Math Pulse

Mathematics, rightly viewed, possesses not only truth, but supreme beauty.

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Is this how mixed quantum state looks like?

ρ = ∑ pₙ |ψₙ⟩⟨ψₙ|

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Watch hydrogen wavefunctions evolve through quantum states (n, l, m).

These aren't fixed orbitals, but the electron's probability density reshaping as it transitions between energy levels.

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The Equation of Life

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Classic derivative notations through the eyes of their creators. Leibniz, Euler, Lagrange, and Newton.

Which one’s your go-to?

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Stokes' Theorem in pure visual glory:

The circulation of F around any closed curve C ∮_C F · dr

equals the flux of its curl through any surface S bounded by C:

∬_S (∇ × F) · dS

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The limit as n → ∞ is a circle.

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Harvard SEAS prismatic architected metamaterial: extruded cubic cells (24 faces, 36 edges) tessellated via snapology origami.

Reconfigures volume, shape, and effective stiffness on demand through hinge folding; scale-independent from nano to meter.

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3D visualization of Euler’s formula: e^{jθ} = cos(θ) + j sin(θ)

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Harmonious Geometry: The Hirajoshi Wave

Gravity-defying spheres trace the haunting paths of the C Hirajoshi scale (C, D, Eb, G, Ab), forming a mesmerizing polyrhythmic pendulum.

Credit: project.jdm

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You’re looking at a 2D projection of the 3D shadow of a 4D Tesseract.

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This is the single most beautiful moment in all of calculus.

The limit as h → 0... Watch the secant line collapse into the exact tangent at that point.

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MIT Entrance Examination

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The Clay Institute offered $1 million for solutions to 7 of the hardest problems in math. Only one has fallen.
Here are the ones that still remain unsolved:

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In 1666, Newton used just 22 terms of this series to calculate π to 16 decimal places. He later admitted, “I’m ashamed to say how many digits I worked out, since I had nothing better to do at the time.”

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This is a prime.

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The largest known prime, discovered in Oct 2024, is 2¹³⁶²⁷⁹⁸⁴¹−1. It’s a Mersenne prime (2ᵖ−1), a special type that’s easier to search for. GIMPS took nearly 6 years to find it after the last record, and it’s the first Mersenne prime discovered using GPUs.

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The Lo Shu Square is an ancient Chinese 3×3 magic square where each row, column, and diagonal sums to 15.

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When you square a number made up of n consecutive ones, the result forms a perfect palindrome that increases from 1 up to n and then decreases back to 1.

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Last digit of any integer n⁵ is n itself