Matheorems

Lightning can be modeled as the shortest path through a random maze from a top point to the ground. A frontier expands until it reaches the bottom, then the path is traced back to form the strike.

What's Stopping You?

Euler's formula: e^{iθ} = cos(θ) + j sin(θ). 3D animation of the unit helix in complex plane vs θ, with real projection as cosine wave and imaginary as sine wave. Core to Fourier analysis in linear electrical systems and control theory.

This ball just flipped upside down in the mirror. When an object passes the focal point of a concave mirror, the image inverts. Beyond the focus it forms a real inverted image; closer than the focus it stays virtual and upright.

“God used beautiful mathematics in creating the world.” — Paul Dirac

Unit quaternion q = cos(θ/2) + u sin(θ/2) encodes a 3-D rotation by angle θ about unit axis u. Four components, closed under multiplication, singularity-free on S³, and renormalizable in O(1). Enables constant-speed SLERP on SO(3).

Topologists: “We optimized the surface.” Kids: crying on the underside

Taylor approximating sin(x) one battle rope wave at a time.

A fractal spirograph formed by nested circles that roll around each other recursively. A pen fixed to the smallest circle tra ces a self-similar path, gradually building intricate branching arms into a symmetric multi-pointed star.

Sphere surface area via latitude zones: circumference 2πr sinθ, width r dθ. Integrate ∫₀^π 2πr² sinθ dθ = 4πr².

Snapology-inspired reconfigurable metamaterial of extruded cubes. Each unit cell has 24 rigid faces & 36 hinged edges, giving 3 degrees of freedom. Varying angles γ1,γ2,γ3 continuously transforms its shape and internal volume from 0 to 7L³.

ℂ × ℝ³ × ℤ ⟶ ℝ⁴: Complex numbers (ℂ) encode phases/amplitudes for interference patterns; ℝ³ gives 3D spatial coordinates; ℤ indexes discrete layers or harmonic modes; maps to ℝ⁴ for intensity, time or color.

3D hydrogen wavefunctions Ψ_{nlm}: probability density |Ψ|^{2} evolves in superposition α|n l m⟩ + β|n′ l′ m′⟩ as quantum numbers n,l,m change. Orbital lobes/nodes morph; diagram tracks allowed transitions (time-independent).

A physical wireframe model of the tesseract (4-cube) whose boundary is given by max(|x|,|y|,|z|,|w|)=1 is rotated; its shadow is the 2D projection of the successive 3D projections of the 4D hypercube.

"I am not a woman mathematician, I am a mathematician." - Emmy Noether