The Math Flow

@TheMathFlow • 48,066 subscribers

All about Mathematics: Books • Pictures • Proofs • Animations• Memes • & • History.

Shorts

The Geometry of Integration by Parts.

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Sine and Cosine waves are...nothing but the shadows of a spinning circle.

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Rotation Matrices in Motion.

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The difference between Δy/Δx and dy/dx.

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The Geometry of Square Waves.

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The Beautiful Gaussian Integral.

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The Hidden Geometry of Trigonometric Functions and the Unit Circle.

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The Fibonacci’s Elephant.🐘

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A circle is secretly a triangle!

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In 1963, mathematician Stanislaw Ulam got bored during a meeting and started doodling a grid spiral of sequential numbers. By highlighting only the primes, he uncovered what remains one of the most mesmerising visual mysteries in number theory: the Ulam Spiral.

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Viviani’s Theorem, named after the Italian mathematician Vincenzo Viviani, states that for any point inside an equilateral triangle, the sum of the perpendicular distances from the point to the three sides is always equal to the altitude (height) of the triangle.

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a² - b² = (a+b)(a-b)

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The geometry of eⁱᶿ = cos(θ) + isin(θ).

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Anatomy of a Definite Integral.

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The Basics of Electromagnetic Waves: Electricity and magnetism can sit still, like static electricity in your hair or a magnet stuck to your fridge. But when they move and change, they actually create each other. Together, they team up to form invisible ripples of energy called electromagnetic waves. Unlike ocean waves or sound waves, which need water or air to ripple through, electromagnetic waves don't need any material at all. They can easily travel through the completely empty vacuum of space. Maxwell's Big Idea: In the 1860s and 1870s, a Scottish scientist named James Clerk Maxwell figured out how this works. He wrote down the math showing exactly how electricity and magnetism link together to make these travelling waves. Today, scientists call his famous rules Maxwell's Equations. Hertz Proves It: Later, a German physicist named Heinrich Hertz took Maxwell's ideas and brought them to life. He was the first person to actually create and catch radio waves. To honour his work, we use the word hertz to measure how fast a wave vibrates (one cycle per second). Hertz's experiments proved two massive ideas: Radio waves are just invisible light: He showed that radio waves travel at the exact same speed as light, proving that they are actually a form of light we just can't see. Going wireless: He finally figured out how to detach these energy fields from physical wires, allowing the waves to fly freely through the air exactly as Maxwell had predicted.

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$We were taught the derivative as a formula to memorise. A definition to recite. A rule to apply. Something that "gives you the slope." But nobody told us what the formula was actually saying. Every symbol is a sentence. Every fraction is a question. Every limit is a story about getting closer and closer to something you can never quite touch. The top of the fraction? That's a change. A difference. A before and after. The bottom? That's how long you waited to see it. The limit? That's you, zooming in, refusing to settle for an approximation - chasing the truth all the way down to an interval so small it almost disappears. Put it all together, and you get the most honest question in calculus: How fast is something changing - right now, in this exact instant? Not on average. Not over a minute. Not eventually. Right now. That's it. That's the derivative. It's not a trick. It's not a rule. It's a beautifully precise way of asking a very human question: what's happening, in this moment? We spent years solving these. Maybe it's time we actually understood them.$