Notes on mathematics.
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2026.03.20
Interesting Number Plates
Taxicab numbers, Carmichael liars, Kaprekar attractors, Wieferich primes, and Grothendieck's 57.
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2026.03.20
Ham Sandwich Theorem
One plane that bisects three objects at once: the intermediate value theorem meets Borsuk–Ulam.
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2026.03.20
Jordan Curve Theorem
Every simple closed curve splits the plane into inside and outside, and why proving it is hard.
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2026.03.20
Quaternions
From Broom Bridge to modern attitude control: smooth 3D rotations without gimbal lock.
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2026.03.20
How Many Times Do You Need to Shuffle a Deck of Cards Before It Is Truly Random?
Diaconis, riffle shuffles, and the cutoff phenomenon behind the famous threshold of seven.
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2026.03.20
Bounded Gaps Between Primes
Unbounded prime gaps, bounded liminf, and how Zhang, Maynard, and Polymath pushed the frontier.
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2026.03.20
Dirac Delta Function
A perfectly sharp tap, its sifting property, and why its Fourier transform is constant.
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2026.03.20
Quantum Zeno Effect
Frequent measurement that freezes decay, and the intermediate-rate anti-Zeno acceleration.
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2026.03.20
The Lady, or the Tiger?
Ill-posed inverse problems and why one bit of information cannot justify the ending.
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2026.03.20
Lie Groups
Symmetries that vary smoothly: groups like rotations, and their infinitesimal Lie algebras.
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2026.03.20
Green's Functions
How systems react to a single point poke, and why that response gives the solution to any forcing.
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2026.03.20
Fourier Series
Turning jagged periodic patterns into weighted sums of smooth waves, and what happens near jumps.
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2026.03.20
Gödel's Proof of the Existence of God
Gödel, modal logic, and the ontological argument: formal assumptions that force a conclusion.
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2026.03.20
Arnold's Cat Map
Stretching and folding a square: a perfectly deterministic transformation that looks, for all practical purposes, like chaos.
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2026.03.20
Isospectral Shapes
Drums that sound the same: isospectral polygons and the transplantation idea that preserves spectra.
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2025.12.17
On Recording Patterns
"Mathematics is not an artistic overlay placed upon reality, but the grammar through which reality permits itself to be analysed at all."