Mathematics on Rafiul Alam

Go Concurrency Pattern: The Sieve of Eratosthenes Pipeline

Sun, 02 Feb 2025 00:00:00 +0000

← Monte Carlo Pi | Series Overview | Mandelbrot Set →

The Problem: Finding Primes with Filters

The Sieve of Eratosthenes is an ancient algorithm for finding prime numbers. The concurrent version creates a pipeline of filters: each prime spawns a goroutine that filters out its multiples.

The Algorithm:

Generate sequence: 2, 3, 4, 5, 6, 7, 8, 9, 10, …
Take first number (2), it’s prime, filter all multiples of 2
Take next number (3), it’s prime, filter all multiples of 3
Take next number (5), it’s prime, filter all multiples of 5
Repeat until desired count

The Beauty: Each prime creates its own filter. Numbers flow through a pipeline of increasingly selective filters. What passes through all filters must be prime.

Go Concurrency Pattern: Monte Carlo Pi Estimation

Thu, 30 Jan 2025 00:00:00 +0000

← Login Counter | Series Overview | Sieve of Eratosthenes →

The Problem: Computing Pi by Throwing Darts

Imagine a square dartboard with a circle inscribed inside it. Throw random darts at the square. The ratio of darts landing inside the circle to total darts thrown approaches π/4.

Why? Mathematics:

Square side length: 2 (from -1 to 1)
Square area: 4
Circle radius: 1
Circle area: π × 1² = π
Ratio: π/4

Throw 1 million darts, multiply by 4, and you’ve estimated π. More darts = better estimate. This is Monte Carlo simulation: using randomness to solve deterministic problems.

Go Concurrency Pattern: The Mandelbrot Set

Mon, 27 Jan 2025 00:00:00 +0000

← Sieve of Eratosthenes | Series Overview | Collatz Explorer →

The Problem: Rendering Fractals in Parallel

The Mandelbrot set is defined by a simple iterative formula:

Start with z = 0
Repeatedly compute z = z² + c
If |z| exceeds 2, the point escapes (not in the set)
Color each pixel by iteration count

The beauty: Each pixel is completely independent. Perfect for parallelism!

The challenge: Some pixels escape in 5 iterations, others take 1000+. This creates load imbalance-some workers finish instantly while others grind away.

Go Concurrency Pattern: The Collatz Explorer

Mon, 20 Jan 2025 00:00:00 +0000

← Mandelbrot Set | Series Overview

The Problem: The Simplest Unsolved Math Problem

The Collatz conjecture (3n+1 problem) is deceptively simple:

Start with any positive integer n
If n is even: divide by 2
If n is odd: multiply by 3 and add 1
Repeat until you reach 1

The conjecture: Every positive integer eventually reaches 1.

Example (n=12):

12 → 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1

The mystery: This has been verified for numbers up to 2^68, but never proven. It’s one of mathematics’ most famous unsolved problems.