Monocurl Essays
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monocurl essays

Mathematical writing with rendered figures and live Monocurl scenes.

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Geometry of Curves and Maps

How local measurements and transformations turn curves, circles, and lines into usable mental models.

  1. Curvature Is Turning Per Unit DistanceA curve is not just a path through points. It carries a hidden clock that measures how quickly its tangent direction turns.
  2. Inversion Turns Lines Into CirclesThe map z -> 1/z looks strange until you watch a straight line become a circle tangent to the origin.
  3. Conformal Maps Keep Tiny AnglesComplex functions can twist and stretch the plane while preserving the angle between infinitesimal curves.
  4. Gauss Bonnet Counts TurningThe total curvature of a surface is tied to topology because local turning has a global accounting rule.

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Dynamics of Smoothing

Equations that turn local rules into global behavior, from diffusion to frequency filtering.

  1. The Heat Equation Sorts Waves By SizeHeat flow looks like blur, but underneath it is a precise frequency filter that erases small wiggles first.

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Linear Algebra as Geometry

Matrices are not tables of numbers. They are geometric machines whose hidden structure becomes visible when you watch what they do to space.

  1. Eigenvectors Are Directions That SurviveA matrix can shear, rotate, and stretch the plane, but a few special directions pass through unchanged except for scale.
  2. Singular Values Measure StretchEvery linear transformation sends the unit circle to an ellipse, and the axes of that ellipse reveal its cleanest possible geometry.

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Signals and Spectra

Waves, filters, and transforms turn complicated shapes into collections of simpler vibrations.

  1. Fourier Series Listen To ShapesA jagged function can be understood as a chord of pure waves, each carrying a frequency and an amplitude.
  2. The Laplace Transform Turns Memory Into AlgebraExponential weights convert differential equations into algebra by asking how a system responds at every decay rate.

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Calculus Without Fog

Derivatives, series, and integrals become clearer when treated as local geometry rather than symbolic rituals.

  1. Taylor Series Are Local MicroscopesA Taylor polynomial records what a function looks like under stronger and stronger magnification near a point.

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Probability as Geometry

Probability distributions are shapes that move, concentrate, and smooth under repeated operations.

  1. Random Walks Become BellsRepeated tiny independent steps produce a bell curve because convolution smooths and spreads probability in a rigid way.
  2. Bayes Tilts BeliefBayes' rule is not a magic formula. It is a geometric reweighting of a prior shape by the evidence it predicts.