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
This spiral draws concentric polygons where the number of sides
- 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.
Visualization of the Riemann zeta function
- 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
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ISC