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The Math Flow

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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.

11,985,045 Aufrufe

“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

62,112 Aufrufe

Rotation matrices at work.

Rotation matrices at work.

88,520 Aufrufe

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

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

64,082 Aufrufe

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

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

404,783 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.

1,070,321 Aufrufe

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 Aufrufe

The Geometry of Integration by Parts.

The Geometry of Integration by Parts.

494,914 Aufrufe

Unrolling Euler's Formula.

Unrolling Euler's Formula.

284,328 Aufrufe

a³−b³ = (a−b)(a²+ab+b²)

a³−b³ = (a−b)(a²+ab+b²)

152,353 Aufrufe

You Can Watch Euler's Formula, Unrolled.

You Can Watch Euler's Formula, Unrolled.

154,314 Aufrufe

Rotation Matrices in Motion.

Rotation Matrices in Motion.

145,670 Aufrufe

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

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

48,856 Aufrufe

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 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.

51,256 Aufrufe

The Beautiful Gaussian Integral.

The Beautiful Gaussian Integral.

86,650 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.

28,118 Aufrufe

The Hidden Geometry of Trigonometric Functions and the Unit Circle.

The Hidden Geometry of Trigonometric Functions and the Unit Circle.

66,830 Aufrufe

The difference between Δy/Δx and dy/dx.

The difference between Δy/Δx and dy/dx.

50,561 Aufrufe

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 Aufrufe

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The Fibonacci Spiral Clock.

The Fibonacci Spiral Clock.

129,437 Aufrufe • vor 1 Tag

Fibonacci: From curves to cubes and back to curves.

Fibonacci: From curves to cubes and back to curves.

174,715 Aufrufe • vor 10 Tagen

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

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

25,115 Aufrufe • vor 2 Tagen

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

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

67,633 Aufrufe • vor 8 Tagen

1³ + 2³ + … + n³ = (1 + 2 + … + n)²

1³ + 2³ + … + n³ = (1 + 2 + … + n)²

37,762 Aufrufe • vor 5 Tagen

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

41,676 Aufrufe • vor 7 Tagen

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

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

40,620 Aufrufe • vor 10 Tagen

Beauty is inseparable from mathematics...ᝰ.

Beauty is inseparable from mathematics...ᝰ.

114,076 Aufrufe • vor 1 Monat

All Six Trig Functions Are Just Lengths on One Circle.

All Six Trig Functions Are Just Lengths on One Circle.

239,043 Aufrufe • vor 2 Monaten

You Can Watch Euler's Formula, Unrolled.

You Can Watch Euler's Formula, Unrolled.

154,314 Aufrufe • vor 1 Monat

The golden spiral, set in motion.

The golden spiral, set in motion.

85,124 Aufrufe • vor 1 Monat

Fractal Spirograph.

Fractal Spirograph.

127,630 Aufrufe • vor 1 Monat

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 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.

23,061 Aufrufe • vor 10 Tagen

Mathematics. Functions. Dance.

Mathematics. Functions. Dance.

326,356 Aufrufe • vor 6 Monaten

The Geometry of Perspective.

The Geometry of Perspective.

106,246 Aufrufe • vor 2 Monaten

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.

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.

43,661 Aufrufe • vor 1 Monat

The Geometry Behind Trigonometry’s Most Famous Formula.

The Geometry Behind Trigonometry’s Most Famous Formula.

114,176 Aufrufe • vor 3 Monaten

Every Curve Hiding Inside a Circle.

Every Curve Hiding Inside a Circle.

104,707 Aufrufe • vor 3 Monaten

(𝑎 + 𝑏)³ = 𝑎³ + 3𝑎²𝑏 + 3𝑎𝑏² + 𝑏³

(𝑎 + 𝑏)³ = 𝑎³ + 3𝑎²𝑏 + 3𝑎𝑏² + 𝑏³

45,769 Aufrufe • vor 1 Monat

This is how a circle produces sine and cosine waves.

This is how a circle produces sine and cosine waves.

98,726 Aufrufe • vor 3 Monaten