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Visualizing Euler's formula provides a stunning geometric interpretation of what is widely considered the most beautiful equation in all of mathematics, seamlessly linking complex exponential growth to periodic trigonometric functions. When plotted on a two-dimensional complex coordinate plane, the formula elegantly demonstrates that raising the mathematical constant base of...

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Math Visualizations

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159,200 просмотров • 4 месяцев назад •via TikTok

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Most people first see Euler’s Formula as a strange equation in a textbook. Then years later, they realize it quietly powers the modern world. Leonhard Euler discovered that these seemingly unrelated mathematical ideas: → exponential growth → imaginary numbers → sine waves → cosine waves → rotation … are all deeply connected through one identity: e^(iθ) = cos(θ) + i·sin(θ) At first glance it looks impossible. How can an exponential function suddenly produce circles and waves? The key insight is that multiplying by a complex exponential creates rotation. As the angle θ changes: - the cosine term tracks horizontal motion - the sine term tracks vertical motion - together they trace a perfect circle in the complex plane Euler showed that waves and rotation are mathematically the same phenomenon viewed differently. That single idea changed science and engineering forever. Today, Euler’s Formula sits underneath: → Fourier Transforms → signal processing → wireless communication → MRI scanners → quantum mechanics → electrical engineering → audio compression → radar systems → GPS → neural network frequency analysis Even modern AI systems indirectly rely on mathematics built on top of these foundations. The famous special case is Euler’s Identity: e^(iπ) + 1 = 0 Richard Feynman reportedly called it: "our jewel." Because it revealed that mathematics is not a collection of separate topics. It is one connected language describing reality itself.
1:27

Most people first see Euler’s Formula as a strange equation in a textbook. Then years later, they realize it quietly powers the modern world. Leonhard Euler discovered that these seemingly unrelated mathematical ideas: → exponential growth → imaginary numbers → sine waves → cosine waves → rotation … are all deeply connected through one identity: e^(iθ) = cos(θ) + i·sin(θ) At first glance it looks impossible. How can an exponential function suddenly produce circles and waves? The key insight is that multiplying by a complex exponential creates rotation. As the angle θ changes: - the cosine term tracks horizontal motion - the sine term tracks vertical motion - together they trace a perfect circle in the complex plane Euler showed that waves and rotation are mathematically the same phenomenon viewed differently. That single idea changed science and engineering forever. Today, Euler’s Formula sits underneath: → Fourier Transforms → signal processing → wireless communication → MRI scanners → quantum mechanics → electrical engineering → audio compression → radar systems → GPS → neural network frequency analysis Even modern AI systems indirectly rely on mathematics built on top of these foundations. The famous special case is Euler’s Identity: e^(iπ) + 1 = 0 Richard Feynman reportedly called it: "our jewel." Because it revealed that mathematics is not a collection of separate topics. It is one connected language describing reality itself.

Tech with Mak

25,539 просмотров • 5 месяцев назад

Functions are one of the most fascinating connections between mathematics and music. A mathematical function assigns an output to every input, and when those outputs are interpreted as sound frequencies, amplitudes, or waveforms, they can create everything from simple tones to complex musical compositions. For example, a sine function produces a pure sound wave, while more complicated functions can generate rich harmonics, evolving textures, and unique timbres. This concept is widely used in digital audio synthesis, signal processing, and electronic music production, making mathematical functions a powerful tool for transforming equations into sound. The idea of hearing functions has become increasingly popular in mathematics education, data sonification, and music technology. By mapping the graph of a function to pitch over time, listeners can literally hear the shape of the equation. Linear functions may create steadily rising or falling notes, quadratic functions can produce sweeping melodic curves, and periodic functions generate repeating musical patterns. Exploring how functions sound helps students visualize mathematical concepts while showcasing the deep relationship between mathematics, sound waves, Fourier analysis, and modern audio engineering. Like and follow for more! #math #audio #funny #function #calculus
0:32

Functions are one of the most fascinating connections between mathematics and music. A mathematical function assigns an output to every input, and when those outputs are interpreted as sound frequencies, amplitudes, or waveforms, they can create everything from simple tones to complex musical compositions. For example, a sine function produces a pure sound wave, while more complicated functions can generate rich harmonics, evolving textures, and unique timbres. This concept is widely used in digital audio synthesis, signal processing, and electronic music production, making mathematical functions a powerful tool for transforming equations into sound. The idea of hearing functions has become increasingly popular in mathematics education, data sonification, and music technology. By mapping the graph of a function to pitch over time, listeners can literally hear the shape of the equation. Linear functions may create steadily rising or falling notes, quadratic functions can produce sweeping melodic curves, and periodic functions generate repeating musical patterns. Exploring how functions sound helps students visualize mathematical concepts while showcasing the deep relationship between mathematics, sound waves, Fourier analysis, and modern audio engineering. Like and follow for more! #math #audio #funny #function #calculus

mathswithmuza

700,600 просмотров • 5 месяцев назад

In 1747, Euler spent 133 pages proving that logarithms have infinitely many solutions - and in doing so accidentally revealed why every LLM, every signal processor, and every Fourier transform works the way it does. The imaginary unit I doesn't break exponential functions. It rotates them, turning exponential growth into the sine and cosine waves that live inside every transformer attention layer. The proof runs through Briggs' 1617 zooming algorithm, complex polar coordinates, and a thin-triangle argument that connects imaginary exponents to angles in radians. When you plug I into the exponential function, you don't get growth - you get rotation confined to the unit circle. The same unit circle that softmax maps every attention score onto before the GPU multiplies them. 27 minutes. Bookmark & watch today. The most beautiful equation in mathematics and why it powers every model you have ever used.
26:53

In 1747, Euler spent 133 pages proving that logarithms have infinitely many solutions - and in doing so accidentally revealed why every LLM, every signal processor, and every Fourier transform works the way it does. The imaginary unit I doesn't break exponential functions. It rotates them, turning exponential growth into the sine and cosine waves that live inside every transformer attention layer. The proof runs through Briggs' 1617 zooming algorithm, complex polar coordinates, and a thin-triangle argument that connects imaginary exponents to angles in radians. When you plug I into the exponential function, you don't get growth - you get rotation confined to the unit circle. The same unit circle that softmax maps every attention score onto before the GPU multiplies them. 27 minutes. Bookmark & watch today. The most beautiful equation in mathematics and why it powers every model you have ever used.

Deliriumm

128,190 просмотров • 1 месяц назад

Ever wondered what neural networks are and how they work? Systems like ChatGPT use neural networks to work as well as they do. Neural networks are composed of "layers" of neurons, layers with different functions; connections between layers called "weights"; and mathematical functions called "activation functions". If you’re interested in learning about these systems, check the comments. Ultimately, the neural network structure of the model serves to visually demonstrate that it is, in fact, a complex mathematical equation. When companies release the model's weights, they are releasing a key component needed to run the model's complete equation. Without the weights, the equation is incomplete. For the math-minded: the weights of a model are the learned numbers (they are variables during training) that are then used as constants in the mathematical functions that make up the model. Neural networks are ultimately just one big, hyper-complex mathematical function, and when a model is trained, it learns the constants associated with the high-dimensional input.
2:37

Ever wondered what neural networks are and how they work? Systems like ChatGPT use neural networks to work as well as they do. Neural networks are composed of "layers" of neurons, layers with different functions; connections between layers called "weights"; and mathematical functions called "activation functions". If you’re interested in learning about these systems, check the comments. Ultimately, the neural network structure of the model serves to visually demonstrate that it is, in fact, a complex mathematical equation. When companies release the model's weights, they are releasing a key component needed to run the model's complete equation. Without the weights, the equation is incomplete. For the math-minded: the weights of a model are the learned numbers (they are variables during training) that are then used as constants in the mathematical functions that make up the model. Neural networks are ultimately just one big, hyper-complex mathematical function, and when a model is trained, it learns the constants associated with the high-dimensional input.

Harper Carroll

50,905 просмотров • 1 год назад

Exponential growth and decay formulas #math #iteachalgebra #algebra #mathematics
1:05

Exponential growth and decay formulas #math #iteachalgebra #algebra #mathematics

iteachalgebra

62,600 просмотров • 2 лет назад

The video visually demonstrates the time-dependent Schrödinger equation, showing the mathematical formula along with an animation of a quantum wave packet evolving over time:
1:27

The video visually demonstrates the time-dependent Schrödinger equation, showing the mathematical formula along with an animation of a quantum wave packet evolving over time:

Physics_ Lab

13,534 просмотров • 3 месяцев назад

What Does a Quantum Particle's Wavefunction Look Like? A quantum wavefunction is not simply an ordinary wave oscillating up and down. It is a complex-valued mathematical function. Its real and imaginary parts can be visualized as two perpendicular oscillations, forming a rotating helix when plotted together. This is a representation of quantum phase, not the physical shape of a particle. For a free, non-relativistic particle, the dispersion relation is ω(k) = ℏk²/2m. Its wavepacket travels at the group velocity, twice the phase velocity. Because different momentum components travel at different speeds, a localized Gaussian wavepacket spreads over time: σ(t) = σ₀√[1 + (t/τd)²]. This mathematics has experimental foundations. Electron diffraction, C₆₀ fullerene interference and ultracold atomic gases demonstrate the wave nature and evolution of quantum matter. What we observe experimentally are interference patterns and probability distributions, not the wavefunction itself.
0:24

What Does a Quantum Particle's Wavefunction Look Like? A quantum wavefunction is not simply an ordinary wave oscillating up and down. It is a complex-valued mathematical function. Its real and imaginary parts can be visualized as two perpendicular oscillations, forming a rotating helix when plotted together. This is a representation of quantum phase, not the physical shape of a particle. For a free, non-relativistic particle, the dispersion relation is ω(k) = ℏk²/2m. Its wavepacket travels at the group velocity, twice the phase velocity. Because different momentum components travel at different speeds, a localized Gaussian wavepacket spreads over time: σ(t) = σ₀√[1 + (t/τd)²]. This mathematics has experimental foundations. Electron diffraction, C₆₀ fullerene interference and ultracold atomic gases demonstrate the wave nature and evolution of quantum matter. What we observe experimentally are interference patterns and probability distributions, not the wavefunction itself.

Philosophy Of Physics

27,182 просмотров • 3 дней назад

Fourier Transform - Perfect and Final Version When you plot 225 functions, each constructed by up to 1000 complex exponential functions (i.e., rotating vectors) you get the following drawing. Math is Beautiful Math is Art
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Fourier Transform - Perfect and Final Version When you plot 225 functions, each constructed by up to 1000 complex exponential functions (i.e., rotating vectors) you get the following drawing. Math is Beautiful Math is Art

Carl Friedrich Gauss

45,503 просмотров • 10 месяцев назад

Euler's formula is not a formula. An MIT professor spent a whole lecture proving it is a definition, not a discovery. This is 18.03, Lecture 6, Arthur Mattuck. Free on MIT OpenCourseWare. The formula is e^(iθ) = cos θ + i sin θ. Most people memorize it and move on. Mattuck refused to. He asked one thing. What gives that the right to be called an exponential at all? An exponential has to obey one law: e^a · e^b = e^(a+b). So he checked it. He multiplied e^(iθ₁) by e^(iθ₂), expanded the sines and cosines, and looked at what fell out. The angle-addition identities. cos(A+B) and sin(A+B), the two formulas you were forced to memorize in school. They were never two formulas. They are one line: e^(iθ) obeying the law of exponents. Then the payoff. An integral that normally needs integration by parts twice and a trick, ∫ e^(-x) cos x dx, drops out in a few lines once you let the numbers go complex. Nothing is easier to integrate than an exponential. Someone once asked him what beauty in math looks like. He pointed at the board. Something long became something short, and lost nothing.
45:18

Euler's formula is not a formula. An MIT professor spent a whole lecture proving it is a definition, not a discovery. This is 18.03, Lecture 6, Arthur Mattuck. Free on MIT OpenCourseWare. The formula is e^(iθ) = cos θ + i sin θ. Most people memorize it and move on. Mattuck refused to. He asked one thing. What gives that the right to be called an exponential at all? An exponential has to obey one law: e^a · e^b = e^(a+b). So he checked it. He multiplied e^(iθ₁) by e^(iθ₂), expanded the sines and cosines, and looked at what fell out. The angle-addition identities. cos(A+B) and sin(A+B), the two formulas you were forced to memorize in school. They were never two formulas. They are one line: e^(iθ) obeying the law of exponents. Then the payoff. An integral that normally needs integration by parts twice and a trick, ∫ e^(-x) cos x dx, drops out in a few lines once you let the numbers go complex. Nothing is easier to integrate than an exponential. Someone once asked him what beauty in math looks like. He pointed at the board. Something long became something short, and lost nothing.

Dominique

89,379 просмотров • 3 месяцев назад

Graphing exponential functions Part 1 #math #iteachalgebra #algebra #mathematics #exponents
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Graphing exponential functions Part 1 #math #iteachalgebra #algebra #mathematics #exponents

iteachalgebra

149,100 просмотров • 2 лет назад

The equation of a circle actually originates from one of the most beautiful ideas in mathematics- the Pythagorean theorem 🔺
2:02

The equation of a circle actually originates from one of the most beautiful ideas in mathematics- the Pythagorean theorem 🔺

S.T.E.M Explorer

12,915 просмотров • 7 месяцев назад

Exponential Growth Functions! #math #maths #mathematics #mathtricks #mathteacher #fyp #fy #fypage #tutorial #middleschoolmath #algebra #algebra2 #algebra1 #exponent #exponentialfunctions #exponential #exponentialgrowth #exponentialdecay
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Exponential Growth Functions! #math #maths #mathematics #mathtricks #mathteacher #fyp #fy #fypage #tutorial #middleschoolmath #algebra #algebra2 #algebra1 #exponent #exponentialfunctions #exponential #exponentialgrowth #exponentialdecay

Count on Me Math

35,200 просмотров • 2 лет назад

How do mathematicians reveal patterns in structures too complex to see directly? That is at the heart of the work of 2026 Wolf Prize in Mathematics Laureates David Kazhdan and Joseph Bernstein. Their discoveries helped build what is now known as geometric representation theory — a field that connects algebra, geometry and symmetry in powerful new ways. Kazhdan’s work introduced new tools for understanding groups and symmetry, including the influential Kazhdan–Lusztig theory. Bernstein developed ideas such as D-modules and helped create mathematical frameworks that linked representation theory with geometry in entirely new ways. The result is a mathematical language that allows researchers to move between different fields and uncover relationships that were previously hidden. Watch the video for a clear visual introduction to their breakthrough — and see how abstract mathematics can connect geometry, algebra and symmetry in surprisingly elegant ways.
1:13

How do mathematicians reveal patterns in structures too complex to see directly? That is at the heart of the work of 2026 Wolf Prize in Mathematics Laureates David Kazhdan and Joseph Bernstein. Their discoveries helped build what is now known as geometric representation theory — a field that connects algebra, geometry and symmetry in powerful new ways. Kazhdan’s work introduced new tools for understanding groups and symmetry, including the influential Kazhdan–Lusztig theory. Bernstein developed ideas such as D-modules and helped create mathematical frameworks that linked representation theory with geometry in entirely new ways. The result is a mathematical language that allows researchers to move between different fields and uncover relationships that were previously hidden. Watch the video for a clear visual introduction to their breakthrough — and see how abstract mathematics can connect geometry, algebra and symmetry in surprisingly elegant ways.

Wolf Prize

93,657 просмотров • 1 месяц назад

Tesla's "Egg of Columbus," demonstrated at the 1893 World's Columbian Exposition, illustrated a fundamental advance in the transmission and conversion of electrical energy. Tesla's polyphase system used multiple alternating currents, offset in phase from one another, to produce a continuously rotating magnetic field. Rather than requiring mechanical switching to create rotation, the changing electrical phases generated rotation within the field itself. The copper egg responded through electromagnetic induction, spinning and eventually rising onto its axis. This same principle became the foundation of the AC induction motor and helped establish polyphase alternating current as a practical system for generating, transmitting, and converting electrical power on a large scale.
0:52

Tesla's "Egg of Columbus," demonstrated at the 1893 World's Columbian Exposition, illustrated a fundamental advance in the transmission and conversion of electrical energy. Tesla's polyphase system used multiple alternating currents, offset in phase from one another, to produce a continuously rotating magnetic field. Rather than requiring mechanical switching to create rotation, the changing electrical phases generated rotation within the field itself. The copper egg responded through electromagnetic induction, spinning and eventually rising onto its axis. This same principle became the foundation of the AC induction motor and helped establish polyphase alternating current as a practical system for generating, transmitting, and converting electrical power on a large scale.

Mechanical Knowledge

82,879 просмотров • 2 месяцев назад

The quadratic formula looks ridiculous. Every piece of it has a reason. Here’s where the √, ±, and 2a actually come from. #MathTok #Algebra #QuadraticFormula #LearnOnTikTokContest #Math
2:38

The quadratic formula looks ridiculous. Every piece of it has a reason. Here’s where the √, ±, and 2a actually come from. #MathTok #Algebra #QuadraticFormula #LearnOnTikTokContest #Math

Gravity Assist

386,800 просмотров • 1 месяц назад

Alan Oppenheim, MIT professor: "Jump Trading pays quants $650K to pull one real signal out of market noise. I wrote the textbook on that exact math and taught it for free." this free MIT lecture is the math the whole "quant edge" is built on, and almost nobody selling you a course will name it. the man teaching it wrote the signal-processing textbook every engineer on earth learned from, and he put the lectures online for nothing. here is the idea. a price chart is a signal buried in noise. most of what you see is random static, and hidden inside is a faint, repeating pattern with real mathematics for pulling it out. that is not a metaphor. jim simons was a codebreaker who did exactly this to enemy transmissions, then pointed the same math at the market and made a hundred billion dollars. none of it is new. oppenheim taught this in 1987, the math is older, and every lecture is free. same story i told in the article above. the edge was never a secret formula. it was knowing a market is a noisy signal, and having the tools to read it. here is what the gurus skip. the math to separate signal from noise is free and everyone has it. the edge is cleaner data than the next desk, and the discipline to trust a faint signal across thousands of trades instead of chasing the static. the filter is free. the patience to use it is not.
29:44

Alan Oppenheim, MIT professor: "Jump Trading pays quants $650K to pull one real signal out of market noise. I wrote the textbook on that exact math and taught it for free." this free MIT lecture is the math the whole "quant edge" is built on, and almost nobody selling you a course will name it. the man teaching it wrote the signal-processing textbook every engineer on earth learned from, and he put the lectures online for nothing. here is the idea. a price chart is a signal buried in noise. most of what you see is random static, and hidden inside is a faint, repeating pattern with real mathematics for pulling it out. that is not a metaphor. jim simons was a codebreaker who did exactly this to enemy transmissions, then pointed the same math at the market and made a hundred billion dollars. none of it is new. oppenheim taught this in 1987, the math is older, and every lecture is free. same story i told in the article above. the edge was never a secret formula. it was knowing a market is a noisy signal, and having the tools to read it. here is what the gurus skip. the math to separate signal from noise is free and everyone has it. the edge is cleaner data than the next desk, and the discipline to trust a faint signal across thousands of trades instead of chasing the static. the filter is free. the patience to use it is not.

Rossst.03

281,905 просмотров • 4 месяцев назад

Math in Sanskrit: Quadratic formula Did you know that the quadratic formula—widely used in mathematics, science, and engineering—was explicitly stated for the first time in a Sanskrit verse by the Indian mathematician Brahmagupta? Watch to know more.
2:52

Math in Sanskrit: Quadratic formula Did you know that the quadratic formula—widely used in mathematics, science, and engineering—was explicitly stated for the first time in a Sanskrit verse by the Indian mathematician Brahmagupta? Watch to know more.

Nityānanda Miśra (मिश्रोपाख्यो नित्यानन्दः)

66,847 просмотров • 11 месяцев назад

We often call Euler’s identity the most beautiful equation in mathematics. But sometimes we forget the beauty of the Gaussian integral. It is absolutely gorgeous. And if you don’t know this integral yet, can you really call yourself a math nerd? 😉
0:54

We often call Euler’s identity the most beautiful equation in mathematics. But sometimes we forget the beauty of the Gaussian integral. It is absolutely gorgeous. And if you don’t know this integral yet, can you really call yourself a math nerd? 😉

Math Files

94,810 просмотров • 1 месяц назад

Math shouldn’t be illegal, but this equation feels like a cheat code. 🤯 Euler’s formula somehow builds a perfect bridge between compound growth and circles, and brings the 5 most important numbers in the universe together into one tiny equation. Did you learn this in school? 🤔 #math #stem #eulersformula #learnontiktok #mathtok #education #science
1:56

Math shouldn’t be illegal, but this equation feels like a cheat code. 🤯 Euler’s formula somehow builds a perfect bridge between compound growth and circles, and brings the 5 most important numbers in the universe together into one tiny equation. Did you learn this in school? 🤔 #math #stem #eulersformula #learnontiktok #mathtok #education #science

Gravity Assist

143,400 просмотров • 7 месяцев назад