“Modular residues are the invisible gears of numerical symmetry — turning chaos into predictable cycles.”
🔄 Residue Fields
Explore Euler’s Totient Function & Modular Residues
Coprime residues shown in cyan
❓ What does this mean?
Think of a 12-hour clock — it wraps around. That’s modular arithmetic: numbers wrap when they exceed a limit.
For example,
Totatives (coprimes) are numbers with no common divisors with the modulus. They’re the keys that unlock the full circle.
For example,
14 mod 12 = 2. It’s like walking in circles and ending up where you started.
Totatives (coprimes) are numbers with no common divisors with the modulus. They’re the keys that unlock the full circle.
🧠 Euler’s Theorem — Why Totatives Matter
Euler’s Theorem states:
aφ(m) ≡ 1 mod m (if gcd(a, m) = 1)
This underpins cryptographic systems like RSA where numbers “loop back” in predictable cycles. Try it below — choose a totative and raise it to φ(m)!
⚡ Totient Power Result
Visual Field Flow
🌈 What is the Particle Flow Field?
This animated canvas simulates a radial flow field using particles orbiting a center point.
Each particle follows a unique spiral trajectory based on trigonometric transformations of its angle and phase.
🌀 How it works:
– Each particle is assigned a radius, speed, and color
– Its position is recalculated frame-by-frame using sine and cosine functions
– The result is a smooth, colorful dance mimicking field lines or orbital flow
🎯 Why it matters:
Flow fields model continuous motion, resonant systems, and even modular energy states in visual terms. They’re often used to visualize dynamic systems—just like how modular mathematics reveals structure through cycles.
🌀 How it works:
– Each particle is assigned a radius, speed, and color
– Its position is recalculated frame-by-frame using sine and cosine functions
– The result is a smooth, colorful dance mimicking field lines or orbital flow
🎯 Why it matters:
Flow fields model continuous motion, resonant systems, and even modular energy states in visual terms. They’re often used to visualize dynamic systems—just like how modular mathematics reveals structure through cycles.