matharium

lim (confusion → 0) mathematics

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We live in a chaotic, unpredictable universe.

This is a grid of 812 double pendulums (28x29), each with slightly different initial conditions for the two angles. Neighbors start almost the same — then the field tears itself into chaos.

Each glyph is one double pendulum drawn in angle space:
x-axis → initial Angle 2
y-axis → initial Angle 1

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Epitrochoids reverse the Spirograph: a circle rolls on the outside of a fixed circle.

With outer radius R, rolling radius r, and pen offset d,

x = (R + r) cos t − d cos ((R + r) t / r)
y = (R + r) sin t − d sin ((R + r) t / r)

The path can look like a flower, a shell, or a dense mandala depending on the ratio R/r and the offset d. The cardioid and nephroid are famous special cases.

These curves are not random ornament, but they are the exact trajectories of rigid-body rolling the same kinematics used in planetary gear trains and cam design.

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Euler's formula turns rotation into algebra:

e^{iθ} = cos θ + i sin θ

For any real angle θ, the complex exponential lands on the unit circle. Its real coordinate is cos θ and its imaginary coordinate is sin θ.

That single identity connects exponentials, trigonometry, and complex numbers. It is why rotations and oscillations can be handled so cleanly in signal processing, AC circuits, wave physics, and quantum mechanics

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The Lorenz attractor: a butterfly drawn by differential equations.

In 1963 Edward Lorenz studied a simplified model of atmospheric convection:

dx/dt = σ(y − x)
dy/dt = x(ρ − z) − y
dz/dt = xy − βz

With classic parameters σ = 10, ρ = 28, β = 8/3, solutions do not settle to a point or a loop. They orbit forever on a fractal-looking set — the famous double-wing attractor.

Ten starts that differ by only a thousandth soon occupy different parts of the wings. That insight helped birth modern chaos theory and the popular phrase “butterfly effect.”

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A Julia set is a portrait of iteration in the complex plane.

Pick a complex constant c. Start from each point z₀ in the plane and repeat z ← z² + c. The Julia set marks the boundary between points that escape to infinity and points that remain trapped.

Here c itself walks a circle: c = 0.7885 e^{iφ} as φ advances. Each frame is a different Julia set; together they morph through dendritic islands, fattened circles, and fractured dust.

Complex dynamics links these images to the Mandelbrot set (the catalog of c values) and to deep questions about chaos, topology, and computability.

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Watch a single complex exponential write the entire language of oscillation.

Euler’s formula states e^{iθ} = cos θ + i sin θ. As θ advances, the tip of the radius traces the unit circle while its projections on the real and imaginary axes become pure cosine and sine waves.

At θ = π the identity collapses to the famous e^{iπ} + 1 = 0, binding five fundamental constants in one line. Engineers encode phase and amplitude of AC signals as complex numbers for exactly this reason: rotation is multiplication.

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A geometric series multiplies each term by a fixed ratio r.

The sum of the first n terms is

S_n = a (1 − r^n) / (1 − r) for r ≠ 1.

If |r| < 1 the infinite sum converges to S = a / (1 − r). Classic example: 1 + 1/2 + 1/4 + 1/8 + … = 2.

Present-value calculations in finance, repeating decimals, and fractal measures all rest on this sum.

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Hypotrochoids: the mathematics behind the Spirograph.

A circle of radius r rolls inside a fixed circle of radius R. A pen at distance d from the moving center traces the curve

x = (R − r) cos t + d cos((R − r)t / r)
y = (R − r) sin t − d sin((R − r)t / r)

Change R, r, and d and the pattern blooms into stars, rosettes, or dense bands. When d = r you recover a hypocycloid; special ratios yield deltoids and astroids.

This animation layers three related parameter sets so you can see how small design choices reshape the whole figure. The same rolling-circle idea appears in gear design, rotary engines, etc.

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A quick-reference map of the most important elementary functions.

From the constant and linear maps, through absolute value, polynomials, roots, reciprocal, exponential and log, to the trigonometric family sin, cos, and tan — each shape is a first-year toolkit piece.

Recognizing these graphs is the first step in modeling growth, oscillation, optimization, and asymptotic behavior across science and engineering.

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Watch a circle stretch into an ellipse and see the directions that only scale.

Apply a linear map A continuously from the identity to A = [[2, 0.5], [0.5, 1]]. Most directions rotate and stretch. Eigenvectors (red and green axes) keep their direction; only their lengths change by the eigenvalues.

A v = λ v

Principal axes of stress, PCA in data science, and stability analysis in dynamical systems all hunt these special directions.

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Euler's identity unites five of the most important constants in mathematics in a single equation:

e^{iπ} + 1 = 0.

It is a special case of Euler's formula e^{iθ} = cos θ + i sin θ. Setting θ = π gives cos π + i sin π = −1, which rearranges to e^{iπ} + 1 = 0.

The identity appears throughout complex analysis, Fourier analysis, and quantum mechanics whenever rotations and oscillations are written in exponential form.

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Four control points. One smooth curve that never fights you.

A cubic Bézier curve is

B(t) = (1−t)³ P₀ + 3(1−t)² t P₁ + 3(1−t) t² P₂ + t³ P₃, t ∈ [0,1]

The video builds the curve while de Casteljau’s intermediate segments show how the point is interpolated. The path starts at P₀, ends at P₃, and is pulled toward P₁ and P₂.

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In a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides.

If the legs have lengths a and b and the hypotenuse has length c, then

a² + b² = c².

This is pure geometry: the three outer squares in the diagram have areas a², b², and c². Rearrangement proofs show the two smaller squares tile the large one.

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Two sources. One field. Patterns of calm and chaos.

Each source radiates circular waves. Where crests meet crests the amplitude doubles (constructive interference); where crest meets trough they cancel (destructive interference). The bright and dark fringes are pure geometry of path difference.

Same math as Young’s double-slit, antenna arrays, noise-cancelling headphones, and water-wave tanks.

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On the unit circle every angle θ maps to a unique point.

Because the radius is 1, that point is exactly (cos θ, sin θ): the x-coordinate is the adjacent side of the right triangle, and the y-coordinate is the opposite side. Dropping a perpendicular from the point to the x-axis recovers the familiar triangle with hypotenuse 1, so Pythagoras collapses to the identity

cos²θ + sin²θ = 1.

The same point is the complex number e^{iθ} = cos θ + i sin θ, with real part cos θ and imaginary part sin θ.

Engineers rely on this identity to split waves into horizontal and vertical components when analyzing AC circuits, oscillations, and phase relationships in signal processing.