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The Math Flow

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

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(a + b + c)² = a² + b² + c² + 2(ab + ac + bc)

(a + b + c)² = a² + b² + c² + 2(ab + ac + bc)

82,239 Aufrufe

Unrolling Euler's Formula.

Unrolling Euler's Formula.

273,610 Aufrufe

A plane sliding through a double cone traces every conic section.

A plane sliding through a double cone traces every conic section.

39,662 Aufrufe

You Can Watch Euler's Formula, Unrolled.

You Can Watch Euler's Formula, Unrolled.

94,397 Aufrufe

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 Aufrufe

The Geometry of Integration by Parts.

The Geometry of Integration by Parts.

481,620 Aufrufe

The distance a circle travels in a single revolution without slipping equals π times its diameter, regardless of the wheel's size.

The distance a circle travels in a single revolution without slipping equals π times its diameter, regardless of the wheel's size.

27,362 Aufrufe

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 Aufrufe

Rotation Matrices in Motion.

Rotation Matrices in Motion.

144,777 Aufrufe

The Beautiful Gaussian Integral.

The Beautiful Gaussian Integral.

86,489 Aufrufe

The Hidden Geometry of Trigonometric Functions and the Unit Circle.

The Hidden Geometry of Trigonometric Functions and the Unit Circle.

66,179 Aufrufe

The difference between Δy/Δx and dy/dx.

The difference between Δy/Δx and dy/dx.

50,226 Aufrufe

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 Aufrufe

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 Aufrufe

a² - b² = (a+b)(a-b)

a² - b² = (a+b)(a-b)

46,344 Aufrufe

The Geometry of Square Waves.

The Geometry of Square Waves.

30,386 Aufrufe

The Fibonacci’s Elephant.🐘

The Fibonacci’s Elephant.🐘

27,310 Aufrufe

A circle is secretly a triangle!

A circle is secretly a triangle!

27,386 Aufrufe

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 Aufrufe

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

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

18,322 Aufrufe

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You Can Watch Euler's Formula, Unrolled.

You Can Watch Euler's Formula, Unrolled.

94,397 Aufrufe • vor 5 Tagen

All Six Trig Functions Are Just Lengths on One Circle.

All Six Trig Functions Are Just Lengths on One Circle.

236,488 Aufrufe • vor 23 Tagen

Fractal Spirograph.

Fractal Spirograph.

126,513 Aufrufe • vor 14 Tagen

The Geometry of Perspective.

The Geometry of Perspective.

105,230 Aufrufe • vor 1 Monat

Mathematics. Functions. Dance.

Mathematics. Functions. Dance.

322,490 Aufrufe • vor 5 Monaten

The Geometry Behind Trigonometry’s Most Famous Formula.

The Geometry Behind Trigonometry’s Most Famous Formula.

113,587 Aufrufe • vor 1 Monat

Every Curve Hiding Inside a Circle.

Every Curve Hiding Inside a Circle.

104,707 Aufrufe • vor 1 Monat

The chaos of double pendulums.

The chaos of double pendulums.

32,280 Aufrufe • vor 16 Tagen

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

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

54,963 Aufrufe • vor 1 Monat

This is how a circle produces sine and cosine waves.

This is how a circle produces sine and cosine waves.

98,291 Aufrufe • vor 2 Monaten

Systems of Equations in Three Dimensions.

Systems of Equations in Three Dimensions.

40,086 Aufrufe • vor 29 Tagen

Sine and Cosine, Built from the Unit Circle.

Sine and Cosine, Built from the Unit Circle.

39,131 Aufrufe • vor 1 Monat

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:👇

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:👇

19,525 Aufrufe • vor 17 Tagen

The Unit Circle: Sines and Cosines in Motion

The Unit Circle: Sines and Cosines in Motion

91,343 Aufrufe • vor 3 Monaten

Sine & Cosine on the Unit Circle.

Sine & Cosine on the Unit Circle.

59,829 Aufrufe • vor 2 Monaten

How the Unit Circle Generates Sine and Cosine Waves:👇

How the Unit Circle Generates Sine and Cosine Waves:👇

87,668 Aufrufe • vor 4 Monaten

The Rössler Attractor.

The Rössler Attractor.

89,233 Aufrufe • vor 4 Monaten

(a+b)³ = a³ + b³ + 3a²b + 3ab²

(a+b)³ = a³ + b³ + 3a²b + 3ab²

60,276 Aufrufe • vor 2 Monaten

Law of Sines: Visual proof

Law of Sines: Visual proof

14,790 Aufrufe • vor 20 Tagen

The Geometry Behind Integration by Parts.

The Geometry Behind Integration by Parts.

26,012 Aufrufe • vor 1 Monat