, to compute two million digits of pi.

People were astonished that anyone would bother to do it, but that was only the beginning of the affair. In , Kanada and his team got sixteen million digits of pi, noticing nothing remarkable. A year later, William Gosper, a mathematician and distinguished hacker employed at Symbolics, Inc. Gosper saw nothing of interest. The next year, David H. Bailey, at the National Aeronautics and Space Administration, used a Cray 2 supercomputer and a formula discovered by two brothers, Jonathan and Peter Borwein, to scoop twenty-nine million digits of pi.

Bailey found nothing unusual. A year after that, in , Yasumasa Kanada and his team got a hundred and thirty-four million digits of pi, using a nec SX-2 supercomputer. They saw nothing of interest. In , Kanada kept going, past two hundred million digits, and saw further amounts of nothing. Then, in the spring of , the Chudnovsky brothers who had not previously been known to have any interest in calculating pi suddenly announced that they had obtained four hundred and eighty million digits of pi—a world record—using supercomputers at two sites in the United States, and had seen nothing.

Kanada and his team were a little surprised to learn of unknown competition operating in American cyberspace, and they got on a Hitachi supercomputer and ripped through five hundred and thirty-six million digits, beating the Chudnovksys, setting a new world record, and seeing nothing. The Chudnovskys pressed onward, too, and by the fall of they had squeaked past Kanada again, having computed pi to one billion one hundred and thirty million one hundred and sixty thousand six hundred and sixty-four decimal places, without finding anything special. It was another world record.

At that point, the brothers gave up, out of boredom. If a billion decimals of pi were printed in ordinary type, they would stretch from New York City to the middle of Kansas. This notion raises the question: What is the point of computing pi from New York to Kansas? Because once you know the solution to a problem it usually is trivial.

Gregory did the calculation from his bed in New York, working through cyberspace on a Cray 2 at the Minnesota Supercomputer Center, in Minneapolis, and on an I. Thomas J. The calculation triggered some dramatic crashes, and took half a year, because the brothers could get time on the supercomputers only in bits and pieces, usually during holidays and in the dead of night.

It was also quite expensive—the use of the Cray cost them seven hundred and fifty dollars an hour, and the money came from the National Science Foundation. Then they could crash their own machine all they wanted, while they opened doors in the house of numbers. The brothers planned to compute two billion digits of pi on their new machine to try to double their old world record. They thought it would be a good way to test their supercomputer: a maiden voyage into pi would put a terrible strain on their machine, might blow it up. He pulled the door open a few inches, and then it stopped, jammed against an empty cardboard box and a wad of hanging coats.

He nudged the box out of the way with his foot. We will not turn you into digits. We were standing in a long, dark hallway. The lights were off, and it was hard to see anything. The hall is lined on both sides with bookshelves, and they hold a mixture of paper and books. The shelves leave a passage about two feet wide down the length of the hallway.

At the end of the hallway is a French door, its mullioned glass covered with translucent paper, and it glowed. We passed a bathroom and a bedroom. The bedroom belonged to Malka Benjaminovna Chudnovsky. We passed a small kitchen, our feet rolling on computer cables. David opened the French door, and we entered the room of the supercomputer. A bare light bulb burned in a ceiling fixture. The room contained seven display screens: two of them were filled with numbers; the others were turned off.

The windows were closed and the shades were drawn. Gregory Chudnovsky sat on a chair facing the screens. From his toes trailed a pair of heelless leather slippers. His cane was hooked over his shoulder, hung there for convenience. I shook his hand. You do it yourself. A steel frame stood in the center of the room, screwed together with bolts. It held split shells of mail-order personal computers—cheap P. The brothers had crammed special logic boards inside the personal computers. Red lights on the boards blinked. The floor was a quagmire of cables. The brothers had also managed to fit into the room masses of empty cardboard boxes, and lots of books Russian classics, with Cyrillic lettering on their spines , and screwdrivers, and data-storage tapes, and software manuals by the cubic yard, and stalagmites of obscure trade magazines, and a twenty-thousand-dollar computer workstation that the brothers no longer used.

From an oval photograph on the wall, the face of their late father—a robust man, squinting thoughtfully—looked down on the scene. They resembled cities seen from the air: the brothers had big plans for m zero. Computer disk drives stood around the room. The drives hummed, and there was a continuous whirr of fans, and a strong warmth emanated from the equipment, as if a steam radiator were going in the room. The brothers heat their apartment largely with chips.

Would you like a Coca-Cola? The interviewer becomes a person in the story. Supercomputers are evolving incredibly fast. The notion of what a supercomputer is and what it can do changes from year to year, if not from month to month, as new machines arise. The definition of a supercomputer is simply this: one of the fastest and most powerful scientific computers in the world for its time. The power of a supercomputer is revealed, generally speaking, in its ability to solve tough problems.

A Cray Y-MP8, running at peak working speed, can perform more than two billion floating-point operations per second. Floating-point operations—or flops, as they are called—are a standard measure of speed. Since Cray Y-MP8 can hit two and a half billion flops, it is considered to be gigaflop supercomputer. Like all supercomputers, a Cray often cruises along significantly below its peak working speed. There is a heated controversy in the supercomputer industry over how to measure the typical working performance of any given supercomputer, and there are many claims and counterclaims.

A Cray is a so-called vector-processing machine, but that design is going out of fashion. Cray Research has announced that next year it will begin selling an even more powerful parallel machine. M zero is not the only ultrapowerful silicon engine to gleam in the Chudnovskian oeuvre. The brothers recently fielded a supercomputer named Little Fermat, which they designed with Monty Denneau, an I. Younis did the grunt work: he mapped out circuits containing more than fifteen thousand connections and personally plugged in some five thousand chips.

Little Fermat is seven feet tall, and sits inside a steel frame in a laboratory at M. What m zero consists of is a group of high-speed processors linked by cables which cover the floor of the room. The cables form a network of connections among the processors—a web. Gregory sketched on a piece of paper the layout of the machine. As far as I know, no one except us has built a machine that has this type of web.

In other parallel machines, the processors are connected only to near neighbors, while they have to talk to more distant processors through intervening processors. But the truth is that nobody really knows how the hell parallel machines should perform, or the best design for them. Right now we have eight processors. We plan to have two hundred and fifty-six processors. We will be able to fit them into the apartment.

He said that each processor had its own memory attached to it, so that each processor was in fact a separate computer. After a processor was fed some data and had got a result, it could send the result through the web to another processor. He said that it was very hard to know what exactly was happening in the web when the machine was running—that the web seemed to have a life of its own. Your brain is the same way. It is mostly made of connections. If I may say so, your brain is a liquid-cooled parallel supercomputer.

The design of the web is the key element in the Chudnovskian architecture. Behind the web hide several new findings in number theory, which the Chudnovskys have not yet published. The brothers would not disclose to me the exact shape of the web, or the discoveries behind it, claiming that they needed to protect their competitive edge in a worldwide race to develop faster supercomputers.

The Chudnovskys have formidable competitors. Thinking Machines Corporation, in Cambridge, Massachusetts, sells massively parallel supercomputers. The price of the latest model, the CM-5, starts at one million four hundred thousand dollars and goes up from there. If you had a hundred million dollars, you could order a CM-5 that would be an array of black monoliths the size of a Burger King, and it would burn enough electricity to light up a neighborhood.

Seymour Cray is another competitor of the brothers, as it were. He invented the original Cray series of supercomputers, and is now the head of the Cray Computer Corporation, a spinoff from Cray Research. Seymour Cray has been working to develop his Cray 3 for several years. It would melt instantly if its cooling system were to fail. Intel, together with a consortium of federal agencies, has invested more than twenty-seven million dollars in the Touchstone Delta, a five-foot-high, fifteen-foot-long parallel supercomputer that sits in a computer room at Caltech.

The machine consumes twenty-five thousand watts of power, and is kept from overheating by chilled air flowing through its core. One day, I called Paul Messina, a Caltech research scientist, who is the head of the Touchstone Delta project, to get his opinion of the Chudnovsky brothers. As for their claim to have built a pi-computing gigaflop supercomputer out of mail-order parts for around seventy thousand dollars, he flatly believed it. The Chudnovskys are counting very little of their human time. The Chudnovsky brothers particularly hoped to leave Kanada and his Hitachi in the dust with their mail-order funny car.

The photograph showed a mountain range in cyberspace: bony peaks and ridges cut by valleys. The mountains and valleys were splashed with colors—yellow, green, orange, violet, and blue. It was the first eight million digits of pi, mapped as a fractal landscape by an I. GF supercomputer at Yorktown Heights, which Gregory had programmed from his bed. Apart from its vivid colors, pi looks like the Himalayas. Gregory thought that the mountains of pi seemed to contain structure.

As he gazed into the nature beyond nature, he wondered if he stood close to a revelation about the circle and its diameter. You need a flashlight.

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## The Complete Idiot's Guide to Understanding Einstein

Our computer is the flashlight. A fax machine in a corner beeped and emitted paper. It was a message from a hardware dealer in Atlanta. David tore off the paper and stared at it. This a service economy. Of course, you know what that means— the service is terrible. He switched on the lights in the hallway and began to shift boxes. Gregory rifled bookshelves. Finally, the brothers confessed that they had temporarily lost their pi. It was a naked hard-disk drive, studded with chips. He handed me the object. It weighs six pounds.

Months passed before I visited the Chudnovskys again. The brothers had been tinkering with their machine and getting it ready to go for two billion digits of pi, when Gregory developed an abnormally related to one of his kidneys. He went to the hospital and had some cat scans made of his torso, to see what things looked like, but the brothers were disappointed in the pictures, and persuaded the doctors to give them the cat data on a magnetic tape.

The images showed cross-sectional slices of his body, viewed through different angles, and they were far more detailed than any image from a cat scanner. Gregory wrote the imaging software. It took him a few weeks. Then the brothers began to calculate pi, slowly at first, more intensely as they gained confidence in their machine, but in May the weather warmed up and Con Edison betrayed the brothers. A heat wave caused a brownout in New York City, and as it struck, m zero automatically shut itself down, to protect its circuits, and died. They spent two weeks restarting it, piece by piece.

Then, on Memorial Day weekend, as the calculation was beginning to progress, Malka Benjaminovna suffered a heart attack. Gregory was alone with his mother in the apartment. An ambulance rushed her to St. The brothers were terrified that they would lose her, and the strain almost killed David. He had developed a bleeding ulcer.

### Math, art, and ideas

Malka Benjaminovna improved slowly. When St. I visited them shortly after that, on a hot day in early summer. David answered the door. There were blue half circles under his eyes, and he had lost weight. Malka Benjaminovna was asleep in her bedroom, and the nurse was sitting beside her. Her room was lined with bookshelves, packed with paper—it was an overflow repository. As long as I am alive, we will not cool a machine with water. Above ninety Fahrenheit.

This is not good. Things begin to fry. David took Gregory under the arm, and we passed through the French door into gloom and pestilential heat. The shades were drawn, the lights were off, and an air-conditioner in a window ran in vain. Sweat immediately began to pour down my body. The steel frame in the center of the room—the heart of m zero—had acquired more logic boards, and more red lights blinked inside the machine.

I could hear disk drives murmuring. The drives were copying and recopying segments of transcendental numbers, to check the digits for perfect accuracy. Gregory knelt on the floor, facing the steel frame. David opened a cardboard box and removed an electronic board. He began to fit it into m zero. David pulled the Mini Mag-Lite from his shirt pocket and handed it to Gregory. The brothers knelt beside each other, Gregory shining the flashlight into the supercomputer. David reached inside with his fingers and palpated a logic board. David adjusted an electric fan.

It saves money. The brothers had thrust the thermometer between two circuit boards in order to look for hot spots inside m zero. He lifted a keyboard out of the steel frame and typed something on it, staring at a display screen. Numbers filled the screen. A buzzer sounded. The stupid thing I obviously has problems. He went over to a bookshelf and picked up a hunting knife.

I thought he was going to plunge it into the supercomputer, but he used it to rip open a cardboard box. Now you know the reason this apartment is full of empty boxes. We have to save them. Gregory, I wonder if you are tired. I will maintain my center of gravity. Let me see, meanwhile, what is happening with this machine. David dialled a mail-order house in Nevada that here will be called Searchlight Computers. I need a fifteen-forty controller. Just the controller! Just a naked unit! How much you charge?. Two hundred and fifty-seven dollars?

For tomorrow morning. How much?. Thirty-nine dollars for Fed Ex? Come on! What about afternoon delivery? What is your name?. Another display screen came alive and filled with numbers. Look, any device might not work. Can you send it today Fed Ex?. I want a naked unit! Not in a shoebox, nothing! A rhythmic clicking sound came from one of the disk drives.

His illness came with the package. As we talked, though, pyramids of boxes and stacks of paper leaned against the dining-room windows, pressing against the glass and blocking daylight, and a smell of hot electrical gear crept through the room. At night, she dreams that she is dancing from room to room through an empty apartment that has parquet floors.

David brought his mother out of her bedroom, settled her at the table, and kissed her on the cheek. Malka Benjaminovna seemed frail but alert. She is a small, white-haired woman with a fresh face and clear blue eyes, who speaks limited English. A mathematician once described Malka Benjaminovna as the glue that holds the Chudnovsky family together. She was an engineer during the Second World War, when she designed buildings, laboratories, and proving grounds in the Urals for testing the Katyusha rocket; later, she taught engineering at schools around Kiev.

## A Gentle Introduction To Learning Calculus – BetterExplained

She handed me plates of roast chicken, kasha, pickles, cream cheese, brown bread, and little wedges of The Laughing Cow cheese in foil. Malka Benjaminovna slid a jug of Gatorade across the table at me. After lunch, and fortified with Gatorade, the brothers and I went into the chamber of m zero, into a pool of thick heat. The room enveloped us like noon on the Amazon, and it teemed with hidden activity.

The disk drives clicked, the red lights flashed, the air-conditioner hummed, and you could hear dozens of whispering fans. Gregory leaned on his cane and contemplated the machine. Gregory sat on a stool and tugged at his beard. David slashed open a cardboard box with his hunting knife and lifted out a board studded with chips, for making color images on a display screen, and plugged it into m zero. Gregory crawled under a table. David handed his Mini Mag-Lite under the table. Gregory joined some cables together and stood up. Very uncomfortable.

David, boot it up. The drives began to click, and the parallel processors silently multiplied and conjoined huge numbers. Gregory headed for bed, David holding him by the arm. What is needed is a teraflop machine. One such design for teraflop machine, by Monty Denneau at I. You want to have at least sixty-four thousand processors in the machine, each of which has the power of a Cray. And the processors will be joined by a network that has the total switching capacity of the entire telephone network in the United States.

### Gödel’s Incompleteness: The #1 Mathematical Breakthrough of the 20th Century

I think a teraflop machine will exist by Now, a better machine is a petaflop machine. A petaflop is a quadrillion flops, a quadrillion floating-point operations per second, so a petaflop machine is a thousand times as fast as a teraflop machine, or a million times as fast as a Cray Y-MP8. The petaflop machine will exist by the year , or soon afterward. It will fit into a sphere less than a hundred feet in diameter. I think that the petaflop machine will be used mainly to simulate machines like itself, so that we can begin to design some real machines.

In the nineteenth century, mathematicians attacked pi with the help of human computers. Dase could multiply large numbers in his head, and he made a living exhibiting himself to crowds in Germany, Denmark, and England, and hiring himself out to mathematicians. A mathematician once asked Dase to multiply 79,, by 93,,, and Dase gave the right answer in fifty-four seconds. Dase extracted the square root of a hundred-digit number in fifty-two minutes, and he was able to multiply a couple of hundred-digit numbers in his head during a period of eight and three-quarters hours.

Dase could do this kind of thing for weeks on end, running as an unattended supercomputer. He would break off a calculation at bedtime, store everything in his memory for the night, and resume calculation in the morning. Occasionally, Dase had a system crash.

In , he bombed while trying to demonstrate his powers to a mathematician and astronomer named Heinrich Christian Schumacher, reckoning wrongly every multiplication that he attempted. He explained to Schumacher that he had a headache. Schumacher also noted that Dase did not in the least understand theoretical mathematics. A mathematician named Julius Petersen once tried in vain for six weeks to teach Dase the rudiments of Euclidean geometry, but they absolutely baffled Dase. Large numbers Dase could handle, and in L. Schulz von Strassnitsky hired him to compute pi.

Dase ran the job for almost two months in his brain, and at the end of the time he wrote down pi correctly to the first two hundred decimal places— then a world record. To many mathematicians, mathematical objects such as the number pi seem to exist in an external, objective reality. Numbers seem to exist apart from time or the world; numbers seem to transcend the universe; numbers might exist even if the universe did not.

I suspect that in their hearts most working mathematicians are Platonists, in that they take it as a matter of unassailable if unprovable fact that mathematical reality stands apart from the world, and is at least as real as the world, and possibly gives shape to the world, as Plato suggested. Most mathematicians would probably agree that the ratio of the circle to its diameter exists brilliantly in the nature beyond nature, and would exist even if the human mind was not aware of it, and might exist even if God had not bothered to create it.

One could imagine that pi existed before the universe came into being and will exist after the universe is gone. Pi may even exist apart from God, in the opinion of some mathematicians, for while there is reason to doubt the existence of God, by their way of thinking there is no good reason to doubt the existence of the circle. I am a Platonist.

Of course pi is a natural object. Since pi is there, it exists. And, unfortunately, the laws of physics change once every generation. You put it through your mind in order to make sense of it. Mathematicians have sorted numbers into classes in order to make sense of them. One class of numbers is that of the rational numbers. Next, we come to the irrational numbers.

Hippasus of Metapontum supposedly made this discovery in the fifth century B. The Pythagoreans believed that everything in nature could be reduced to a ratio of two whole numbers, and they threw Hippasus overboard for his discovery, since he had wrecked their universe. One night he went to sleep and began dreaming about atoms. He saw the nucleus of the atom, with electrons spinning around it, much as planets spin around their sun.

Immediately on awakening, Bohr felt the vision was accurate. But as a scientist he knew the importance of validating his idea before announcing it to the world. He returned to his lab and searched for evidence to support his theory. It held true - and Bohr's vision of atomic structure turned out to be one of the greatest breakthroughs of his day. Bohr was later awarded a Nobel Prize for Physics as a result of this leap in creative thinking while asleep. In , Howe invented the sewing machine based on a famous dream that helped him understand the mechanical penetration of the needle.

He was not the first to conceive the idea of a sewing machine, however Howe made significant refinements to the design and was awarded the first US patent for a sewing machine using a lockstitch design. According to family history records:. Howe worked and worked, and puzzled, and finally gave it up. Then he thought he was taken out to be executed. He noticed that the warriors carried spears that were pierced near the head. Instantly came the solution of the difficulty, and while the inventor was begging for time, he awoke.

It was 4 o'clock in the morning. He jumped out of bed, ran to his workshop, and by 9, a needle with an eye at the point had been rudely modeled. After that it was easy. That is the true story of an important incident in the invention of the sewing machine. Albert Einstein: The Speed of Light. Einstein is famous for his genius insights into the nature of the universe - but what about his dreams? As it happens, he came to the extraordinary scientific achievement - discovering the principle of relativity - after having a vivid dream.

As a young man, Einstein dreamed he was sledding down a steep mountainside, going so fast that eventually he approached the speed of light. As this moment, the stars in his dream changed their appearance in relation to him. He awoke and meditated on this idea, soon formulating what would become one of the most famous scientific theories in the history of mankind.

Einstein's Dreams by Alan Lightman is now a modern classic - a fictional collage of stories dreamed by Albert Einstein in on the brink of his breakthrough discoveries. In one, time is circular, so that people are fated to repeat their triumphs and failures over and over. In another, time stands still, where lovers cling together in eternity.

In another yet, time is a nightingale, trapped by a bell jar. The mathematical genius made substantial contributions to analytical theory of numbers, elliptical functions, continued fractions, and infinite series, and proved more than 3, mathematical theorems in his lifetime. Ramanujan stated that the insight for his work came to him in his dreams on many occasions. Ramanujan said that, throughout his life, he repeatedly dreamed of a Hindu goddess known as Namakkal.

She presented him with complex mathematical formulas over and over, which he could then test and verify upon waking. Once such example was the infinite series for Pi:. There is a fascinating story behind the prolific dreamer. Learn more about the life of Ramanujan - a self-taught math prodigy from India - and how how he formed his brilliant theories.

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From the temples and slums of Madras to the courts and chapels of Cambridge University, it's an unlikely story that saw Ramanujan rise from underprivileged obscurity to one of the greatest geniuses of the 20th century. A sick man for most of his life, he wrote mainly to support his family. Stevenson said:. While recovering in bed from a hemorrhage, Fanny Stevenson heard his screams resulting from an opium-induced nightmare. He promptly awoke and complained:.

Fanny later discovered she had woken him at the first transformation scene. The next morning, Stevenson began scribbling furiously, and three days later he had written a 30,word draft. But when Fanny noted it was an allegory, contrary to his original intent, he threw it on the fire and started over. For a further three days his family tip-toed around him while he sat in bed, writing, surrounded by torn up pages, until at last the final draft was ready.

In all he wrote 64, words in six days - some kind of miracle in his time, without typewriters or computer power to speak of. The success of his book was phenomenal. The story has also inspired modern day spin-offs, such as Bradley Cooper in Limitless , in which a would-be writer is transformed into a "perfect" version of himself.

Otto Loewi was a German-born pharmacologist whose discovery of acetylcholine ironically, a neurotransmitter which promotes dreaming helped advance medical therapy. The discovery earned him a Nobel Prize 13 years later. However, he is almost as famous for the means by which he discovered it, as he is for the discovery itself. In , Loewi dreamed of an experiment that would prove once and for all that transmission of nerve impulses was chemical -- not electrical. He woke up, scribbled the experiment down, and went back to sleep. The next morning, he arose excited to try his experiment but was horrified to find he couldn't read his midnight ramblings.

That day, he said, was the longest day of his life, as he tried but failed to recall his dream.

The following night, however, he had the same dream repeat itself and upon awakening went directly to his lab to prove the Noble Prize-winning theory of chemical transmission of the nervous impulse. He said he discovered the ring shape of the Benzene molecule after having a reverie of a snake seizing its own tail - a common ancient symbol known as the ouroboros:.

After his mother passed away from diabetes, Frederick Banting was motivated to find a cure. Eventually he found the next best thing: a treatment using insulin injections which, though not a true cure, could at least significantly extend the lifespan of sufferers. The discovery won him a Nobel Prize in Medicine at just 32 years old. Although he lacked knowledge of diabetes and clinical research, his unique knowledge of surgery combined with his assistant's Charles Best's knowledge of diabetes made the ideal research team.

While seeking to isolate the exact cause of diabetes, Banting had a dream telling him to surgically ligate tie up the pancreas of a diabetic dog in order to stop the flow of nourishment. He did - and discovered a disproportionate balance between sugar and insulin. This breakthrough lead to another dream that revealed how to develop insulin as a drug to treat the condition.

Banting was named Canada's first Professor of Medical Research and by , he was the most famous man in the country. He received letters and gifts from hundreds of grateful diabetics all over the world, and since then insulin has saved or transformed the lives of millions of people. But when you learn the art of dream control , anything is possible. So give it a go - tap into your own hidden insights - and perhaps you'll come up with some inventions of your own She is currently studying for a biology degree in Auckland and blogging at her site Science Me.

A lot has happened in the last 5 months. But how did we go from business as usual to changing the face of the entire lucid dreaming supplements industry? When I was first taken on-board as Chief Lucidity Officer in , one of the first things I was tasked with was taking a good look at our operations and giving things a bit of an overhaul. To lucid dream is to examine an intensely heightened state of self awareness, with all the senses activated - a uniquely human experience.

What's more, lucid dreaming offers profound benefits that touch all of us, no matter our culture, beliefs or life circumstances.