Experiments around spiral patterns inspired by the Ulam spiral.

https://alganet.github.io/spiral/

This is the classic Ulam spiral on a square grid.

• It starts at the center and moves in a spiral pattern: Right, Up, Left, Down.
• The step size increases by 1 every two turns (1 step Right, 1 step Up, 2 steps Left, 2 steps Down, etc.).

This visualization plots natural numbers using polar coordinates based on the formula:

• $r = c \sqrt{n}$
• $\theta = \sqrt{n} \times 2\pi$

The polar coordinates are converted to Cartesian $(x, y)$ to plot each number $n$.

This spiral draws concentric polygons where the number of sides $S$ is configurable.

• The spiral is built in layers.
• Each layer consists of a polygon with $S$ sides.
• Numbers are mapped sequentially along the perimeter of these polygons.

A recursive visualization of prime gaps using a Polar Coordinate System with a configurable modulus (defaulting to 18).

• Structure: The plane is divided into slices (rays). Primes predominantly cluster on specific rays based on modular arithmetic (e.g., Mod 6, Mod 18).
• Drill Down: Users can select specific gap types (e.g., "Twin Primes") to filter the view.
• Visuals:
  • Arcs: Connect prime pairs across different rays (jumps).
  • Petals: Connect prime pairs on the same ray (loops/resonance).
  • Massive Pioneers: Rare, high-density gaps are highlighted to show the "skeleton" of the prime distribution.

Lays the non-zero numbers 1 … p-1 around a ring and shows that multiplication mod a prime is a rotation.

• When p is prime, repeatedly multiplying by a single step-maker g (a primitive root) visits every non-zero number exactly once.
• Order the ring by that walk and a number's angle is "how many g-steps reach it". Then a × b mod p is just adding those step counts — a constant rotation around the ring.
• Toggle Arrange: steps ↔ value to contrast the neat rotation against how tangled the same multiplication looks in the ordinary by-value order.
• Toggle Orbit to trace a, a×b, a×b², …, which draws a regular star polygon — the same constant-angle stepping as the polygonal spiral, but driven by × instead of +.
• Toggle Slices to cut the one big ring of p-1 slots into several small rings (e.g. 12 → 3,4). A number sits at slot step mod size on each ring, and multiplication rotates each ring independently. The cut is clean only when the sizes multiply to p-1 and share no common factors (so every number lands on a unique combination of slots). Try p=13 with cut 3,4 (clean), then 2,6 (two numbers collide), then p=17 (whose 16 = 2⁴ is a prime power and can't be split at all).

Keeps the Möbius spiral as the base (what each number is built from: prime / μ) and adds sparse markers for significant landings on the multiplication clock mod p — chosen because coloring every landing just replaces the Möbius view with an orthogonal one, whereas a sparse overlay can coexist with it.

• Laps — n ≡ 1 (mod p): where the × clock completes a full turn back to the identity.
• Squares — n mod p is a quadratic residue (the square landings; a subgroup of half the residues).
• Roots — n mod p is a primitive root (a generator of the whole × clock).
• Toggle each independently against the factorization texture. The question: do these multiplicative classes align with the Möbius / twin-prime slices, or cut across them independently?

A single spiral shows one prime's × clock; this stacks many primes to surface what the stack knows that one cut cannot. Pick a number g and keep multiplying around each prime's clock; sweeping primes p:

• how often g takes the full tour (visits every number = is a primitive root) settles near 37% — and almost any g lands there;
• how often g is a perfect square on the clock (a quadratic residue) settles at exactly 0.5.
• g = 5 runs stubbornly high (~0.394) — some numbers are just special. Curiosity: the ~37% is the unsolved Artin's constant ≈ 0.3739; the exact 50% follows from quadratic reciprocity. Computed live in JS. (Analysis scripts: analysis/.)

Visualization of the Riemann zeta function $\zeta(s)$ on the critical line $Re(s) = 0.5$.

• Traces the path of $\zeta(0.5 + it)$ in the complex plane as $t$ increases.
• Visualizes the partial sum spiral $S_N = \sum_{n=1}^N n^{-s}$ tending towards the function's value.
• Includes a background colored by the Möbius function $\mu(n)$ mapped to the horizontal range.

All visualizations use a shared coloring strategy based on number properties:

• Primes: Highlighted with a specific color. Twin primes may have a distinct shade.
• Composite Numbers: Colored based on the Möbius function $\mu(n)$:
  • $\mu(n) = -1$: Reddish
  • $\mu(n) = 0$: Greenish (contains a squared prime factor)
  • $\mu(n) = 1$: Bluish

ISC