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Fermat's Library

6 years

We launched Margins A free online repository where you can 📝 Save and annotate your papers 🗣 Share papers and collaborate with anyone 💻 Annotate using LaTeX, Code, Markdown and more

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Fermat's Library

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5 years

Here's a useful counterintuitive fact: one 18 inch pizza has more 'pizza' than two 12 inch pizzas

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4 years

From Ramanujan

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6 years

Cancellation of Momentum: A ball is fired at 50 mph, out of a cannon from a truck that is going at 50 mph. The two momenta cancel each other and in the Earth reference frame the ball has 0 momentum.

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5 years

A demonstration of the conservation of angular momentum using a Hoberman sphere. Angular momentum depends on the rotational velocity of an object, and on its rotational inertia. When an object changes its shape, its angular velocity will also change.

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4 years

Start with 82 and go backwards to 1 8281807978777675747372717069686 7666564636261605958575655545352 5150494847464544434241403938373 6353433323130292827262524232221 2019181716151413121110987654321 ...this number is prime!

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3 years

The time period of a pendulum with infinite length is 84.6 minutes.

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2 years

In linguistics, Escher sentences are sentences which initially seem acceptable but upon further reflection have no well-formed meaning.

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3 years

List of people that have walked on the Moon: Neil Armstrong 🇺🇸🪦 Buzz Aldrin 🇺🇸 91 yo Pete Conrad 🇺🇸🪦 Alan Bean 🇺🇸🪦 Alan Shepard 🇺🇸🪦 Edgar Mitchell 🇺🇸🪦 David Scott 🇺🇸 89 yo James Irwin 🇺🇸🪦 John Young 🇺🇸🪦 Charles Duke 🇺🇸 86 yo Eugene Cernan 🇺🇸🪦 Harrison Schmitt 🇺🇸 86 yo

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5 years

Here's an interesting approximation for the number 8

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5 years

Here's a demonstration that concave mirrors invert images pass their focal point more here:

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4 years

A visual proof why 1+3+5+...+(2n-1)=n² ⚪⚫⚪⚫⚪⚫ ⚫⚫⚪⚫⚪⚫ ⚪⚪⚪⚫⚪⚫ ⚫⚫⚫⚫⚪⚫ ⚪⚪⚪⚪⚪⚫ ⚫⚫⚫⚫⚫⚫

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7 months

In the heart of World War II, as the Nazis took control of Copenhagen, a peculiar situation took place at the Institute of Theoretical Physics, led by physicist Niels Bohr. Two Nobel laureates Max von Laue and James Franck, fearing the confiscation of their gold Nobel Prize

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4 years

How to derive Electromagnetic Waves from Maxwell's eq. in 1 Tweet 1)∇.B = 0 2)∇.E = 0 3)∇xE = -∂B/∂t 4)∇xB = μ₀ε₀∂E/∂t Using ∇x(∇xA) = ∇(∇⋅A)-∇²A for E & B ∂²E/∂t² = 1/μ₀ε₀∇²E ∂²B/∂t² = 1/μ₀ε₀∇²B E and B obey the wave eq. with c = √1/μ₀ε₀💡!

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5 years

In 1683 the Swiss mathematician Jacob Bernoulli found the first approximation of the mathematical constant e while studying continuous compound interest. e is one of the most important numbers in mathematics alongside 0, 1, π and i.

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9 months

On November 12, 1833, there was a meteor shower so intense that it was possible to see up to 100,000 meteors crossing the sky every hour. Many at the time believed it was the end of the world, leading Adolf Vollmy to create a woodcut inspired by the event.

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5 years

Here's why the sun turns red near the sunset on Earth

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2 years

The first time Albert Einstein wrote down E=mc²

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2 years

Newton was born 378 years ago on Christmas Day. His 3 laws laid the foundation for classical mechanics.

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6 years

FUN FACT J.J. Thomson won the Nobel in Physics in 1906 when he showed electrons were particles. His son G. P. Thomson won the Nobel Prize in Physics in 1937 for showing that electrons are waves.

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5 years

The Tsiolkovsky Rocket Equation derived by the Russian schoolteacher Tsiolkovsky in 1898 describes how rockets move in space - expelling part of its mass. "Mankind will not forever remain on Earth" - Tsiolkovsky (1911)

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4 years

Beautiful shapes created by simple harmonic motion 🧐

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4 years

Here's a great photo of the ISS by J. Mccarthy. The reason why the ISS keeps its relative size against both the Sun and the Moon is due to a remarkable coincidence: the diameter of the Moon is 400x smaller than the diameter of the Sun, but it is also 400x closer to us!

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2 years

Although Georg Ohm discovered one of the most fundamental laws in the field of electricity, his work was ignored by his colleagues, and he lived in poverty for much of his life.

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1 year

How do Maxwell's equations predict that the speed of light is constant

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4 years

Newton was born 377 years ago on Christmas Day. His 3 laws laid the foundation for classical mechanics.

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3 years

Brute-forcing a 256-bit key would require a minimum energy of 590000000000000000000 Suns during 1 year!

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3 years

A proof without words that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 1 🤓

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6 years

The Geneva Drive: a mechanism that translates a continuous rotation movement into intermittent discrete motion. (used in mechanical watches, movie projectors, banknote counting machines)

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5 years

Mars' Olympus Mons, the tallest known volcano in the solar system, has such a gradual slope that someone standing at the base couldn't see the summit because it's beyond the horizon.

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4 years

It was only 282 years ago that Euler presented in his textbooks the exact formula for the volume of a sphere

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4 years

Highest points on Earth 1 - Mount Everest🇳🇵: Highest altitude above sea level 2 - Mauna Kea 🇺🇸: Highest altitude from base to summit 3 - Mount Chimborazo 🇪🇨: Highest distance from Earth's center

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4 years

Universe ↑ ↑ 10²⁴ ↑ ↑ 🌌 Milky Way ↑ ↑ 10¹⁸ ↑ ↑ 10¹⁵ ↑ ↑ 💫 Solar System ↑ ↑ ☀️ Sun ↑ 🌍 Earth 10⁶ ↑ ↑ 🏔 Mountains ↑ 🐋 Whales 👫 Humans ↓ 🐞 Ladybug 10⁻³ ↓ ↓ 10⁻⁶ 🦠 Virus ↓ 🧬 DNA ⚛️ Atoms ↓ 10⁻¹² ↓ ↓ 10⁻¹⁵ ↓ ⚡️ Electrons ⚫️ Quarks

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5 years

These diagrams show the paths traced by Mercury, Venus, Mars, Jupiter and Saturn as seen from Earth.

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9 months

This is the last photo taken from the surface of Venus, 40 years ago. The Venera-14 lander reached the surface in 1982, lasting 52 minutes in Venus' temperature of 450°C (847°F).

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5 months

The length of the Christmas tree lights to wrap around the tree perfectly

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3 years

Marie Curie was exposed to such high doses of radiation that 100 years later her research notebooks need to be kept sealed in a lead-lined box. She is considered the mother of modern physics and is the only person to win a Nobel Prize in 2 different sciences: Physics, Chemistry

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1 year

Newton was born 380 years ago on Christmas Day. His 3 laws laid the foundation for classical mechanics.

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1 year

The following formula yields the correct decimal digits of π to 42 billion digits

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4 years

Take the alphabet and do the following conversion: A = 1 B = 2 C = 3 ... Z = 26 Words are sums of the corresponding letter values. The word PRIME is prime.

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5 years

In the fall of 1972, President Nixon announced that "the rate of increase of inflation was decreasing". Probably the first time a sitting president used the 3rd derivative to increase the chances of reelection.

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3 years

FUN FACT J.J. Thomson won the Nobel in Physics in 1906 when he showed electrons were particles. His son G. P. Thomson won the Nobel Prize in Physics in 1937 for showing that electrons are waves. 🧐

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5 years

John Wallis devised this beautiful formula for π in 1655. It is the infinite product of even numbers squared divided by their two adjacent odd numbers.

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5 years

The infinite gift 🎁is an interesting object where the side of the nth box is 1/√n. As n→+∞, the gift has infinite surface area and length but finite volume!

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6 years

All the following symbols and notations were introduced by Leohard Euler

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3 years

John Nash recommendation letter

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4 years

FUN FACT J.J. Thomson won the Nobel in Physics in 1906 when he showed electrons were particles. His son G. P. Thomson won the Nobel Prize in Physics in 1937 for showing that electrons are waves.

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5 years

Geneva Drive: A mechanism that converts continuous motion into discrete motion. The name was derived from its usage in mechanical watches which were popularized in Geneva. This mechanism can also be found in movie projectors, banknote counting machines...

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4 years

Here's why the circle has 360 degrees: around 2400 B.C., the ancient Sumerians noticed the Sun's annual path across the sky was ~ 360 days. In order to track the Sun's motion, they decided to divide the circle in 360 degrees.

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7 years

A visualization of chaos: 41 triple pendulums with very slightly different initial conditions

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4 years

These diagrams show the paths traced by Mercury, Venus, Mars, Jupiter and Saturn as seen from Earth.

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2 years

Tupper's self-referential formula is a formula that visually represents itself when graphed at a specific location in the (x, y) plane.

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3 years

All the basic trigonometric identities in one figure

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4 years

Euler's identity displays a profound connection between fundamental numbers π, e (the base of natural logarithms), and i (the imaginary unit). It is considered one of the most beautiful formulas in mathematics.

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1 year

In the fall of 1972, President Nixon announced that "the rate of increase of inflation was decreasing". Probably the first time a sitting president used the 3rd derivative to increase the chances of reelection.

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3 years

Euler's number "e" In 1683 the Swiss mathematician Jacob Bernoulli found the first approximation of the mathematical constant e while studying continuous compound interest. e is one of the most important numbers in mathematics alongside 0, 1, π and i.

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4 years

Clever algorithm to find out whether or not 2 words are anagrams

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6 years

In 1982, 11,000 people planted 11,000 trees on an artificial mountain near Ylöjärvi, Finland forming a mathematical pattern based on the golden section. The project was conceived by the artist Agnes Denes and the site is legally protected for the next 400 years.

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5 years

Viviani's Theorem: in an equilateral △, the sum of the distances from any interior point to the 3 sides is equal to the height of the △

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6 years

The fifth hyperfactorial: 5⁵ × 4⁴ × 3³ × 2² × 1¹ = 86400000 milliseconds is exactly 1 day!

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9 months

In 2000, Faber and Faber offered a $1,000,000 prize to anyone who proved Goldbach's conjecture: every integer greater than 5 can be written as the sum of three primes (21=11+7+3). To this day, it remains unsolved.

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4 years

"Chaos: When the present determines the future, but the approximate present does not approximately determine the future." Edward Lorenz

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5 years

René Descartes was born exactly 423 years ago. He is responsible for proposing the use of letters at the end of the alphabet for unknowns and letters at the beginning for known quantities

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4 years

Happy π day! 3.14141592653589 793238462643383 27 950 288 4 197 16 939 93 751 05 8 2097 49445 9230 781

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4 years

"You know, people think mathematics is complicated. Mathematics is the simple bit. It's the stuff we can understand. It's cats that are complicated". RIP John Conway

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3 years

Start with 82 and go backwards to 1 8281807978777675747372717069686 7666564636261605958575655545352 5150494847464544434241403938373 6353433323130292827262524232221 2019181716151413121110987654321 ...this number is prime!

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6 years

Comparison between 5,000 and 50,000 prime numbers plotted in polar coordinates

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1 year

Which is greater 1000¹⁰⁰¹ or 1001¹⁰⁰⁰?

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6 years

Individual dots moving along straight lines. 🧐

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8 months

The following formula yields the correct decimal digits of π to 42 billion digits

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5 months

Ramanujan's most beautiful formula

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3 years

Clever algorithm to find out whether or not 2 words are anagrams

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5 years

Euler was born exactly 312 years ago. All the following symbols and notations were introduced by Euler.

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1 year

How many π digits do we need? 3.1415 ➡️ design the finest engines 3.1415926535 ➡️ obtain the circumference of the Earth within a fraction of an inch 3.1415926535897932384626433832795028842 ➡️ measure the radius of the universe to an accuracy equal to the size of a hydrogen atom

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6 years

Today is the 311th year since the birth of Leonhard Euler. Euler published more than 500 books and papers during his lifetime, while 400 more appeared after his death.

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2 years

The French mathematician André Weil was imprisoned for failure to report for duty in 1940. While in jail for six months at Rouen, he proved the Riemann hypothesis for curves over finite fields.

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3 years

This is one of the biggest numbers you can find in a physics paper. It's the expected distance one would travel before encountering another visible-universe-sized region of space with an identical quantum state as ours.

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5 years

It was only 282 years ago that Euler presented in his textbooks the exact formula for the volume of a sphere

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5 years

Here's how to represent the multiplication of two numbers on the real line

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6 years

Area of the Circle

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4 years

"There are two possible outcomes: if the result confirms the hypothesis, then you've made a measurement. If the result is contrary to the hypothesis, then you've made a discovery" - Enrico Fermi

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5 years

A visual proof that 1/4 + 1/4² + 1/4³ + ... = 1/3 ⚪⚪⚪⚪⚪⚪⚪⚪ ⚪⚫⚪⚪⚪⚪⚪⚪ ⚪⚪⚫⚫⚪⚪⚪⚪ ⚪⚪⚫⚫⚪⚪⚪⚪ ⚪⚪⚪⚪⚫⚫⚫⚫ ⚪⚪⚪⚪⚫⚫⚫⚫ ⚪⚪⚪⚪⚫⚫⚫⚫ ⚪⚪⚪⚪⚫⚫⚫⚫

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5 years

Happy π day! π=3️⃣,1⃣4️⃣15926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712⃣0⃣1⃣9⃣

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4 years

Here's a special countdown for 2020. Happy New Year!

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11 months

Prime Numbers in Polar Coordinates 5,000 (left) Prime Numbers and 50,000 (right) Prime Numbers plotted in polar coordinates.

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3 years

"If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?" - David Hilbert

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6 years

Here is a visual proof of square binomial: (a+b)²= a²+2ab+b²

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11 months

A Proof withouth words (a+b)² =a² + 2ab + b² ⬛️

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6 years

Motion produced by fifteen uncoupled pendulums of different lengths. Order and chaos visualized. 😮 Learn more here:

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6 years

Here's a quick method to calculate the square root of a number (√y) by hand

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4 years

Here is how you can fold a piece of paper into an ellipse: 1) Cut a circle from a regular piece of paper 2) Draw a dot inside the paper circle 3) Fold it so that the circumference falls on that point 4) Repeat this procedure using random folds

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6 years

Here's a cool way to paper-fold an ellipse: 1) Cut a circle and fold it so that the circumference falls on a fixed point inside 2) Repeat this procedure using random folds

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3 years

A visual proof of the square binomial: (a + b)² = a² + 2ab + b² 🤓

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4 years

FUN FACT: The planet Venus has a rotation period of 243 Earth days and it orbits the Sun every 225 Earth days. Venus takes longer to rotate about its axis than any other planet in the Solar System. One day on planet Venus is longer than one year.

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1 year

The length of the Christmas tree lights to wrap around the tree perfectly

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2 years

A visual representation of the first 100,000 digits of π

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4 years

In a rectangular billiard table whose sides are integers and coprime, a ball hit from a corner at a 45 degree angle will end up in a corner after LENGTH+WIDTH-2 rebounds.

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5 years

The reason why the shape of a lot of pianos looks like an exponential is because for waves on a string, doubling length means reducing a music pitch by an octave (2x the frequency).

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9 months

The time period of a pendulum with infinite length is 84.6 minutes

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5 years

The reason why the shape of a lot of pianos looks like an exponential is because for waves on a string, doubling length means reducing a music pitch by an octave (2x the frequency).

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1 year

This is the only pattern of its kind

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