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The Math Flow

@TheMathFlow • 57,019 subscribers

Mathematics insights presented through animations, formulas, pictures, graphics, quotes, books, history, and memes. DM for inquiries.

Shorts

You Can Watch Euler's Formula, Unrolled.

You Can Watch Euler's Formula, Unrolled.

93,030 views

Unrolling Euler's Formula.

Unrolling Euler's Formula.

273,307 views

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

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

1,068,500 views

The Geometry of Integration by Parts.

The Geometry of Integration by Parts.

481,607 views

At any point on a curve, there's one circle that fits the curve better than any other. It touches the curve at that point, leans in the same direction, and bends by the same amount. It tells you how sharply the curve is bending at that exact spot, and which way it's leaning. If the curve is bending sharply, the circle is small. If the curve is nearly straight, the circle is huge. The osculating circle of the curve.

At any point on a curve, there's one circle that fits the curve better than any other. It touches the curve at that point, leans in the same direction, and bends by the same amount. It tells you how sharply the curve is bending at that exact spot, and which way it's leaning. If the curve is bending sharply, the circle is small. If the curve is nearly straight, the circle is huge. The osculating circle of the curve.

46,308 views

Rotation Matrices in Motion.

Rotation Matrices in Motion.

144,777 views

The Beautiful Gaussian Integral.

The Beautiful Gaussian Integral.

86,489 views

The Hidden Geometry of Trigonometric Functions and the Unit Circle.

The Hidden Geometry of Trigonometric Functions and the Unit Circle.

66,179 views

The difference between Δy/Δx and dy/dx.

The difference between Δy/Δx and dy/dx.

50,226 views

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.

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.

52,594 views

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.

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.

39,726 views

The Geometry of Square Waves.

The Geometry of Square Waves.

30,386 views

a² - b² = (a+b)(a-b)

a² - b² = (a+b)(a-b)

46,344 views

The Fibonacci’s Elephant.🐘

The Fibonacci’s Elephant.🐘

27,310 views

A circle is secretly a triangle!

A circle is secretly a triangle!

27,386 views

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.

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.

37,313 views

Visualization of equation: 1 + tan² (θ) = sec² (θ).

Visualization of equation: 1 + tan² (θ) = sec² (θ).

18,322 views

Anatomy of a Definite Integral.

Anatomy of a Definite Integral.

28,908 views

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

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

19,986 views

17 equations that changed the world.

17 equations that changed the world.

30,160 views

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You Can Watch Euler's Formula, Unrolled.
0:20
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You Can Watch Euler's Formula, Unrolled.

The Math Flow

93,030 views • 2 days ago

All Six Trig Functions Are Just Lengths on One Circle.
0:28
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All Six Trig Functions Are Just Lengths on One Circle.

The Math Flow

236,009 views • 21 days ago

Fractal Spirograph.
0:22
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Fractal Spirograph.

The Math Flow

126,322 views • 12 days ago

The Geometry of Perspective.
0:39
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The Geometry of Perspective.

The Math Flow

105,230 views • 1 month ago

Mathematics. Functions. Dance.
0:29
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Mathematics. Functions. Dance.

The Math Flow

322,337 views • 4 months ago

The Geometry Behind Trigonometry’s Most Famous Formula.
0:27
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The Geometry Behind Trigonometry’s Most Famous Formula.

The Math Flow

113,587 views • 1 month ago

The chaos of double pendulums.
0:30
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The chaos of double pendulums.

The Math Flow

32,280 views • 14 days ago

Every Curve Hiding Inside a Circle.
0:54
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Every Curve Hiding Inside a Circle.

The Math Flow

104,707 views • 1 month ago

Nature's Only Five Perfect Shapes: Five Platonic Solids.
0:45
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Nature's Only Five Perfect Shapes: Five Platonic Solids.

The Math Flow

54,906 views • 27 days ago

This is how a circle produces sine and cosine waves.
0:30
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This is how a circle produces sine and cosine waves.

The Math Flow

98,291 views • 2 months ago

Systems of Equations in Three Dimensions.
0:27
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Systems of Equations in Three Dimensions.

The Math Flow

35,307 views • 27 days ago

Sine and Cosine, Built from the Unit Circle.
0:32
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Sine and Cosine, Built from the Unit Circle.

The Math Flow

39,051 views • 1 month ago

On this day in 1685, Brook Taylor was born, the English mathematician whose name became inseparable from one of calculus’s most powerful ideas: the Taylor series. He showed how a sufficiently smooth function can be represented locally using its derivatives:👇
0:26
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On this day in 1685, Brook Taylor was born, the English mathematician whose name became inseparable from one of calculus’s most powerful ideas: the Taylor series. He showed how a sufficiently smooth function can be represented locally using its derivatives:👇

The Math Flow

19,525 views • 15 days ago

The Unit Circle: Sines and Cosines in Motion
0:28
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The Unit Circle: Sines and Cosines in Motion

The Math Flow

91,343 views • 3 months ago

Sine & Cosine on the Unit Circle.
0:27
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Sine & Cosine on the Unit Circle.

The Math Flow

59,829 views • 2 months ago

How the Unit Circle Generates Sine and Cosine Waves:👇
0:21
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How the Unit Circle Generates Sine and Cosine Waves:👇

The Math Flow

87,668 views • 4 months ago

The Rössler Attractor.
0:30
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The Rössler Attractor.

The Math Flow

89,233 views • 4 months ago

(a+b)³ = a³ + b³ + 3a²b + 3ab²
0:33
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(a+b)³ = a³ + b³ + 3a²b + 3ab²

The Math Flow

60,276 views • 2 months ago

The Geometry Behind Integration by Parts.
0:58
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The Geometry Behind Integration by Parts.

The Math Flow

25,958 views • 1 month ago

Law of Sines: Visual proof
0:28
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Law of Sines: Visual proof

The Math Flow

13,630 views • 18 days ago