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The Math Flow

@TheMathFlow • 89,391 subscribers

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

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Solving a Rubik's Cube with graph theory.

Solving a Rubik's Cube with graph theory.

11,984,909 views

“Philosophy is written in this grand book, the universe, which stands continually open before our eyes… It is written in the language of mathematics.” - Galileo Galilei

“Philosophy is written in this grand book, the universe, which stands continually open before our eyes… It is written in the language of mathematics.” - Galileo Galilei

40,160 views

Rotation matrices at work.

Rotation matrices at work.

88,520 views

Pythagorean theorem, written on the unit circle: sin²θ + cos²θ = 1.

Pythagorean theorem, written on the unit circle: sin²θ + cos²θ = 1.

64,082 views

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

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

404,783 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.

1,070,321 views

Three golden rectangles, set mutually perpendicular, quietly define a regular icosahedron. Their twelve corners are exactly its twelve vertices.

Three golden rectangles, set mutually perpendicular, quietly define a regular icosahedron. Their twelve corners are exactly its twelve vertices.

26,351 views

The Geometry of Integration by Parts.

The Geometry of Integration by Parts.

494,914 views

Unrolling Euler's Formula.

Unrolling Euler's Formula.

284,328 views

a³−b³ = (a−b)(a²+ab+b²)

a³−b³ = (a−b)(a²+ab+b²)

152,353 views

You Can Watch Euler's Formula, Unrolled.

You Can Watch Euler's Formula, Unrolled.

154,314 views

Rotation Matrices in Motion.

Rotation Matrices in Motion.

145,670 views

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

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

48,856 views

Viviani’s theorem demonstrates spatial conservation, showing that the three perpendicular lines behave like a zero-sum game: moving closer to one side requires corresponding movement away from the other two. Local changes maintain a global constant.

Viviani’s theorem demonstrates spatial conservation, showing that the three perpendicular lines behave like a zero-sum game: moving closer to one side requires corresponding movement away from the other two. Local changes maintain a global constant.

30,582 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.

51,256 views

The Beautiful Gaussian Integral.

The Beautiful Gaussian Integral.

86,650 views

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.

28,118 views

The Hidden Geometry of Trigonometric Functions and the Unit Circle.

The Hidden Geometry of Trigonometric Functions and the Unit Circle.

66,830 views

The difference between Δy/Δx and dy/dx.

The difference between Δy/Δx and dy/dx.

50,561 views

A square wave in time is a set of amplitudes in frequency, and the Fourier series is the bridge between the two.

A square wave in time is a set of amplitudes in frequency, and the Fourier series is the bridge between the two.

19,828 views

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The Fibonacci Spiral Clock.
1:00
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The Fibonacci Spiral Clock.

The Math Flow

126,156 views • 1 day ago

Fibonacci: From curves to cubes and back to curves.
0:21
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Fibonacci: From curves to cubes and back to curves.

The Math Flow

174,715 views • 10 days ago

“The infinite! No other question has ever moved so profoundly the spirit of man.” - David Hilbert
0:24
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“The infinite! No other question has ever moved so profoundly the spirit of man.” - David Hilbert

The Math Flow

25,115 views • 2 days ago

"The eternal silence of these infinite spaces frightens me." - Blaise Pascal
0:24
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"The eternal silence of these infinite spaces frightens me." - Blaise Pascal

The Math Flow

67,633 views • 8 days ago

1³ + 2³ + … + n³ = (1 + 2 + … + n)²
0:22
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1³ + 2³ + … + n³ = (1 + 2 + … + n)²

The Math Flow

37,762 views • 5 days ago

The Aizawa attractor: It's a distinctive three-dimensional chaotic shape traced out by a system of three nonlinear differential equations with six parameters.
1:00
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The Aizawa attractor: It's a distinctive three-dimensional chaotic shape traced out by a system of three nonlinear differential equations with six parameters.

The Math Flow

41,676 views • 7 days ago

"The universe is written in the language of mathematics." – Galileo Galilei.
0:32
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"The universe is written in the language of mathematics." – Galileo Galilei.

The Math Flow

40,620 views • 9 days ago

Beauty is inseparable from mathematics...ᝰ.
0:25
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Beauty is inseparable from mathematics...ᝰ.

The Math Flow

114,076 views • 1 month 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

239,043 views • 2 months ago

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

The Math Flow

154,314 views • 1 month ago

The golden spiral, set in motion.
0:25
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The golden spiral, set in motion.

The Math Flow

85,124 views • 1 month ago

Fractal Spirograph.
0:22
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Fractal Spirograph.

The Math Flow

127,630 views • 1 month ago

The Archimedean spiral is one of the oldest curves in mathematics, with turns that stay evenly spaced. Here, a straight line slowly turns into this spiral, and four hundred connected points follow its shape.
0:23
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The Archimedean spiral is one of the oldest curves in mathematics, with turns that stay evenly spaced. Here, a straight line slowly turns into this spiral, and four hundred connected points follow its shape.

The Math Flow

23,061 views • 10 days ago

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

The Math Flow

326,356 views • 6 months ago

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

The Math Flow

106,246 views • 2 months ago

Matrix multiplication seems complex until you observe it step by step. Watch how each of the nine entries is formed: one row from matrix A, one column from matrix B, three products, and one sum.
0:37
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Matrix multiplication seems complex until you observe it step by step. Watch how each of the nine entries is formed: one row from matrix A, one column from matrix B, three products, and one sum.

The Math Flow

43,661 views • 1 month 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

114,176 views • 3 months 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 • 3 months ago

(𝑎 + 𝑏)³ = 𝑎³ + 3𝑎²𝑏 + 3𝑎𝑏² + 𝑏³
0:27
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(𝑎 + 𝑏)³ = 𝑎³ + 3𝑎²𝑏 + 3𝑎𝑏² + 𝑏³

The Math Flow

45,769 views • 1 month 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

98,726 views • 3 months ago